André
Schlichting
WWU Münster
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February 2025
Raffaele
Vitolo
University of Salento
Bi-Hamiltonian geometry of WDVV equations: general results
Abstract.
It is known (work by Ferapontov and Mokhov) that a system of N-dimensional WDVV equations can be written as a pair of N-2 commuting quasilinear systems (first-order WDVV systems). In recent years, particular examples of such systems were shown to possess two compatible Hamiltonian operators, of the first and third order. It was also shown that all $3$-dimensional first-order WDVV systems possess such bi-Hamiltonian formalism. We prove that, for arbitrary N, if one first-order WDVV system has the above bi-Hamiltonian formalism, than all other commuting systems do. The proof needs some interesting results on the structure of the WDVV equations that will be discussed as well. (Joint work with S. Opanasenko).
January 2025
Stefan
Weber
Leibniz Universität Hannover
Robust Portfolio Selection Under Recovery Average Value at Risk
Abstract.
We study mean-risk optimal portfolio problems where risk is measured by Recovery Average Value at Risk, a prominent example in the class of recovery risk measures. We establish existence results in the situation where the joint distribution of portfolio assets is known as well as in the situation where it is uncertain and only assumed to belong to a set of mixtures of benchmark distributions (mixture uncertainty) or to a cloud around a benchmark distribution (box uncertainty). The comparison with the classical Average Value at Risk shows that portfolio selection under its recovery version allows financial institutions to better control the recovery of liabilities while still allowing for tractable computations. The talk is based on joint work with Cosimo Munari, Justin Plückebaum and Lutz Wilhelmy.
Coherent Transport of Semiconductor Spin-Qubits: Modeling, Simulation and Optimal Control
Abstract
Jan-Frederik
Pietschmann
Universität Augsburg
Olivier
Marchal
Univ. St-Etienne
Ilya
Cheyrev
U Edinburgh
Ilya
Chevyrev
U Edinburgh
Roxana
Radulescu
Joachim
Rehberg
WIAS
December 2024
Pascal
Maillard
CNRS
Zhen-Qing
Chen
U of Washington
Computational Aspects of Quadratic Forms in Determining the Representation Type of Quiver Algebras
Roman
Sauer
KIT
Dante
Luber
Thesis defense
Abstract.
Thesis defense Dante Luber
Hana
Kourimska
Sebastian
Bechtel
Université Paris-Saclay, France
What we know about square roots of elliptic systems -- and a bit more!
Abstract
Patrick
Cheridito
ETH Zürich
Rainer
Sinn
Leipzig University
Positive Geometry in the plane
Abstract.
I want to introduce positive geometries in the plane and discuss examples from discrete, convex, and algebraic geometry. There are open problems in this relatively simple case already and I will present one. This is mostly based on joint work with Kathlén Kohn, Ragni Piene, Kristian Ranestad, Felix Rydell, Boris Shapiro, Miruna-Stefana Sorea, and Simon Telen.
Günter
Last
KIT
Patrick
Cheridito
ETH Zurich
Georg
Heinze
WIAS
Discrete-to-continuum limit for reaction-diffusion systems via variational convergence of gradient systems
Abstract
November 2024
Laura
Ciobanu
HWU
Undecidable problems in group theory
Julien
Chhor
Toulouse School of Economics
Locally sharp goodness-of-fit testing in sup norm for high-dimensional counts
Sandro
Roch
Technische Universität Berlin
Welt der Pseudogeraden
Abstract.
Wir erkunden zusammen die Welt der Pseudogeradenarrangements: Anordnungen von Kurven in der Ebene, von denen sich je zwei in genau einem Punkt kreuzen. Diese einfachen Objekte faszinieren mit ihren vielfältigen Bezügen zu verwandten Strukturen aus Kombinatorik, Geometrie und Informatik, wie zum Beispiel Sortiernetzwerke, Rhombenpflasterungen oder Young-Tableaux. Uns beschäftigt unter anderem die Frage, wie sich diese zufällig mithilfe einer Markov-Kette erzeugen lassen. Lassen sie sich einfach mischen wie ein Kartenstapel?
Denis
Korolev
WIAS
Hybrid physics-informed neural network based dual-scale solver and its applications to learning-informed upscaling
Abstract.
Inspired by the Liquid Composite Molding process for fiber-reinforced composites and its associated multiscale fluid flow structure, we present a novel framework for optimizing physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs), with applications to dual-scale PDE systems. Our hybrid approach approximates the fine-scale PDE using PINNs, producing a PDE residual-based objective subject to a coarse-scale PDE model parameterized by the fine-scale solution. Techniques from materials science introduce feedback mechanisms that yield scale-bridging coupling, resulting in a non-standard PDE-constrained optimization problem. From a discrete standpoint, the formulation represents a hybrid numerical solver that integrates both neural networks and finite elements. In this talk, we present the application setting, mathematical model, and highlights of its analysis, as well as outline perspectives on developing optimization algorithms for the hybrid framework in the infinite-dimensional setting.
Pengfei
Huang
Leipzig
Parahoric reduction theory of formal connections
Abstract.
The celebrated reduction theory of formal connections is due to Hukuhara, Levelt, Turrittin, and Babbitt-Varadarajan, among others. In this talk, we will demonstrate the parahoric reduction theory of formal parahoric connections, which generalizes the aforementioned results and also extends Boalch’s result for the case of regular singularities. As applications, we will establish the equivalence between extrinsic and intrinsic definitions of regular singularities, as well as a parahoric version of Frenkel-Zhu’s Borel reduction theorem for formal connections. This is based on a recent joint work with Z. Hu, R. Sun, and R. Zong.
Rob
Klabbers
HU Berlin
Elliptic long-range quantum integrable systems
Abstract.
There are at least two seemingly distinct realms of quantum integrability. The first domain is formed by the (short-range) Heisenberg spin chains, connected to the quantum inverse scattering method, which play a role in many different contexts both in physics and mathematics. The second domain is formed by the Calogero-Sutherland models and their deformations, which are families of differential or difference operators associated to root systems, with close ties to harmonic analysis, orthogonal Jack and Macdonald polynomials, and Knizhnik-Zamolodchikov equations. Their integrability follows from a connection to affine Hecke algebras. Understanding how these two realms are connected goes through the elliptic CS models and their generalisations, which are also interesting in their own right. I will discuss in what way this bridge between worlds is formed and how far we are in building it. Along the way I will try to point out connections to different research areas.
Filippo
de Feo
Luiss Guido Carli University, Milano
Optimal control of stochastic delay differential equations and applications to financial and economic models
Abstract.
Optimal control problems involving Markovian stochastic differential equations have been extensively studied in the research literature; however, many real-world applications necessitate the consideration of path-dependent non-Markovian dynamics. In this talk, we consider an optimal control problem of (path-dependent) stochastic differential equations with delays in the state. To use the dynamic programming approach, we regain Markovianity by lifting the problem on a suitable Hilbert space. We characterize the value function $V$ of the problem as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equation, which is a fully non-linear second-order partial differential equation on a Hilbert space with an unbounded operator. Since no regularity results are available for viscosity solutions of these kinds of HJB equations, via a new finite-dimensional reduction procedure that allows us to use the regularity theory for finite-dimensional PDEs, we prove partial $C^{1,\alpha}$-regularity of $V$. When the diffusion is independent of the control, this regularity result allows us to define a candidate optimal feedback control. However, due to the lack of $C^2$-regularity of $V$, we cannot prove a verification theorem using standard techniques based on Ito’s formula. Thus, using a technical double approximation procedure, we construct functions approximating $V$, which are supersolutions of perturbed HJB equations and regular enough to satisfy a non-smooth Ito’s formula. This allows us to prove a verification theorem and construct optimal feedback controls. We provide applications to optimal advertising and portfolio optimization. We discuss how these results extend to the case of delays in the control variable (also) and discuss connections with new results of $C^{1,1}$-regularity of the value function and optimal synthesis for optimal control problems of stochastic differential equations on Hilbert spaces via viscosity solutions.
Yonatan
Shadmi
Imperial College London
Fluid limits of fragmented limit order markets
Abstract.
Maglaras, Moallemi and Zheng (2021) have introduced a flexible queueing model for fragmented limit-order markets, whose fluid limit remains remarkably tractable. In this talk I will present the proof that, in the limit of small and frequent orders, the discrete system indeed converges to the fluid limit, which is characterized by a system of coupled nonlinear ODEs with singular coefficients at the origin. Moreover, I will discuss the temporal asymptotic stability for an arbitrary number of limit order books in that, over time, it converges to the stationary equilibrium state studied by Maglaras et al.
Eleonore
Bach
EPFL
Smoothed analysis of the Simplex method: nearly tight noise dependence
Abstract.
Smoothed analysis is a method for analysing the performance of algorithms, used especially for those algorithms whose running time in practice is significantly better than what can be proven through worst-case analysis. Given an arbitrary linear program with $d$ variables and $n$ inequality constraints, we prove that there is a simplex method with smoothed complexity upper bounded by $O(\sigma^{-1/2} d^{11/4} \log(n/\sigma)^{7/4})$ pivot steps improving over the current best bound of $O(\sigma^{-3/2} d^{13/4} \log(n)^{7/4})$ pivot steps due to Huiberts, Lee and Zhang (STOC '23). For the same method we prove a lower bound on its smoothed complexity of $0.03 \sigma^{-1/2} d^{-1/4}\ln(n)^{-1/4}$ pivot steps for $n = (4/\sigma)^d$ inequality constraints. Here $\sigma > 0$ is the standard deviation of Gaussian distributed noise added to the original LP data. This nearly closes the gap between the upper and lower bounds in regards to their dependence on the noise parameter $\sigma$. In this talk we will discuss the algorithmic improvements that we used for the above new upper bound. This is joint work with Sophie Huiberts.
Igor
Makhlin
Brief overview of Brion's theorem
Julia
Komjathy
TU Delft
Bernhard
Stankewitz
Universität Potsdam
Contraction rates for conjugate gradient
and Lanczos approximate posteriors in Gaussian process regression
Addis Belete
Zewde
Dire Dawa University
Solving Potential Multi-Leader-Follower Games via Parametric Optimization
Sonja
Steffensen
RWTH Aachen
Solving Potential Multi-Leader-Follower Games via Parametric Optimization
Ya
Deng
Nancy
Euler Characteristic of Algebraic Varieties
Abstract.
This talk is based on joint works with Botong Wang. A conjecture by Chern-Hopf-Thurston states that an aspherical closed real n-manifold \(X\) satisfies \( (-1)^n\chi(X) \geq 0 \), where \( \chi(X) \) denotes the Euler characteristic of \(X\). I will focus on the case where \(X\) has the structure of a complex algebraic variety, which implies that \(X\) has large fundamental group. Inspired by this, in 1995, Kollár proposed the following conjecture: a complex projective manifold \(X\) satisfies \(\chi(K_X) \geq 0\) if it has generically large fundamental group. In this talk, I will outline the proofs of both conjectures under the assumption that \(\pi_1(X)\) is linear.
Guillaume
Baverez
Aix-Marseille Université
Uniqueness of Malliavin-Kontsevich-Suhov measures
Abstract.
About 20 years ago, Kontsevich & Suhov conjectured the existence and uniqueness of a family of measures on the set of Jordan curves, characterised by conformal invariance and another property called 'conformal restriction'. This conjecture was motivated by (seemingly unrelated) works of Schramm, Lawler & Werner on stochastic Loewner evolutions (SLE), and Malliavin, Airault & Thalmaier on 'unitarising measures'. The existence of this family was settled by works of Werner-Kemppainen and Zhan, using a loop version of SLE. The uniqueness was recently obtained in a joint work with Jego. I will start by reviewing the different notions involved before giving some ideas of our proof of uniqueness: in a nutshell, we construct a family of 'orthogonal polynomials' which completely characterise the measure. In the remaining time, I will discuss the broader context in which our construction fits, namely the conformal field theory associated with SLE.
Certified Reduced-Order Methods for Model Predictive Control of Time-Varying Evolution Processes
Abstract.
In this talk model predictive control (MPC) is utilized to stabilize a class of linear time-varying parabolic partial differential equations (PDEs). In our first example the control input is only finite-dimensional, i.e., it enters as a time-depending linear combination of finitely many indicator functions whose total supports cover only a small part of the spatial domain. In the second example the PDE involve switching coefficient functions. We discuss stabilizability and the application of reduced-order models to derive algorithms with closed-loop guarantees. <p>This is joint work with , and Benjamin Unger (Stuttgart).</p>
Ana Maria
Botero
U Bielefeld
and
José Ignacio
Burgos Gil
ICMAT
and
Jean-Benoît
Bost
U Paris XI
and
Maryna
Viazovska
EPFL
A Day of Arithmetic Geometry (on the occasion of the retirement of Jürg Kramer)
Abstract
Enrico
Bortoletto
ZIB
and
Berenike
Masing
ZIB
On the Tropical and Zonotopal Geometry of Periodic Timetables
Abstract.
The Periodic Event Scheduling Problem (PESP) is a fundamental tool for optimizing periodic timetables in public transport. While periodic tension spaces have been well-studied, this talk focuses on the geometry of timetables and cycle offsets through tropical and zonotopal frameworks. We examine the feasible timetable space as a union of polytropes and propose a heuristic for PESP, based on polytrope neighborhood relations. Additionally, we recognize the space of fractional cycle offsets as a zonotope and analyze its tilings in relation to timetable polytropes. We conclude with new bounds on cycle bases, as well as some new and developing questions.
Free boundary problems as limits of a bulk-surface model for receptor-ligand interactions on evolving domains
Abstract.
We derive various free boundary problems as reaction or singular limits of a coupled bulk-surface reaction-diffusion system on an evolving domain. These limiting free boundary problems may be formulated as Stefan-type problems on an evolving hypersurface. These results, which are new even in the setting where there is no domain evolution, are of particular relevance to an application in cell biology. In this talk, I will discuss the modelling, sketch the analysis, show some numerical simulations and finish with some open questions. Based on a joint work with Charlie Elliott, Chandrasekhar Venkataraman and Diogo Caetano.
Felix
Lotter
MPI-MiS Leipzig
Cyclic polytopes through the lens of iterated integrals
Abstract.
The volume of a cyclic polytope P can be obtained as a linear combination of iterated integrals along any convex piecewise linear path running through the edges of P. We explore the question what other functions on the set of cyclic (or more precisely, alternating) polytopes arise as iterated integrals in this way. In fact, we show that there are infinitely many such features which are algebraically independent. We obtain descriptive rings of functions on the set of alternating d-polytopes with n vertices, compatible with restrictions to subpolytopes.
Alessandro
Verra
Università Roma Tre
An application of the Segre primal to an enumerative problem
Abstract.
The classification of complex, nodal cubic threefolds goes back to Corrado Segre. Among these, a unique one, up to projective equivalence, has the maximal number of ten nodes and it is named the Segre primal. In this talk we describe the solution of the following enumerative problem, where the Segre primal appears. Let \(V\) be a smooth complex cubic threefold and \(x\) a general point of it, then the six lines of \(V\) through \(x\) are in a quadric cone surface and define six points of the projective line \(P\). This defines a rational map from \(V\) to the moduli space of genus 2 curves. What is the degree of this map? Joint work with Ciro Ciliberto.
Gernot
Akemann
Bielefeld Universität
Three universality classes in non-Hermitian random matrices
Abstract.
Non-Hermitian random matrices with complex eigenvalues have important applications, for example in open quantum systems in their chaotic regime. It has been conjectured that amongst all 38 symmetry classes of non-Hermitian random matrices only 3 different local bulk statistics exist. This conjecture has been based on numerically generated nearest-neighbour spacing distributions between complex eigenvalues so far. In this talk I will present first analytic evidence for this conjecture. It is based on expectation values of characteristic polynomials in the three simplest representatives for these statistics: the well-known Ginibre ensemble of complex normal matrices, complex symmetric and complex self-dual random matrices. After giving a basic introduction into the complex eigenvalue statistics of the Ginibre ensemble, I will present results for all three ensembles for finite matrix size N as well as in various large-N limits. These are expected to be universal, that is valid beyond ensembles with Gaussian distribution of matrix elements. This paper is based on joint work with Noah Aygün, Mario Kieburg and Patricia Päßler in arXiv/2410.21032
Vesna
Iršič Chenoweth
Burning game
Johannes
Muhle-Karbe
Imperial College London
Concave Cross Impact
Abstract.
The price impact of large orders is well known to be a concave function of trade size. We discuss how to extend models consistent with this “square-root law” to multivariate settings with cross impact, where trading each asset also impacts the prices of the others. In this context, we derive consistency conditions that rule out price manipulation. These minimal conditions make risk-neutral trading problems tractable and also naturally lead to parsimonious specifications that can be calibrated to historical data. We illustrate this with a case study using proprietary CFM meta order data. (Joint work in progress with Natascha Hey and Iacopo Mastromatteo)
David
Hobson
University of Warwick
Portfolio optimization under transaction costs with recursive preferences
Abstract.
The solution to the investment-consumption problem in a frictionless Black-Scholes market for an investor with additive CRRA preferences is to keep a constant fraction of wealth in the risky asset. But this requires continuous adjustment of the portfolio and as soon as transaction costs are added, any attempt to follow the frictionless strategy will lead to immediate bankruptcy. Instead as many authors have proposed the optimal solution is to keep the pair (cash, value of risky assets) in a no-transaction (NT) wedge. We return to this problem to see what we can say about: When is the problem well-posed? Where does the NT wedge lie? How do the results change if we use recursive preferences? We introduce the shadow fraction of wealth and show how we can make significant progress towards the solution by focussing on this quantity. Indeed many of the qualitative features of the solution can described by looking at a quadratic whose parameters depend on the parameters of the problem. This is joint work with Martin Herdegen and Alex Tse.
Alex
Black
Anna
Zamojska-Dzienio
Warsaw University of Technology
Algebraic approach to barycentric coordinates
Abstract.
Barycentric coordinates provide solutions to the problem of expressing an element of a compact convex set as a convex combination of a finite number of extreme points of the set. They have been studied widely within the geometric literature, typically in response to the demands of numerical analysis and computer graphics. In this talk we bring an algebraic perspective to the problem, based on barycentric algebras. We present some recent results obtained together with A. Romanowska and J.D.H Smith.
David
Hobson
U of Warwick
Benjamin
Gess
TU Berlin
Andrei
Yafaev
University College London
On the Andre-Pink-Zannier conjecture and its generalisations
Abstract.
This is a joint work with Rodolphe Richard (Manchester). The Andre-Pink-Zannier conjecture is a case of Zilber-Pink conjecture on unlikely intersections in Shimura varieties. We will present this conjecture and a strategy for proving it as well as its proof for Shimura varieties of abelian type. In the second talk we present a "hybrid conjecture" combining the recently proved Andre-Oort conjecture and Andre-Pink-Zannier. It is motivated by its analogy with Mordell-Lang for abelian varieties. We will explain this analogy as well as the proof of the hybrid conjecture for Shimura varieties of abelian type. (This is the second talk, the first one taking place on Tuesday at the Arithmetic Geometry Seminar).
A Semismooth Newton Method for Obstacle-Type Quasivariational Inequalities
Abstract
Vladimir
Spokoiny
WIAS und HU Berlin
Andrei
Yafaev
UCL
On the Andre-Pink-Zannier conjecture and its generalisations, part I
Abstract.
This is a joint work with Rodolphe Richard (Manchester). The Andre-Pink-Zannier conjecture is a case of Zilber-Pink conjecture on unlikely intersections in Shimura varieties. We will present this conjecture and a strategy for proving it as well as its proof for Shimura varieties of abelian type. In the second talk, which will be in the Algebraic Geometry Seminar on Wednesday, we present a 'hybrid conjecture' combining the recently proved Andre-Oort conjecture and Andre-Pink-Zannier. It is motivated by its analogy with Mordell-Lang for abelian varieties. We will explain this analogy as well as the proof of the hybrid conjecture for Shimura varieties of abelian type.
Guilherme
Feitosa de Almeida
Hannover Universität
1D Landau-Ginzburg superpotential of big quantum cohomology of CP2
Abstract.
Using the inverse period map of the Gauss-Manin connection associated with QH∗(CP2) and the Dubrovin construction of Landau-Ginzburg superpotential for Dubrovin Frobenius manifolds, we construct a one-dimensional Landau-Ginzburg superpotential for the quantum cohomology of CP2. In the case of small quantum cohomology, the Landau-Ginzburg superpotential is expressed in terms of the cubic root of the j-invariant function. For big quantum cohomology, the one-dimensional Landau-Ginzburg superpotential is given by Taylor series expansions whose coefficients are expressed in terms of quasi-modular forms. Furthermore, we express the Landau-Ginzburg superpotential for both small and big quantum cohomology of QH∗(CP2) in closed form as the composition of the Weierstrass ℘-function and the universal coverings of C \ (Z ⊕ jZ) and C \ (Z ⊕ zZ) respectively. This seminar is based on the results of arXiv/2402.09574.
Heather
Harrington
MPI Dresden
Topology Data Analysis for Multiscale Biology
Nicolas Alexander
Weiss
HU Berlin
What is Topological Data Analysis?
Abstract.
Going back to work by Edelsbrunner, Carlsson and others in computational as well as "applied" topology, the field of Topological Data Analysis (TDA) is by now widely established. It provides a toolbox that is already getting applied to various fields outside mathematics, in particular to the life sciences. Assuming no prior knowledge about topology, we will begin by pointing to examples of how a very classical invariant, homology, can be applied (outside math) as well as computed in practice. We will then see how applying homology to growing $\epsilon$-neighborhoods of point cloud data in $\mathbb{R}^n$ leads to a multiscale data descriptor, called the barcode. If time permits we mention some representation theoretic perspectives on this.
October 2024
Sebastian
Meyer
TU Dresden
A topological CSP dichotomy
Abstract.
Many decision problems from mathematics and theoretical computer science, such as graph colorings and solving linear equations, can be modeled as constraint satisfaction problems (CSPs). It is known that finite CSPs are NP-complete or in P and how to distinguish them algebraically. We give a topological description of this dichotomy. This implies a dichotomy for simplicial complexes and a possibility to reprove the Hell-Nešetřil-Theorem.
Rainer
Sinn
Leipzig University
Positive Geometry in the plane
Abstract.
I want to introduce positive geometries in the plane and discuss examples from discrete, convex, and algebraic geometry. There are open problems in this relatively simple case already and I will present one. This is mostly based on joint work with Kathlén Kohn, Ragni Piene, Kristian Ranestad, Felix Rydell, Boris Shapiro, Miruna-Stefana Sorea, and Simon Telen.
Tomáš
Roubíček
Charles University Prague and Inst. of Thermomechanics, Czech Acad. Sci.
Time discretization in visco-elastodynamics at large displacements and strains in the Eulerian frame
Abstract
Olga
Klopp
ESSEC Business School, Paris
Adaptive density estimation under low-rank constraints
Abstract.
In this talk, we address the challenge of bivariate probability density estimation under low-rank constraints for both discrete and continuous distributions. For discrete distributions, we model the target as a low-rank probability matrix. In the continuous case, we assume the density function is Lipschitz continuous over an unknown compact rectangular support and can be decomposed into a sum of K separable components, each represented as a product of two one-dimensional functions. We introduce an estimator that leverages these low-rank constraints, achieving significantly improved convergence rates. We also derive lower bounds for both discrete and continuous cases, demonstrating that our estimators achieve minimax optimal convergence rates within logarithmic factors.
Bernadette
Lessel
Bonn Universität
On the early history of quantum gravity
Abstract.
Quantum gravity, in the sense of a formal quantization of general relativity, had a first beginning in the year 1930. That year, the Belgian physicist Léon Rosenfeld published a seminal paper called "Zur Quantelung der Wellenfelder" in which he developed Heisenberg and Pauli's recently constructed method to quantize the electromagnetic field in order to apply it to the tetrad formulation of general relativity. In my talk, I aim to shed light on a perhaps surprising crucial historical influence that made this piece of intellectual work possible: that of unified field theory. A purely classical program, most prominently pursued by Albert Einstein and Hermann Weyl, to formally reduce the (classical) electromagnetic field to the gravitational field as described by general relativity.
Bernd
Sturmfels
MPI-MiS Leipzig
The Two Lives of the Grassmannian
Abstract.
The Grassmannian parametrizes linear subspaces of a real vector space. It is both a projective variety (via Plücker coordinates) and an affine variety (via orthogonal projections). We examine these two representations, through the lenses of linear algebra, commutative algebra, and statistics.
Ronald
de Wolf
QuSoft, CWI and University of Amsterdam
Quantum Algorithms for Optimization
Abstract.
Faster algorithms for optimization problems are among the main potential applications for future quantum computers. There has been interesting progress in this area in recent years, for instance improved quantum algorithms for gradient descent and for solving linear and semidefinite programs. In this talk I will survey what we know about quantum speed-ups both for discrete and for continuous optimization, with a bit more detail about two speed-ups I worked on recently: for regularized linear regression and for Principal Component Analysis. I'll also discuss some issues with this line of work, in particular that quadratic or subquadratic quantum speed-ups will only kick in for very large instance sizes and that many of these algorithms require some kind of quantum RAM.
Matthew
Dupraz
FU Berlin
Tropical linear series
Abstract.
Tropicalization is a process that allows us to obtain from algebro-geometric objects combinatorial ones, while preserving many interesting properties. For example, we may tropicalize an algebraic curve to obtain a metric graph, and study divisors on the latter. Surprisingly, certain properties, like the Riemann-Roch theorem stay true for systems of divisors on a metric graph, and so we may wonder just how much are these objects related. In this talk I would like to introduce the tropical analog of linear series, and mention some interesting properties that they exhibit.
Evgeny
Feigin
Tel Aviv University
Generalized chain polytopes, Grassmannians and representations
Abstract.
For a finite poset P we introduce a two-parameter family X(m,M) of polytopes defined by the set of inequalities labeled by subposets of P. The polytopes enjoy Minkowski sum property and are important for geometric and algebraic applications. We will discuss the connection of X(m,M) with the geometry of the classical Grassmann varieties. We will also construct a family of representations of the degenerate sl(n) Lie algebra admitting monomial bases labeled by integer points of X(m,M) (for the Grassmann poset P). The talk is based on https://arxiv.org/abs/2403.10074 (joint with Wojciech Samotij).
Arno
Siri-Jegousse
UNAM
Jean Baptiste
Teyssier
Paris
Pedro
Montero
Universidad Técnica Valparaíso
Additive structures on quintic del Pezzo varieties
Abstract.
A classical problem of F. Hirzebruch concerns the classification of compactifications of affine space into smooth projective varieties with Picard rank one. It turns out that any such compactification must be a Fano manifold, i.e., it has an ample anti-canonical divisor. After reviewing some known results, I will focus on the specific case of equivariant compactifications of affine space (i.e., of the "vector group" \(\mathcal{G}_a^n\)), particularly in the case of del Pezzo varieties.<br>We will recall that del Pezzo varieties are a natural higher-dimensional generalization of classical del Pezzo surfaces. Over the field of complex numbers, these varieties were extensively studied by T. Fujita in the 1980s, who classified them by their degree.<br>I will present a result on the existence and uniqueness of "additive structures" on del Pezzo quintic varieties. Specifically, we determine when and how many distinct ways they can be obtained as equivariant compactifications of the commutative unipotent group. As an application, we obtain results on the k-forms of quintic del Pezzo varieties over an arbitrary field k of characteristic zero, as well as for singular quintic varieties. This is a joint work with Adrien Dubouloz and Takashi Kishimoto.
Weining
Wang
University of Groningen
Conditional nonparametric variable screening by neural factor regression
Abstract.
High-dimensional covariates often admit linear factor structure. To effectively screen correlated covariates in high-dimension, we propose a conditional variable screening test based on non-parametric regression using neural networks due to their representation power. We ask the question whether individual covariates have additional contributions given the latent factors or more generally a set of variables. Our test statistics are based on the estimated partial derivative of the regression function of the candidate variable for screening and a observable proxy for the latent factors. Hence, our test reveals how much predictors contribute additionally to the non-parametric regression after accounting for the latent factors. Our derivative estimator is the convolution of a deep neural network regression estimator and a smoothing kernel. We demonstrate that when the neural network size diverges with the sample size, unlike estimating the regression function itself, it is necessary to smooth the partial derivative of the neural network estimator to recover the desired convergence rate for the derivative. Moreover, our screening test achieves asymptotic normality under the null after finely centering our test statistics that makes the biases negligible, as well as consistency for local alternatives under mild conditions. We demonstrate the performance of our test in a simulation study and two real world applications.
A semismooth Newton method for obstacle-type quasivariational inequalities
Abstract.
Quasivariational inequalities (QVIs) are ubiquitous but, in particular, arise in PDE-constrained optimization in cases where the constraint set depends on the solution itself. In obstacle-type QVIs, this manifests as an obstacle that bends according to the state of the system. QVIs are notoriously hard to analyze, especially in the infinite-dimensional setting, and developing fast solvers posed in infinite dimensions has proven particularly challenging. As such most solvers in the literature rely on fixed point algorithms which can be slow to converge. In this talk, we introduce the first semismooth Newton method, posed in a Banach space setting, for such problems. We will see that the solver enjoys favourable properties such as local superlinear convergence and mesh independence.
Jean-Baptiste
Teyssier
Paris
Boundedness for Betti numbers of étale sheaves in positive characteristic
Abstract.
Let X be a smooth proper variety over an algebraically closed field of characteristic p>0 and let D be a divisor of X. In this talk, we will advertise the existence of bounds for the Betti numbers of a local system L on X-D depending only on local numerical data: the rank of L and the ramification conductors of L at the generic points of D. If time permits, we will explain a consequence of these bounds to a form of the wild Lefschetz theorem envisioned by Deligne. This is joint work with Haoyu Hu.
Rolf
Krause
Università della Svizzera italiana
Multilevel methods for non-smooth minimization and variational inequalities
Olexandr
Burylko
Institute of Mathematics, Kyiv, Ukraine
Reversible saddle-node separatrix-loop bifurcation
Stefan
Felsner
TU Berlin
On the number of arrangements of pseudolines: upper and lower bounds
Abstract.
Arrangements of pseudolines are classic objects in discrete and computational geometry. They have been studied with increasing intensity since their introduction almost 100 years ago. The study of the number Bn of non-isomorphic simple arrangements of n pseudolines goes back to Goodman and Pollack, Knuth, and others. It is known that Bn is in the order of 2^Θ(n^2) and finding asymptotic bounds on bn = log2(Bn)/n^2 remains a challenging task. I will present the ideas that lead to improvements of upper and lower bounds over the last 15 years. In particular I will present the recent progress obtained with Fernando Cortés Kühnast, Justin Dallant, and Manfred Scheucher (SoCG'24).
Roan
Talbut
Imperial College
Tropical Gradient Descent
Abstract.
The field of tropical statistics - motivated by the identification of the tropical Grassmannian and the space of phylogenetic trees - has produced a range of unconstrained optimisation problems over the tropical projective torus. We will review the types of convexity exhibited by tropical loss functions in statistics, and we propose a new gradient descent method for solving tropical optimisation problems. Theoretical results establish global solvability for tropically star-quasi-convex problems, and numerical experiments demonstrate the method's superior performance over classical descent for tropical optimisation problems which exhibit tropical quasi-convexity but not classical convexity. Notably, tropical gradient descent seamlessly integrates into advanced optimisation methods, such as Adam, offering improved overall performance.
Janusz
Ginster
WIAS
Formation of microstructure for singularly perturbed problems related to helimagnets and shape-memory alloys
Abstract
Learning to Compute Gröbner bases
Abstract.
Solving a polynomial system, or computing an associated \gb basis, has been a fundamental task in computational algebra. However, it is also known for its notorious doubly exponential time complexity in the number of variables in the worst case. In this talk, I present a new paradigm for addressing such problems, i.e., a machine-learning approach using a Transformer. The learning approach does not require an explicit algorithm design and can return the solutions in (roughly) constant time. This talk covers our initial results on this approach and relevant computational algebraic and machine learning challenges.
Irene
Spelta
HU Berlin
Decomposable G-curves and special subvarieties of the Torelli locus
Abstract.
As it is well known, the Torelli morphism \(j:\mathcal{M}_g\to\mathcal{A}_g\) sends an algebraic curve \(C\) to its Jacobian variety \(JC\). The (closure of the) image inside \(\mathcal{A}_g\) is the so-called Torelli locus. In this talk, we will discuss on the extrinsic geometry of this locus. In particular, we will consider certain special subvarieties coming from families of decomposable G-curves.
Botond
Szabo
Bocconi Milan
Privacy constrained semiparametric inference
Abstract.
For semi-parametric problems differential private estimators are typically constructed in a case-by-case basis. In this work we develop a privacy constrained semi-parametric plug-in approach, which can be used in general, over a collection of semi-parametric problems. We derive minimax lower and matching upper bounds for this approach and provide an adaptive procedure in case of irregular (atomic) functionals. Joint work with Lukas Steinberger (Vienna) and Thibault Randrianarisoa (Toronto, Vector Institute).
History of Mathematics in the Indian Subcontinent
Abstract.
The astral sciences in the Indian subcontinent—mathematics, astronomy, and related disciplines—have thrived for over two and a half millennia. This rich culture of inquiry has yielded profound insights and techniques that form the bedrock of modern scientific practices, including the base-ten decimal place value system and trigonometry. Despite India’s prolific production of millions of manuscripts over this period, a mere fraction have been identified and studied in depth. This talk will explore some of the mathematical highlights of this scientifically dynamic tradition and address the challenges faced by historians of mathematics in comprehensively accounting for the scientific legacy of this extensive and rich culture of inquiry.
Vincent
Pilaud
Universitat de Barcelona
Wigglyhedra
Abstract.
I will present combinatorial and geometric properties of the wiggly complex and wiggly permutations, which are sort of degenerations of the pseudotriangulation complex and contain copies of all Cambrian associahedra. This is joint work with Asilata Bapat (arXiv:2407.11632).
Tim
Seynnaeve
KU Leuven
The translation-invariant Bell polytope
Abstract.
Bell's theorem, which states that the predictions of quantum theory cannot be accounted for by any classical theory, is a foundational result in quantum physics. In modern language, it can be formulated as a strict inclusion between two geometric objects: the Bell polytope and the convex body of quantum behaviours. Describing these objects leads to a deeper understanding of the nonlocality of quantum theory, and has been a central research theme is quantum information theory for several decades. After giving an introduction to the topic, I will focus on the so-called translation-invariant Bell polytope. Physically, this object describes Bell inequalities of a translation-invariant system; mathematically it is obtained as a certain projection of the ordinary Bell polytope. Studying the facet inequalities of this polytopes naturally leads into the realm of tensor networks, combinatorics, and tropical algebra. This talk is based on joint work with Jordi Tura, Mengyao Hu, Eloic Vallée, and Patrick Emonts.
Dongmeng
Xi
Shanghai University
Minkowski Problems: old and new
Abstract.
The classical Minkowski problem asks both the necessary and sufficient conditions for a spherical measure to be the surface area measure of a convex body. Minkowski and Aleksandrov made landmark contributions to this area. We then review the dual Minkowski problems introduced by Huang-Lutwak-Yang-Zhang, in which they opened the door to the study of general geometric measures. Finally, we present some recent developments in Minkowski problems, including those in integral geometry, Gaussian probabilistic space, and affine convex geometry. We will introduce the new concepts and problems, explore the connections between old and new ones, and discuss the new developments of tools.
September 2024
Sylvain
Spitz
Thesis Defense
Abstract.
Thesis Defense of Sylvain Spitz
Matthias
Grützner
HU Berlin
On a variational model for microstructures in shape-memory alloys: energy scaling behaviour
Benjamin
Schröter
KTH
Classes of matroids for which Tutte polynomials are universal valuative invariants
Abstract.
Matroids are generalization of both graphs and hyperplane arrangements. Many interesting invariants of these combinatorial objects are valuative. Two prominent examples of valuative matroid invariants are the Tutte polynomial and the \mathcal{G}-invariant. The relevance of the \mathcal{G}-invariant steams from its universal property that any other valuative invariant can be obtained as a specialization. Nevertheless, the most intense studied invariant of matroids is clearly the Tutte polynomial as it respects deletion and contraction. An interesting question therefore is on which minor and duality closed classes of matroids is the Tutte polynomial universal. In my talk I will give a complete answer to this question. This talk is based on joint work with Luis Ferroni.
Hannes
Reichle
HU Berlin
Mathematical Analysis of Minimizing Movement Schemes for Oscillatory Energies
Dimitri
Loutchko
University of Tokyo, Institute of Industrial Science, Japan
Information geometrical entropy production decompositions in chemical reaction networks
Abstract
Verifying the equivalence or non-equivalence of quantum circuits with tensor networks
Abstract.
The development of quantum computers and algorithms is currently rapidly accelerating and will likely continue to do so in the next few years and even decades. As these systems continue to increase in size and complexity, there is an increasing need for methods to aid in their design. In particular, there is a need for debugging tools at each level of the quantum computing stack, specifically in the compilation and optimization of quantum circuits. In this work, we bridge the gap between quantum many-body physics and computer science by using tensor network techniques to verify the equivalence or non-equivalence of quantum circuits in order to detect errors that may occur during the many steps of this process.
Ingo
Meise
ZIB
/
TU Berlin
A Comparison of Simple and Repeated Weighted Majority Voting
Abstract.
While weighted majority voting (WMV) is a popular method in ensemble learning for classification, more complex aggregation rules have recently been shown to have beneficial theoretical properties such as better convergence in the number of calls to a weak learning oracle and better PAC-bounds. After a brief discussion of the literature, this talk compares simple WMV to repeated WMV. In particular, a tight approximability analysis of 1) empirical error minimization, 2) regularization by sparsity and 3) sparsity maximization subject to zero training error is provided. It reveals that, under polynomial time transformations, the respective problems for simple WMV are equivalent to a hard version of Minimum Weighted Set Cover (MWSC) while they are equivalent to a simple version of MWSC for repeated WMV. This analysis of approximability implies two lower bounds on convergence in the number of calls to a weak learning oracle until perfect fit for 1) simple WMV and 2) arbitrary aggregation rules which emphasizes the worse scalability of simple WMV. These theoretical results are complemented by experiments on medium-sized data sets from LIBSVM showing that repeated WMV needs fewer oracle calls compared to AdaBoost and XGBoost while having even or better test accuracy on average.
August 2024
Anna M.
Limbach
Czech Academy of Sciences
Graphon Branching Processes and Fractional Isomorphism
Abstract.
In their study of the giant component in inhomogeneous random graphs, Bollobás, Janson, and Riordan introduced a class of branching processes parametrized by an L¹-graphon. We prove that two such branching processes have the same distribution if and only if the corresponding graphons are fractionally isomorphic, a notion introduced by Grebík and Rocha. https://arxiv.org/abs/2408.02528
Gregory
Sorkin
London School of Economics and Political Science
Matchings and Loose Cycles in the Semirandom Hypergraph Model
Anita
Liebenau
UNSW Sydney
Ramsey with purple edges
Shagnik
Das
National Taiwan University
Fractionally Intersecting Families
Michael
Anastos
Institute of Science and Technology Austria
Climbing up a random subgraph of the hypercube
Hugo
Abreu
ZIB
Bell inequalities with classical communication via Frank-Wolfe algorithms
Abstract.
Bell's 1964 theorem established that the predictions of quantum theory cannot be explained by any local theory, making nonlocality a pivotal concept in the understanding of quantum mechanics. Quantifying the nonlocality of a quantum state in a given scenario is challenging, so we propose examining the amount of classical communication required to simulate the correlations obtained by a quantum state, and thereby indirectly characterizing its nonlocality. We explore Bell scenarios augmented with one or more bits of classical communication with Frank-Wolfe algorithms, and extend the BellPolytopes.jl Julia library to handle those scenarios. In this talk, I will introduce the theoretical framework, cover implementation details, and present some preliminary results.
Andreas
Amann
University College Cork, Ireland and Potsdam Institute for Climate Impact Research
Dynamics of adaptive networks with Hebbian and Anti-Hebbian plasticity
July 2024
Ansgar
Freyer
TU Vienna
Lattice Points in Hyperplane Sections
Abstract.
The flatness theorem due to Khinchine states that a convex body without interior lattice points cannot be too large, in the sense that its lattice width is bounded by a constant depending only on the dimension, the so-called flatness constant. Khinchine's flatness theorem played a key role for integer programming in fixed dimension and thus inspired a lot of research on the flatness constant, in particular its asymptotic growth. In the talk we present another application of the flatness theorem in discrete geometry. We will use it to compare the number of lattice points in a (centrally symmetric) convex body to the maximum number of lattice points in one of its (central) hyperplane sections. Moreover, we discuss methods of determining the flatness constant in low dimensions. The talk is based on joint works with Giulia Codenotti and Martin Henk.
Adélie
Garin
EPFL
Topological Data Analysis meets Geometric Group Theory: Stratifying the space of barcodes using Coxeter complexes
Abstract.
At the intersection of data science and algebraic topology, topological data analysis (TDA) is a recent field of study, which provides robust mathematical, statistical and algorithmic methods to analyze the topological and geometric structures underlying complex data. TDA has proved its utility in many applications, including biology, material science and climate science, and it is still rapidly evolving. Barcodes are frequently used invariants in TDA. They provide topological summaries of the persistent homology of a filtered space. Understanding the structure and geometry of the space of barcodes is hence crucial for applications.<br><br>In this talk, we use Coxeter complexes to define new coordinates on the space of barcodes.These coordinates define a stratification of the space of barcodes with n bars where the highest dimensional strata are indexed by the symmetric group. This creates a bridge between the fields of TDA, geometric group theory and permutation statistics, which could be exploited by researchers from each field.<br><br>No prerequisite on TDA or Coxeter complexes are required.
Chris
Huggle
Sheaves and Cosheaves on continuous posets
Daniel
Matthes
TU München
Covariance modulated optimal transport: geometry and gradient flows
Stefan
Sauter
Uni Zürich
The acoustic half space Green's function with
impedance boundary condition in d spatial dimensions:
Fast evaluation and numerical quadrature
Abstract.
In our talk, we introduce a representation of the acoustic half space Green's function with impedance boundary conditions in d space dimensions which avoids oscillatory Fourier integrals. A numerical quadrature method is developed for its fast evaluation. In the context of boundary element methods this function must be integrated over pairs of simplices and we present an efficient approximation method.
Jürg
Kramer
HU Berlin
Arithmetic intersections of line bundles with singular metrics
Abstract.
In our talk, we will present an extension of arithmetic intersection theory of adelic divisors on quasi-projective varieties introduced by Yuan-Zhang to the case where these divisors are not necessarily arithmetically nef. The key tool to realize this extension is the concept of relative finite energy established by T. Darvas et al.. In particular, our theory will allow to compute heights on mixed Shimura varieties, e.g. the arithmetic self-intersection number of the line bundle of Siegel-Jacobi forms on the universal abelian variety. This is joint work with José Burgos Gil.
Florentin
Seifert
On variational models for biological membranes
Martin
Burger
DESY, Helmholtz Imaging
Particle Methods in Machine Learning and Inverse Problems
Jia-Jie
Zhu
WIAS
$\vec{w}h\alpha\mathfrak{t}\;\; \forall\mathbb{R}\varepsilon\ldots$gradient flows and optimal transport for machine learning and optimization?
Abstract.
In this talk, I will provide an overview of gradient flows over non-negative and probability measures, e.g. in the Wasserstein space, and their application in modern machine learning tasks, such as variational inference, sampling, training of over-parameterized models, and robust optimization. Then, I will present the high level idea of theoretical analysis as well as our recent work in unbalanced transport, such as Hellinger-Kantorovich (a.k.a. Wasserstein-Fisher-Rao), and its particle approximation for machine learning. The talk is mainly based on the recent joint works with Alexander Mielke, Pavel Dvurechensky, and Egor Gladin.
Jianniao
Qui
University of Calgary
Consensus-based optimization for equilibrium points of games
Abstract.
In this talk, we will introduce Consensus-Based Optimization (CBO) for min-max problems, a novel multi-particle, derivative-free optimization method that can provably identify global equilibrium points. This paradigm facilitates the transition to the mean-field limit, making the method amenable to theoretical analysis and providing rigorous convergence guarantees under reasonable assumptions about the initialization and the objective function, including nonconvex-nonconcave objectives. Additionally, numerical evidence will be presented to demonstrate the algorithm’s effectiveness. This talk is based on joint works with Giacomo Borghi, Enis Chenchene, Hui Huang, and Konstantin Riedl.
Tiziano
de Angelis
University of Turin
Linear-quadratic stochastic control with state constraints on finite-time horizon
Abstract.
We obtain a probabilistic solution to linear-quadratic optimal control problems with state constraints. Given a closed set D ⊆ [0, T] × Rd, a diffusion X in Rd must be linearly controlled in order to keep the time-space process (t, Xt) inside the set C := ([0, T] × Rd) \ D, while at the same time minimising an expected cost that depends on the state (t, Xt) and it is quadratic in the speed of the control exerted. We find an explicit probabilistic representation for the value function and the optimal control under a set of mild sufficient conditions concerning the coefficients of the underlying dynamics and the regularity of the set C. Fully explicit formulae are presented in some relevant examples. (Joint work with Erik Ekström, University of Uppsala, Sweden)
Germain
Poullot
Uni Osnabrück
Deformed Permutahedra: Let's visit the submodular cone together!
Abstract.
The permutahedron is a famous polytope whose combinatorics encapsulates the combinatorics of permutations (seen as a Coxeter group). This talk aims at illustrating what happens when we "deform" the permutahedron, that is to say we glide the facets of the permutahedron, keeping the same normal vectors. Each deformation can hence be thought as a sub-combinatorics of the combinatorics of permutations, endowed with a polytopal realization. We will first give two ways of thinking about deformations in general, and then look at the case of the permutahedron. The set of deformations of a polytope forms a cone: in the case of the permutahedron, this cone is named the "submodular cone", and has a lot of nice but complicated properties. The last part of the talk will be devoted to the illustration of some fun facts, obtained thanks to numerical experiments, on the submodular cone.
Chris
Huggle
Sheaves and Cosheaves on continuous posets
Vittorio
Di Fraia
Localizing Invariants
Jiniao
Qiu
U of Calgary
Dan
Crisan
Imperial College London
Caroline
Lasser
Fakultät für Mathematik, TU München
Variational Gaussian approximation for quantum dynamics
Abstract
Lara
Theallier
HU Berlin
Lower energy bounds in the Landau-de Gennes model for nematic liquid crystals
Does Gender Still Matter? Perspectives of Scientists in Leadership Position and Early Career Researchers on Academic Careers in Mathematics.
Abstract
Anya
Katsevich
MIT, Cambridge, MA
Laplace asymptotics in high-dimensional Bayesian inference
Abstract.
Computing integrals against a high-dimensional posterior is the major computational bottleneck in Bayesian inference. A popular technique to reduce this computational burden is to use the Laplace approximation (LA), a Gaussian distribution, in place of the true posterior. We derive a new, leading order asymptotic decomposition of integrals against a high-dimensional Laplace-type posterior which sheds valuable insight on the accuracy of the LA in high dimensions. In particular, we determine the tight dimension dependence of the approximation error, leading to the tightest known Bernstein von Mises result on the asymptotic normality of the posterior. The decomposition also leads to a simple modification to the LA which yields a higher-order accurate approximation to the posterior. Finally, we prove the validity of the high-dimensional Laplace asymptotic expansion to arbitrary order, which opens the door to approximating the partition function, of use in high-dimensional model selection and many other applications beyond statistics.
Richard von Mises Lectureregister here
Gertrud
Graser
Lösen von Multi-Leader-Follower Spielen über nichtglatte Nash Equilibrium Probleme
Eleonore
Bach
EPFL
Georg
Lehner
Limits of dualizable infinity-categories
Vittorio
Di Fraia
Localizing Invariants
Jie
Han
Beijing Institute of Technology
Graph Theory theorems in random graphs
Andreas
Vikelis
Faculty of Mathematics, University of Vienna, Austria
Measure-valued solutions for non-associative finite plasticity
Ngoc Tien
Tran
Uni Augsburg
Hybrid high-order method for the biharmonic eigenvalue problem
Generative Modelling with Tensor Train Approximations of Hamilton–Jacobi–Bellman Equations
Abstract
Celine
Duval
Université de Lille
Geometry of excursion sets: Computing the surface area from discretized points
Abstract.
The excursion sets of a smooth random field carries relevant information in its various geometric measures. After an introduction of these geometrical quantities showing how they are related to the parameters of the field, we focus on the problem of discretization. From a computational viewpoint, one never has access to the continuous observation of the excursion set, but rather to observations at discrete points in space. It has been reported that for specific regular lattices of points in dimensions 2 and 3, the usual estimate of the surface area of the excursions remains biased even when the lattice becomes dense in the domain of observation. We show that this limiting bias is invariant to the locations of the observation points and that it only depends on the ambient dimension. (based on joint works with H. Biermé, R. Cotsakis, E. Di Bernardino and A. Estrade).
Remy
van Dobben de Bruyn
Utrecht
Constructible sheaves and exodromy
Abstract.
Locally constant sheaves are most easily understood as representations of the fundamental group, via the monodromy correspondence. In algebraic geometry, it is often preferable to use the larger class of constructible sheaves, as these are stable under (higher) pushforward. In 2018, Barwick, Glasman, and Haine proved an exodromy correspondence for constructible étale sheaves, using ideas from higher topos theory and profinite stratified homotopy theory. In this talk, I will present a more direct geometric proof of the étale and pro-étale exodromy theorems, based on joint work with Sebastian Wolf.
Nicholas
Charron
ZIB
Machine Learned Force Fields, Coarse Graining, HPC, & Beyond
Abstract.
Machine learned force fields (MLFFs), particular those using deep neural networks to model interaction potentials are quickly becoming a powerful tool for modelling complex molecular systems at scale with both classical and quantum accuracy. In this talk, we demonstrate an application of developing transferable coarse grained (CG) MLFFs for proteins using HPC resources to show how machine learning can be used easily and effectively on compute clusters to solve relevant chemical/physical problems. We furthermore discuss growing trends and usage of MLFFs with HPC resources, including emerging datasets, hardware demands, and integrations of machine learned potentials with existing simulation software.
Norbert
Heuer
PUC Chile
Abstrakte hybride Galerkin Methoden
June 2024
On the inverse conductivity problem
Marwa
Zainelabdeen
WIAS
What is PINNs?
Abstract.
In convection-dominated regimes, solutions of convection-diffusion problems often have steep gradients. Traditional numerical methods struggle with these issues. Recently, physics-informed neural networks (PINNs) have emerged, minimizing residuals at collocation points. This talk introduces specialized loss functionals for PINNs tailored to these problems.
Stefan
Felsner
Drawings of products of cycles
Benjamin
Jourdan
University of Paris-Est
Convexity propagation and convex ordering of one-dimensional stochastic differential equations
Abstract.
We consider driftless one-dimensional stochastic differential equations. We first recall how they propagate convexity at the level of single marginals. We show that some spatial convexity of the diffusion coefficient is needed to obtain more general convexity propagation and obtain functional convexity propagation under a slight reinforcement of this necessary condition. Such conditions are not needed for directional convexity. This is a joint work with Gilles Pagès.
Libo
Li
UNSW Sydney
Vulnerable European and American Options in a Market Model with Optional Hazard
Abstract.
We study the upper and lower bounds for prices of European and American style options with the possibility of an external termination, meaning that the contract may be terminated at some random time. Under the assumption that the underlying market model is incomplete and frictionless, we obtain duality results linking the upper price of a vulnerable European option with the price of an American option whose exercise times are constrained to times at which the external termination can happen with a non-zero probability. Similarly, the upper and lower prices for a vulnerable American option are linked to the price of an American option and a game option, respectively. In particular, the minimizer of the game option is only allowed to stop at times which the external termination may occur with a non-zero probability.
Tuan Thanh
Truong
AG H. Gottschalk, TU Berlin
and
Miguel
Matamouros Resende Frade de Campos
AG H. Gottschalk, TU Berlin
Shape definition and shape optimization with invertible neural networks
Karel
Devriendt
MPI-MiS
Spanning trees, effective resistances and curvature on graphs
Abstract.
Kirchhoff's celebrated matrix tree theorem expresses the number of spanning trees of a graph as the maximal minor of the Laplacian matrix of the graph. In modern language, this determinantal counting formula reflects the fact that spanning trees form a regular matroid. In this talk, I will discuss some consequences of this matroidal perspective for the study of a related quantity from electrical circuit theory: the effective resistance. I will give a characterization of effective resistances in terms of a certain polytope associated with the spanning tree matroid and discuss applications to recent work on discrete notions of curvature based on the effective resistance.
Paul
Brommer-Wierig
Tensor products and dualizability of presentable ∞-categories
Martin
Ulirsch
Goethe University Frankfurt am Main
What is the tropicalization of a matrix?
Abstract.
Tropicalization is a process that associates to an algebro-geometric object a piecewise linear polyhedral shadow that captures its essential combinatorial structure. In this talk, I will give an overview of the numerous ways on how to extract tropical information from a matrix over a non-Archimedean field. Each perspective will give rise to inherently quite different phenomena. Central instances of this rich panorama include the tropical geometry of vector bundles, logarithmic concavity results for valuated (bi-)matroids (using techniques from combinatorial Hodge theory), and the geometry of affine buildings.This talk draws from joint work with Andreas Gross and Dmitry Zakharov; Andreas Gross, Inder Kaur, and Annette Werner; Felix Röhrle; Jeff Giansiracusa, Felipe Rincon, and Victoria Schleis; Luca Battistella, Kevin Kühn, Arne Kuhrs, and Alejandro Vargas; as well as with Desmond Coles.
Tibor
Szabó
Freie Universität Berlin
Global rigidity of random graphs in R
Noemi
Kurt
Goethe University Frankfurt
Benjamin
Jourdain
École des Ponts ParisTech
Model predictive control for generalized Nash Equilibrium problems
Abstract.
We study model predictive control (MPC) schemes for non-cooperative dynamic games. The dynamic games are modelled as jointly convex generalized Nash equilibrium problems (GNEP) governed by a jointly controlled linear time-discrete dynamics. We are particularly interested in the asymptotic stability of the resulting closed-loop dynamics. To this end, we introduce a family of auxiliary problems, α-Quasi-GNEPs, which approximate the original GNEPs. For MPC schemes based on α-Quasi-GNEPs, stability guarantees can be derived if stabilizing end-constraints are enforced. This analysis is based on showing that the underlying optimal-value function is a Lyapunov function for the closed-loop. Passing to a limit, we identify a suitable Lyapunov function for MPC schemes based on the original GNEPs.
Swarm-Performance of Multi-Agent Systems and Connections to Equity
Abstract.
Many real-world systems are composed of agents whose interactions result in a collective swarm behavior that may be complex, unexpected, and/or unintended. We highlight intriguing cases of interplay between the micro-scale behavior of agents and the macro-scale performance of the swarm, with a particular emphasis on heterogeneous systems composed of different types of agents, such as: traffic flow (the role of automation/connectivity on the energy footprint of urban traffic flow), mixed human/robotic groups (transportation of supplies to a disaster area), and biological systems (schools of fish and colonies of penguins). We particularly show how behavior interpretable as 'equitable’ or 'altruistic’ is possible to arise from pure survival-of-the-fittest objective functions
Clément
Berenfeld
Universität Potsdam
A theory of stratification learning
Abstract.
Given i.i.d. sample from a stratified mixture of immersed manifolds of different dimensions, we study the minimax estimation of the underlying stratified structure. We provide a constructive algorithm allowing to estimate each mixture component at its optimal dimension-specific rate adaptively. The method is based on an ascending hierarchical co-detection of points belonging to different layers, which also identifies the number of layers and their dimensions, assigns each data point to a layer accurately, and estimates tangent spaces optimally. These results hold regardless of any ambient assumption on the manifolds or on their intersection configurations. They open the way to a broad clustering framework, where each mixture component models a cluster emanating from a specific nonlinear correlation phenomenon.
Gari Yamel
Peralta Alvarez
HU Berlin
Toric arithmetic varieties and adelic intersection numbers
Abstract.
A result of Burgos, Philippon and Sombra states that the height of a toric arithmetic variety with respect to a line bundle equipped with a torus-invariant continuous semipositive metric is given by the integral of a convex function over a polytope. In this talk, I will discuss a generalization of this formula for line bundles with torus-invariant singular metrics. The techniques used to obtain this result are based on Yuan and Zhang's theory of adelic line bundles. Moreover, this formula agrees with the generalized adelic intersection numbers of Burgos and Kramer.
Lídia
Rodríguez
ZIB
/
UPC
A Formal Proof of the Sensitivity Conjecture
Abstract.
The use of proof assistants has been on the rise these past few years thanks to their ability to detect small flaws in mathematical proofs. The integration of AI has also made the use of these kind of tools easier and it has opened a lot of possible advancements for the future. In this talk we will focus on the ITP Lean, with an emphasis on some results in the hypercube graph. The center of these results will be the Sensitivity Conjecture, which had already been formalized previously. Our additions to this problem are the formalization of two extremal examples to prove the tightness of the two inequalities in the conjecture. These new results involved a few challenges, such as having to implement a proof of the inclusion-exclusion principle. Additionally, they highlighted the importance of selecting a good way to express our statement, especially as a number of definitions had to be made and the different data representations came with different lemmas available.
Silas
Rathke
On the maximum diameter of d-dimensional simplicial complexes
Lena
Weis
TU Berlin
Computing lines in tropical cubic surfaces
Abstract.
In the 1840s Arthur Cayley and George Salmon showed that a smooth cubic surface in \(\mathbb{P}^3\) contains exactly 27 lines. With the rise of tropical geometry the question of whether this theorem holds for tropical lines in smooth tropical cubic surfaces was posed. Vigeland (2010) was the first to observe that it can contain in fact infinitly many lines. Nonetheless, he conjectured that the aggregate count of isolated and families of lines remained invariant at 27. This conjecture was refuted by Hampe and Joswig (2017), as they discovered a smooth tropical cubic surface containing 26 isolated lines and 3 families of lines. In this talk we will show that the aggregated count can be also less than 27.
Georg
Lehner
Ramzi’s Theorem and limits of compactly assembled categories
Deborah
Hendrych
ZIB
Solving the Optimal Experiment Design Problem with mixed-integer convex methods
Simon
Schmidt
Ruhr-Universität Bochum
Introduction to nonlocal games and graph isomorphism games
Abstract.
A nonlocal game consists of two players that collaborate to win a game against a referee. Comparing their performance with and without access to shared entangled particles allows us to probe the capabilities of entanglement. In this talk, we will give an introduction to nonlocal games. We will focus on the graph isomorphism game, in which the players try to convince the referee that they know an isomorphism between a fixed pair of graphs.
Tim
Stiebert
HU Berlin
Guaranteed lower eigenvalue bounds for the Schrödinger eigenvalue problem
Carl
Lian
Tufts University
Complete quasimaps to \(Bl_p(\mathbb{P}^2)\)
Abstract.
We consider the problem of counting curves \(C\) of fixed moduli in a target variety \(X\) passing through the maximal number of points ("Tevelev degrees"). A broad program for obtaining such a count is:<br>(i) Establish a Brill-Noether theorem for maps to \(X\).<br>(ii) Use (i) to construct a compact moduli space \(M\), generically parametrizing maps to \(X\), which witnesses (without excess intersections) the desired count.<br>(iii) Compute integrals on \(M\), e.g., by degeneration methods.<br>Typically, the moduli spaces coming from, e.g., Gromov-Witten theory, are not sufficient in step (ii). We review the case \(X=\mathbb{P}^r\), where this program has been carried out. Here, one may take \(M\) to be the moduli space of complete collineations (relative to the space of linear series on \(C\)), which is an iterated blow-up of a Quot scheme. We then report on work in progress with Alessio Cela on the case where \(X\) is a blow-up of \(\mathbb{P}^2\) at a point. Here, a Brill-Noether statement is given (more generally, for the blow-up of \(\mathbb{P}^r\) at any linear space) by a result of Farkas, and we construct a moduli space \(M\) of "complete quasimaps" to \(X\). Degenerations on this space are made possible by ideas from the previous calculation on \(\mathbb{P}^r\) and from Coskun's geometric Littlewood-Richardson rule. Our construction seems to hint toward a more general theory of complete quasimaps to other targets.
Heather
Harrington
MPI Dresden
Topology Data Analysis for Multiscale Biology
Likai
Jiao
HU Berlin
An infinite-dimensional price impact model
Abstract.
In this talk, we introduce an infinite-dimensional price impact process as a kind of Markovian lift of non-Markovian 1-dimensional price impact processes with completely monotone decay kernels. In an additive price impact scenario, the related optimal control problem is extended and transformed into a linear-quadratic framework. The optimal strategy is characterized by an operator-valued Riccati equation and a linear backward stochastic evolution equation (BSEE). By incorporating stochastic in-flow, the BSEE is simplified into an infinite-dimensional ODE. With appropriate penalizations, the well-posedness of the Riccati equation is well-known. This is a joint work with Prof. Dirk Becherer and Prof. Christoph Reisinger.
Eduardo
Abi Jaber
École Polytechnique, Palaiseau
Stochastic Fredholm equations: a passe-partout for propagator models with cross-impact, constraints and mean-field interactions
Abstract.
We will provide explicit solutions to certain systems linear stochastic Fredholm equations. We will then show the versatility of these equations for solving various optimal trading problems with transient impact including: (i) cross-impact (multiple assets), (ii) constraints on the inventory and trading speeds, and (iii) N-player game and mean-field interactions (multiple traders). Based on joint works with Nathan De Carvalho, Eyal Neuman, Huyên Pham, Sturmius Tuschmann, and Moritz Voss.
Vladimir
Shikmann
Technische Universität Chemnitz
Cardinality-constrained optimization problems in general position and beyond
Abstract.
We study cardinality-constrained optimization problems (CCOP) in general position, i.e. those optimization-related properties that are fulfilled for a dense and open subset of their defining functions. We show that the well-known cardinality-constrained linear independence constraint qualification (CC-LICQ) is generic in this sense. For M-stationary points we define nondegeneracy and show that it is a generic property too. In particular, the sparsity constraint turns out to be active at all minimizers of a generic CCOP. Moreover, we describe the global structure of CCOP in the sense of Morse theory, emphasizing the strength of the generic approach. Here, we prove that multiple cells need to be attached, each of dimension coinciding with the proposed M-index of nondegenerate M-stationary points. Beyond this generic viewpoint, we study singularities of CCOP. For that, the relation between nondegeneracy and strong stability in the sense of Kojima is examined. We show that nondegeneracy implies the latter, while the reverse implication is in general not true. To fill the gap, we fully characterize the strong stability of M-stationary points under CC-LICQ by first- and second-order information of CCOP defining functions. Finally, we compare nondegeneracy and strong stability of M-stationary points with second-order sufficient conditions recently introduced in the literature.
Bálint
Zsigri
FU Berlin
Grassmannians, oriented matroids, and MacPherson's conjecture
Matei
Hanu
Optimisation in Bayesian experimental design
Tristram
Bogart
Los Andes
Eduardo
Abi Jaber
Ecole Polytechnique
Ruijie
Yang
HU Berlin
Minimal exponent of a hypersurface
Abstract.
Recently, the minimal exponent of a hypersurface over complex numbers has been understood as a useful refined invariant of the log canonical threshold. It has found many new applications including deformation of Calabi-Yau 3-folds (Friedman-Laza), higher rational/du Bois singularities (Mustata-Popa) and geometric Schottky problem (Schnell-Yang). However, some basic properties of this invariant remain mysterious. In this talk I will discuss the conjecture of Mustata and Popa on birational characterization of the minimal exponent, which is the main obstruction for the computation in practice. I will explain the heuristic of the Mustata-Popa conjecture from Igusa's work on counting integer solutions of congruence equations and Igusa’s strong monodromy conjecture. Then I will discuss how several ideas from mixed Hodge modules and geometric representation theory can lead to a better understanding of the minimal exponents. This is based on two joint works with Christian Schnell and Dougal Davis, respectively.
Marc
Hallin
Université Libre de Bruxelles
The long quest for quantiles and ranks in Rd and on manifolds
Abstract.
Quantiles are a fundamental concept in probability, and an essential tool in statistics, from descriptive to inferential. Still, despite half a century of attempts, no satisfactory and fully agreed-upon definition of the concept, and the dual notion of ranks, is available beyond the well-understood case of univariate variables and distributions. The need for such a definition is particularly critical for varia- bles taking values in Rd, for directional variables (values on the hypersphere), and, more generally, for variables with values on manifolds. Unlike the real line, indeed, no canonical ordering is available on the- se domains. We show how measure transportation brings a solution to this problem by characterizing distribution-specific (data-driven, in the empirical case) orderings and center-outward distribution and quantile functions (ranks and signs in the empirical case) that satisfy all the properties expected from such concepts while reducing, in the case of real-valued variables, to the classical univariate notion.
Marco
Flores
HU Berlin
Modularity of formal Fourier-Jacobi series from a cohomological point of view
Abstract.
In the Kudla program, generating series constructed by means of arithmetic intersection numbers are conjectured to appear as Fourier coefficients of modular forms. One instance of this program was the proof in 2015 by Bruinier and Raum of Kudla's modularity conjecture over \(\mathbb{C}\) for special cycles on a Shimura variety of orthogonal type, obtained by showing that certain formal series of Jacobi forms are in fact modular forms. This result was reformulated and proven over \(\mathbb{Z}\) by Kramer using the Faltings-Chai theory of toroidal compactifications of \(\mathcal{A}_g\). <br>In this talk, we contextualize the above result using cohomology of line bundles on such a toroidal compactification \(\overline{\mathcal{A}}_g\), which we believe is the more natural point of view. We then obtain cohomological vanishing results characterizing the modularity of formal Fourier-Jacobi series, taking advantage of the natural resolution morphism from \(\overline{\mathcal{A}}_g\) to the minimal compactification \(\mathcal{A}^*_g\).
Ehrhard
Behrends
Freie Universität Berlin
Zauberhafte Mathematik
Abstract.
Wenn Kunststücke von Zauberern das Publikum beeindrucken, kann das verschiedene Ursachen haben. Es kann dann um Fingerfertigkeit gehen, oder um Ablenkung, den Einsatz mehr oder weniger komplizierter Apparaturen bis hin zu aufwändigen elektronischen Schaltungen usw. Man kann sich aber auch Tatsachen aus verschiedenen Wissenschaftsgebieten zunutze machen: Physik, Chemie und natürlich auch Mathematik. In dem Vortrag soll gezeigt werden, wie man verschiedene Aspekte unseres Faches für die Zauberei nutzen kann. Eher klassisch (und meiner Meinung nach ein bisschen langweilig) ist der Einsatz elementarer Algebra: "Denk Dir eine Zahl ..." Darum wird es nicht gehen. Vielmehr werden wir Beispiele aus Kombinatorik, Gruppentheorie, Stochastik und Zahlentheorie kennenlernen. Das Niveau reicht von "leicht verständlich" bis "ein bisschen anspruchsvoll". Für Bachelorstudierende sollte es keine Probleme geben. Es empfiehlt sich, etwas zum Schreiben und ein Kartenspiel dabei zu haben. Dann kann man einige der Kunststücke gleich ausprobieren.
Elisabeth
a.d. Siepen
and
Georg
Lehner
Compactly assembled infinity-categories II + Examples
Georg
Lehner
Compactly assembled infinity-categories II
Andre
Zepernick
Quasi-Monte Carlo Methods for PDEs on Random Domains
Rene
Brandenberg
TU München
Tristram
Bogart
Los Andes
Numerical Semigroups via Projections and via Quotients
Abstract.
A numerical semigroup is any cofinite subset of the natural numbers that is closed under addition and contains 0. Numerical semigroups and their higher-dimensional cousins, known as affine semigroups, arise in optimization as well as in the study of toric ideals and varieties in algebraic geometry. Every numerical semigroup S has a unique minimal generating set, whose cardinality is called the embedding dimension e(S). If e(S) is large, it is natural to ask whether S can be ''encapsulated'' by some larger semigroup T with smaller embedding dimension. One notion of ''encapsulation'' is that S is the quotient of a numerical semigroup T by a number d, that is: S = T / d = {x: dx belongs to T\} Another, more geometric notion, is that S can be obtained as a coordinate projection of an affine semigroup T. Our main result is that these two forms of encapsulation are essentially equivalent. We also give various families of semigroups that both do and do not admit encapsulation. This project is joint work with Christopher O'Neill and Kevin Woods.
Dhruv
Mubayi
University of Illinois
New bounds for the Erdős-Rogers problem
Stefan
Sauter
Uni Zürich
The acoustic half space Green's function with
impedance boundary condition in d spatial dimensions:
Fast evaluation and numerical quadrature
Abstract.
In our talk, we introduce a representation of the acoustic half space Green's function with impedance boundary conditions in d space dimensions which avoids oscillatory Fourier integrals. A numerical quadrature method is developed for its fast evaluation. In the context of boundary element methods this function must be integrated over pairs of simplices and we present an efficient approximation method.
Benedikt
Gräßle
HU Berlin
Analysing WOPSIP by supercloseness
Carl
Yerger
Davidson College
Ramsey Theory on the Integer Grid: The "L"-Problem
Abstract.
This talk will begin with a brief introduction to classical Ramsey theory (whose main idea is that complete disorder is impossible) and a discussion of a few well-known results. No specialized background knowledge in combinatorics is required for this talk. Then joint work with Will Smith from University of South Carolina (formerly Davidson) on a Ramsey theory problem based on the integer grid will be presented. Specifically, the L-theorem (an easy corollary of the Gallai-Witt theorem) states that for any integer k, there exists some n such that a k-colored n by n grid must contain a monochromatic \"L\" (a series of points of the form (i, j), (i, j+t), and (i+t, j+t) for some positive integer t. In this talk, we will investigate the upper bound for the smallest integer n such that a 3-colored n by n integer grid is guaranteed to contain a monochromatic L. We use various methods, such as counting intervals on the main diagonal, to improve the upper bound from 2593 to 493. For the lower bound, we match the lower bound of 21 generated by Canacki et al. (2023) and discuss possible improvements.
Jia-Jie
Zhu
WIAS Berlin
Wasserstein and beyond: Optimal transport and gradient flows for machine learning and optimization
Abstract.
In the first part of the talk, I will provide an overview of gradient flows over non-negative and probability measures and their application in modern machine learning tasks, such as variational inference, sampling, training of over-parameterized models, and robust optimization. Then, I will present our recent results on the analysis of a couple of particularly relevant gradient flows, including the settings of Wasserstein, Hellinger/Fisher-Rao, and reproducing kernel Hilbert space. The focus is on the global exponential decay of the entropy functionals along the gradient flows such as Hellinger-Kantorovich (a.k.a. Wasserstein-Fisher-Rao) and a new type of gradient flow geometries that guarantee convergence of minimizing a maximum-mean discrepancy, which we term the interaction-force transport.
Richard
Rimanyi
UNC Chapel Hill and Newton Institute Cambridge
3d mirror symmetry for characteristic classes
Abstract.
In this joint work with Tommaso Botta we study the elliptic characteristic classes called stable envelopes introduced by M. Aganagic and A. Okounkov. Stable envelopes measure singularities, they geometrize quantum group representations, and they can be interpreted as monodromy matrices of certain differential or difference equations. We prove that for a rich class of holomorphic symplectic varieties (called bow varieties) their elliptic stable envelopes display a duality inspired by mirror symmetry in d=3, N=4 quantum field theories. In the key step of our proof, we "resolve" large charge branes to a number of smaller charge branes. This phenomenon turns out to be the geometric counterpart of the algebraic fusion procedure. Along the way we discover more about the rich geometry of bow varieties, such as their Bruhat order and the elliptic Hall algebra structure on their stable envelopes.
May 2024
Anna
Wienhard
MPI Leipzig
Beyond hyperbolic geometry
Kanishka
Katipearachchi
HU Berlin
What is Hyperbolic Geometry?
Abstract.
In this talk I will give a gentle and non-rigourous introduction to hyperbolic geometry. Starting from a slight modification of the parallel postulate, we will build a model for the hyperbolic plane and use its transformation group to deduce a metric for it. We will then use this model to build hyperbolic surfaces and possibly define the Teichmüller space of hyperbolic structures for a topological surface.
Shige
Peng
Shandong University
Solving probability measure uncertainty by nonlinear expectations
Abstract.
In 1921, economist Frank Knight published his famous 'Uncertainty, Risk and Profit' in which his challenging is still largely open. In this talk we explain why nonlinear expectation theory provides a powerful and fundamentally important mathematical tool to this century problem.
Marko
Weber
National University of Singapore
General Equilibrium with Unhedgeable Fundamentals and Heterogeneous Agents
Abstract.
We examine the implications of unhedgeable fundamental risk, combined with agents' heterogeneous preferences and wealth allocations, on dynamic asset pricing and portfolio choice. We solve in closed form a continuous-time general equilibrium model in which unhedgeable fundamental risk affects aggregate consumption dynamics, rendering the market incomplete. Several long-lived agents with heterogeneous risk-aversion and time-preference make consumption and investment decisions, trading risky assets and borrowing from and lending to each other. We find that a representative agent does not exist. Agents trade assets dynamically. Their consumption rates depend on the history of unhedgeable shocks. Consumption volatility is higher for agents with preferences and wealth allocations deviating more from the average. Unhedgeable risk reduces the equilibrium interest rate only through agents' heterogeneity and proportionally to the cross-sectional variance of agents' preferences and allocations.
Lilla
Tothmeresz
ELTE
Relationships between the geometry of graph polytopes and graph structure
Abstract.
The symmetric edge polytope is a lattice polytope associated to a graph, that is actively investigated both because of its beautiful geometry, and because of its connections to the Kuramoto synchronization model of physics. One can also investigate „non-symmetric” edge polytopes, that are assigned to directed graphs instead of undirected ones. Moreover, one can even generalize them to regular (oriented) matroids. For these more general polytopes, many interesting geometric phenomena uncover themselves that are hidden for symmetric edge polytopes. We would like to demonstrate that the geometric properties of (graph and matroid) edge polytopes are connected to deep graph/matroid-theoretic properties. In particular, we show that the co-degree of an edge polytope is equal to the minimal cardinality of a dijoin. Other notions of combinatorial optimization also turn up naturally with respect to edge polytopes. In particular, for an Eulerian directed graph, complements of arborescences rooted at a given vertex give a (unimodular) triangulation of the edge polytope of the cographic matroid. This gives an alternative, geometric proof for the fact that an Eulerian digraph has the same number of arborescences for any choice of root vertex. I will also mention open problems. Based on joint work with Tamás Kálmán.
Elisabeth
a.d. Siepen
Compactly assembled infinity-categories
Geometric Machine Learning and Graph Machine Learning
Konrad
Mundinger
ZIB
Learning Operators via Hypernetworks
Shige
Peng
Shandong University
Nikolay
Barashkov
MPI Leipzig
Carsten
Carstensen
HU Berlin
A simple approach for companion operators for Crouzeix-Raviart finite element spaces with inhomogeneous Dirichlet boundary conditions
Tailen
Hsing
University of Michigan
A functional-data perspective in spatial data analysis
Abstract.
More and more spatiotemporal data nowadays can be viewed as functional data. The first part of the talk focuses on the Argo data, which is a modern oceanography dataset that provides unprecedented global coverage of temperature and salinity measurements in the upper 2,000 meters of depth of the ocean. I will discuss a functional kriging approach to predict temperature and salinity as a smooth function of depth, as well as a co-kriging approach of predicting oxygen concentration based on temperature and salinity data. In the second part of the talk, I will give an overview on some related topics, including spectral density estimation and variable selection for functional data.
Quantum noise characterization with a tensor network quantum jump method
Abstract.
In this talk, we will discuss a novel approach to characterizing the noise in noisy quantum circuits through the Tensorized Quantum Jump Method (TJM). The well-known Quantum Jump Method (Monte Carlo wave function), which is used to approximate Lindbladian dynamics, can be transferred to a tensor network algorithm via Strang splitting of the Lindbladian and the help of a dynamical low-rank approximation through the Time-Dependent Variational Principle. Choosing the sparse Pauli strings of the Sparse Pauli-Lindblad Model (SPLM) as Lindblad operators makes this method a new approach to characterizing quantum noise in large systems by learning the corresponding noise parameters.
Paul
Ziegler
Darmstadt
p-adic integration on Artin stacks
Abstract.
After giving an introduction to the technique of p-adic integration, I will explain how this technique can be extended to Artin stacks, and give an application to BPS invariants. This is joint work with M. Groechenig and D. Wyss.
Andrea
Walther
HU Berlin
Backpropagation and Nonsmooth Optimization for Machine Learning
Abstract.
Backpropagation is the major workhorse for many machine learning algorithms. In this presentation, we will examine the theory behind backpropagation as provided by the technique of algorithmic differentiation. Subsequently, we will discuss how this classic derivative information can be used for nonsmooth optimization. Examples from reail will illustrate the application of the proposed nonsmooth optimization algorithm.
Richard
Lang
University Hamburg
A hypergraph bandwidth theorem
Michael
Zaks
Inst. of Physics, HU Berlin
Continua of equilibrium states in globally coupled ensembles
Robust Multilevel Training of Artificial Neural Networks
Abstract.
In this talk, we will introduce a multilevel optimizier for training of an artificial neural network. We are particularly interested in nerual networks to learn the hidden physical law or nonlinear mapping from the given data using algebraic multigrid strategies. And we would like to give some further insight into the potential of multilevel optimization methods in the end.
Alessio
Figalli
ETH Zürich
Exploring Stability in Geometric and Functional Inequalities
Abstract.
In the realms of analysis and geometry, geometric and functional inequalities are of paramount significance, influencing a variety of problems. Traditionally, the focus has been on determining precise constants and identifying minimizers. More recently, there has been a growing interest in investigating the stability of these inequalities. The central question we aim to explore is: "If a function nearly achieves equality in a known functional inequality, can we demonstrate, in a quantitative way, its proximity to a minimizer?" In this talk I will overview this beautiful topic and discuss some recent results.
Yan
Alves Radtke
Signotopes: Pseudoconfigurations and Extensions
Ilia
Zharkov
KSU
Approximation of the diagonal and A-infinity structures on polytopes
Abstract.
Given a convex polytope P and a choice of a linear function on it one can define the multiplication on its cellular cochains by a cellular approximation of the diagonal in P^2. This is not associative (except for simplices and cubes) and gives rise to so called higher A-infinity products. I will describe a one-parameter family of permuto-associahedra based on a q-version of the Reiner-Ziegler realization. As q tends to zero, the family tends to the dual permutahedron. Restricted to a single permutation chamber, this inscribes the associahedron in the simplex similar to the Loday realization and allows one to give a geometric explanation of the A-infinity relations among the products. This is a part of a joint project with Gabe Kerr in the context of mirror symmetry. But in the talk we will restrict our attention to the combinatorics of the (fiber) polytopes.
Chris
Huggle
Controlled Algebra
Ilja
Klebanov
Sampling in Unit Time with Kernel Fisher-Rao Flow Paper von Aimee Maurais und Youssef Marzouk
Paul
Mücksch
Leibniz Universität Hannover
Combinatorial models of fibrations for hyperplane arrangements and oriented matroids
Abstract.
The complement of an arrangement of hyperplanes in a complex vector space is a much studied interesting topological space. A fundamental problem is to decide when this space is aspherical, i.e. its universal covering space is contractible. For special classes of arrangements, such as the braid arrangements or more generally supersolvable arrangements, this can be achieved by utilizing fibrations which connect complements of arrangements of different rank. Another prominent space associated to an arrangement is its Milnor fiber -- the typical fiber of the evaluation map of the defining polynomial of the arrangement on its complement which is a smooth fibration by Milnor's famous result. This is a much more subtle topological invariant and it is still an open problem to understand its homology or even its first Betti number in conjunction with the combinatorial structure of the arrangement. I will present a new combinatorial approach to study such fibrations for arrangements which can be defined over the reals via oriented matroids. This is partly joint work with Masahiko Yoshinaga (Osaka University).
Jürgen
Sprekels
WIAS
Necessary and sufficient optimality conditions in the sparse optimal control of singular Allen--Cahn systems with dynamic boundary conditions
Marian
Aprodu
University of Bucharest
Resonance, syzygies, and rank-3 Ulrich bundles on the del Pezzo threefold \(V_5\)
Abstract.
This is a joint work with Yeongrak Kim. We investigate a geometric criterion for a smooth curve of genus 14 and degree 18 to be described as the zero locus of a section in an Ulrich bundle of rank 3 on a del Pezzo threefold \(V_5\). The main challenge is to read off the Pfaffian quadrics defining \(V_5\) from geometric properties of the curve. We find that this problem is related to the existence of a special rank-two vector bundle on the curve, with trivial resonance. From an explicit calculation of the Betti table, we also deduce the uniqueness of the del Pezzo threefold.
Alexander
Suciu
Northeastern University
Ruma
Maity
Uni Innsbruck
Finite element methods for the Landau-de Gennes minimization problem of nematic liquid crystals
Abstract.
Nematic liquid crystals represent a transitional state of matter between liquid and crystalline phases that combine the fluidity of liquids with the ordered structure of crystalline solids. These materials are widely utilized in various practical applications, such as display devices, sensors, thermometers, nanoparticle organizations, proteins, and cell membranes. In this talk, we discuss finite element approximation of the nonlinear elliptic partial differential equations associated with the Landau-de Gennes model for nematic liquid crystals. We establish the existence and local uniqueness of the discrete solutions, a priori error estimates, and a posteriori error estimates that steer the adaptive refinement process. Additionally, we explore Ball and Majumdar's modifications of the Landau-de Gennes Q-tensor model that enforces the physically realistic values of the Q tensor eigenvalues. We discuss some numerical experiments that corroborate the theoretical estimates, and adaptive mesh refinements that capture the defect points in nematic profiles.
Alexandru
Suciu
Northeastern University
The Milnor fibrations of hyperplane arrangements
Abstract.
To each multi-arrangement \((A,m)\), there is an associated Milnor fibration of the complement \(M=M(A)\). Although the Betti numbers of the Milnor fiber \(F=F(A,m)\) can be expressed in terms of the jump loci for rank 1 local systems on \(M\), explicit formulas are still lacking in full generality, even for \(b_1(F)\). After introducing these notions and explaining some of the known results, I will consider the "generic" case, in which \(b_1(F)\) is as small as possible. I will describe ways to extract information on the cohomology jump loci, the lower central series quotients, and the Chen ranks of the fundamental group of the Milnor fiber in this situation.
Vladimir
Spokoiny
WIAS Berlin
Gaussian variational inference in high dimension
Abstract.
We consider the problem of approximating a high-dimensional distribution by a Gaussian one by minimizing the Kullback-Leibler divergence. The main result extends Katsevich and Rigollet (2023) and claims that the minimiser can be well approximated by the Gaussian distribution with the mean and variance as for the underlying measure. We also describe the accuracy of approximation and the range of applicability for such approximation in terms of efficient dimension. The obtained results can be used for analysis of various sampling scheme in optimization.
Emiliano
Ambrosi
Strasbourg
Reduction modulo p of the Noether problem
Abstract.
Let k be an algebraically closed field of characteristic \( p \ge 0 \) and V a faithful k-rational representation of an \( l \)-group G. Noether's problem asks whether V/G is (stably) birational to a point. If \( l = p \), then Kuniyoshi proved that this is true, while for \( l \neq p \) Saltman constructed \( l \)-groups for which V/G is not stably rational. Hence, the geometry of V/G depends heavily on the characteristic of the field. We show that for all the groups G constructed by Saltman, one cannot interpolate between the Noether problem in characteristic 0 and p. More precisely, we show that there does not exist a complete valuation ring R of mixed characteristic (0,p) and a smooth proper R-scheme \( X \to \mathrm{Spec}(R) \) whose special fiber and generic fiber are both stably birational to V/G. The proof combines the integral p-adic Hodge theoretic results of Bhatt-Morrow-Scholze, with the study of the Cartier operator on differential forms in positive characteristic. This is a joint work with Domenico Valloni.
Manfred
Scheucher
Saturation results around the Erdős-Szekeres problem
Sina
Dahms
B & W Deloitte GmbH, Berlin
and
Matthias
Drees
B & W Deloitte GmbH, Berlin
and
Lea
Fernandez
B & W Deloitte GmbH, Berlin
Extreme value theory in the insurance sector
Abstract.
Dusty insurance industry or buzzword bingo? Not with us! We work both in the world of insurance industry and management consulting, which means for us no two days are the same. Our practice supports many of the world’s leading organizations by using modern data analytics and complex mathematical models. Thereby we quantify the risks of the insurance industry, making risks visible and manageable. We work together with our clients to assess their strategic priorities, increase economic value, optimize capital, and drive organizational performance. Sina Dahms, Matthias Drees, and Lea Fernandez from Deloitte will talk about extreme value theory and its applications in insurance and will give exclusive insights into the day-to-day work of an actuarial consultant. Sina has a PhD in financial mathematics from HU Berlin, Matthias holds degrees in mathematics and physics from universities in Munich, Cambridge and Tokyo, and Lea recently finished her studies of mathematics and physics at the TU Berlin.
Kathrin
Flaßkamp
Universität des Saarlandes
Optimal Path Planning in Stereotactic Neurosurgery
Mateusz
Skomra
LAAS
Smoothed analysis of deterministic discounted and mean-payoff games
Abstract.
Deterministic turn-based discounted and mean-payoff games are fundamental classes of games with an unsettled complexity. They belong to the complexity classes NP and coNP, but are not known to be polynomial-time solvable. Furthermore, they are at the bottom of a hierarchy of complexity classes that stratifies the NP search problems. Despite these properties, the problem of solving turn-based games efficiently has been open for 35 years. Nevertheless, even though we do not know how to solve these games in polynomial time in the worst case, practical experiments suggest that solving random games is easy. More precisely, the policy iteration methods, which can take exponentially many steps in the worst case, converge quickly to the solution when the weights of the game are taken at random. The aim of my talk is to give an explanation of this phenomenon using the framework of "smoothed analysis" introduced by Spielman and Teng to explain the real-world efficiency of the simplex method. We prove that if the weights of a turn-based deterministic game are perturbed by a Gaussian noise, then the resulting randomized problem can be solved efficiently by a variant of a policy iteration method. This talk is based on a joint work with Bruno Loff.
Daniel
Schlaugies
Sheaves of Manifolds - Introduction
Moritz
Gau
Dimension reduction of a thermo-visco-elastic problem at small strains
Robert
Gruhlke
QMC meets Optimal sampling
John
Abbott
Determinants of Integer Matrices
Abstract.
Computing the determinant of an integer matrix is a fundamental operation, and also the basis for determinant of other types of matrix e.g. with rational number or polynomial entries. Its utility means it is also a well-studied problem with several solutions which are ''good'' for certain classes of matrix. We present a new technique which is markedly better for ''awkward'' matrices where the existing methods all perform poorly. We consider only dense unstructured matrices.We recall briefly the most practical existing methods, then present the new approach. We combine these ''ingredients'' into an algorithm which is never worse than the best existing methods, and is markedly better for ''awkward'' matrices. An implementation will be included in OSCAR 1.1.No special knowledge is required, but it is helpful to know what Hadamard's determinant bound is.
Adrian
Gonzalez Casanova
UC Berkeley
Nazim
Khelifa
IHES
Density criteria for typical Hodge loci and applications
Abstract.
After recalling the Zilber-Pink paradigm introduced in Hodge theory by Klingler and further developed by Baldi-Klingler-Ullmo, I will present joint work with David Urbanik giving sufficient conditions that ensure that the Hodge locus, i.e. the locus in the base of an integral polarized variation of Hodge structures where the fibers acquire non-generic Hodge tensors, is dense for the complex analytic topology in the base. I will then explain how to relate this result to classical results on Noether-Lefschetz loci. Finally, I will explain how the current knowledge of the Hodge locus can be used to revisit and improve classical bounds on the dimension of the image of period maps, studied among others by Carlson, Griffiths, Kasparian, Mayer and Toledo.
Fabian
Telschow
HU Berlin
Estimation of the expected Euler characteristic of excursion sets of random fields and applications to simultaneous confidence bands
Abstract.
The expected Euler characteristic (EEC) of excursion sets of a smooth Gaussian-related random field over a compact manifold can be used to approximate the distribution of its supremum for high thresholds. Viewed as a function of the excursion threshold, the EEC of a Gaussian-related field is expressed by the Gaussian kinematic formula (GKF) as a finite sum of known functions multiplied by the Lipschitz–Killing curvatures (LKCs) of the generating Gaussian field. In the first part of this talk we present consistent estimators of the LKCs as linear projections of ''pinned" Euler characteristic (EC) curves obtained from realizations of zero-mean, unit variance Gaussian processes. As observed data seldom is Gaussian, we generalize these LKC estimators by an unusual use of the Gaussian multiplier bootstrap to obtain consistent estimates of the LKCs of Gaussian limiting fields of non-stationary statistics. In the second part, we explain applications of LKC estimation and the GKF to simultaneous familywise error rate inference, for example, by constructing simultaneous confidence bands and CoPE sets for spatial functional data over complex domains such as fMRI and climate data and discuss their benefits and drawbacks compared to other methodologies.
Anastasija
Pesic
Variational Models for Pattern Formation in Biomembranes
Melanie
Koser
Energy Driven Pattern Formation in a Model for Two-Dimensional Frustrated Spin Systems
Lukas
Abel
A Scaling Law for a Model of Epitaxial Growth with Dislocations
Asha
Dond
IISER TVM
Quasi-optimality of adaptive FEMs for distributed elliptic optimal control problems
Abstract.
In this talk, we will discuss the quasi-optimality of adaptive nonconform- ing finite element methods for distributed optimal control problems governed by m-harmonic operators for m = 1, 2. A variational discretization approach is employed and the state and adjoint variables are discretized using non- conforming finite elements. The general axiomatic framework that includes stability, reduction, discrete reliability, and quasi-orthogonality establishes the quasi-optimality. Numerical results demonstrate the theoretically pre- dicted orders of convergence.
Tobias
Boege
Polyhedra in information theory
Abstract.
A central object in information theory is the entropy region. Its closure in the euclidean topology is a convex cone and the elements of its dual cone are known as ''linear information inequalities''. They form a large portion of the arsenal of information theorists for solving channel capacity problems. In this talk, I will survey techniques for finding new information inequalities via so-called extension properties and conditional information inequalities. All of these techniques are secretly powered by polyhedra geometry. Hence, they can be implemented, automated and freely combined using the common language of linear programming. My vision is that information inequalities will be stored and thoroughly catalogued as discrete geometric objects.
Hendrik
Weber
U of Münster
Mate
Gerencser
TU Vienna
Markus
Schmidtchen
Institut für Wissenschaftliches Rechnen, TU Dresden
Gradient flow solutions for porous medium equations with nonlocal Lévy-type pressure
Abstract
Benedikt
Gräßle
HU Berlin
Inf-sup bounds for semilinear problems from nonconforming discretisations
Solving the Optimal Experiment Design Problem with mixed-integer convex methods
Abstract
Georg
Keilbar
HU Berlin
and
Ratmir
Miftachov
HU Berlin
Rate independent evolutions: some basics, some progress
Abstract.
We discuss some elementary rate independent evolutions, in particular the stop and the play, and offer remarks on the historical development. We also elaborate on issues concerning related optimal control problems.
Victor
Gonzalez Alonso
Hannover
Infinitesimal rigidity of certain modular morphisms
Abstract.
The Torelli morphism maps (the isomorphism class of) a smooth complex projective curve to its polarized jacobian variety. It has been recently proved by Farb that this is the only non-constant holomorphic map from the moduli space of curves to that of principally polarized abelian varieties, and Serván has recently proved a similar result for the Prym morphism. These result can be interpreted by saying that a certain moduli space of morphisms consists of just one point, and it is natural to ask whether this point is reduced. In this talk I will present a joint work with Giulio Codogni and Sara Torelli, where we show that this is indeed the case (in the setting of moduli stacks): These morphisms do not admit non-trivial infinitesimal deformations. The proof uses the Fujita decomposition of the Hodge bundle of a family of curves, and can be applied to other morphisms involving moduli of smooth curves.
Christoph
von Tycowicz
ZIB
Geometric Deep Learning
Abstract.
The increasing success of deep learning techniques during the last decade express a paradigm shift in machine learning and data science. While learning generic functions in high dimensions is a cursed estimation problem, many challenging tasks such as protein folding or image-based diagnosis, have now been shown to be achievable with appropriate computational resources. These breakthroughs can be attributed to the fact that most tasks of interest aren't actually generic; they possess inherent regularities derived from the effective low-dimensionality and structure of the physical world. In this talk, we will see how geometric concepts allow to expose these regularities and how we can use them to incorporate prior (physical) knowledge into neural architectures.
Julian
Sahasrabudhe
U Cambridge
A new lower bound for sphere packing
Ji Hoon
Chun
TU Berlin
$\vec{w}h\alpha\mathfrak{t}\;\; \forall\mathbb{R}\varepsilon\ldots$sphere packing lower and upper bounds?
Abstract.
In Euclidean space, the densest sphere packings and their densities are only known in dimensions 1, 2 (Thue, Fejes Tóth), 3 (Hales), 8 (Viazovska), and 24 (Cohn et al.). However, several nontrivial lower and upper bounds for the density δ(d) of the densest packing in dimension d have been established. A simple "folklore" result states that δ(d) ≥ 1/2^d. In this talk we present the intuition and details of three other lower and upper bounds for δ(d): the Minkowski–Hlawka theorem for a lower bound, Blichfeldt's upper bound, and Rogers's upper bound. These results, among others, place δ(d) within a narrow strip of possible densities.
Robert
Lauff
Trees in Planar Graphs
Katharina
Oberpriller
Universität München
Reduced-form framework and affine processes with jumps under model uncertainty
Abstract.
We introduce a sublinear conditional operator with respect to a family of possibly non-dominated probability measures in presence of multiple ordered default times. In this way we generalize the results in [3] where a consistent reduced-form framework under model uncertainty for a single default is developed. Moreover, we present a probabilistic construction of Rd-valued non-linear affine processes with jumps, which allows to model intensities in a reduced-form framework. This yields a tractable model for Knightian uncertainty for which the sublinear expectation of a Markovian functional can be calculated via a partial integro-differential equation. This talk is based on [1] and [2].
Manuel
Bodirsky
TU Dresden
The Complexity of Constraint Satisfaction with Semilinear Constraints
Abstract.
The linear program feasibility problem is a well-studied example of a constraint satisfaction problem that can be solved in polynomial time. For some other CSPs over numeric domains, the computational complexity is wide open, such as satisfiability of max-plus systems. In this talk I will give a survey on what is known about the border of polynomial-time tractability and NP-hardness for the large class of CSPs where all the allowed constraints come from some fixed set of semilinear relations, i.e., relations that are definable over the rationals with addition and the order.
Mark
Backhaus
Six Functor Formalisms
April 2024
Alessandro
Ghigi
Pavia
Projective structures on Riemann surfaces and metrics on the moduli space of curves
Abstract.
I will describe some recent results on projective structures on Riemann surfaces. After recalling some basic definitions I will explain a correspondence between varying projective structures over the moduli space of curves and (1,1)-forms over it. I will describe explicitely the correspondence in two examples: the projective structure coming from uniformization and a projective structure coming from Hodge theory. Finally I will also describe a new projective structure obtain from the line bundle \( 2 \Theta\).
Olesia
Zavarzina
V.N. Karazin Kharkiv National University, Ukraine
Around the plasticity problem
Hannes
Tarun
Konstruktion und Geometrische Repräsentationen dreiecksfreier Graphen mit hoher chromatischer Zahl (Wiederholung)
Markus
Herrmann-Wicklmayr
Universität des Saarlandes
Accelerating Multi-Objective Model Predictive Control Using High-Order Sensitivity Information
Hankyung
Ko
Uppsala University
Coxeter combinatorics of double cosets
Abstract.
A Coxeter group is a group W together with a generating set S of reflections satisfying the (Coxeter-)braid relations. This talk concerns parabolic double cosets in a Coxeter group, i.e., double cosets with respect to subgroups generated by subsets of S. I will discuss how to express them and show the double coset braid relations from a joint work with Ben Elias. If there is time, I will also give an "atomic" description obtained more recently, partially jointly with Ben Elias, Nico Libedinsky, Leonardo Patimo.
Holger
Reich
Module Theory
Ruming
Zhang
TU Berlin
Direct and inverse problems in periodic waveguides
Christian
Merdon
WIAS Berlin
Pressure-robustness in Navier-Stokes simulations
Nicolas
Verzelen
INRAE Montpellier
Computational trade-offs in high-dimensional clustering
Johannes O.
Royset
USC, University of Southern California, CA
Approximations of Rockafellians, Lagrangians, and Dual Functions. The case for solving surrogates instead of actual optimization problems
Abstract.
Optimization problems are notorious for being unstable in the sense that small changes in their parameters can cause large changes in solutions. However, Rockafellian relaxations, Lagrangian relaxations, and dual problems are typically more stable. While focusing on the nonconvex case, we develop sufficient conditions under which approximations of Rockafellian relaxations, Lagrangian relaxations, and dual problems convergence, epigraphically or hypographically, to limiting counterparts, and quantify the rate of convergence. The conditions are milder than those required by approximations of the actual problems confirming the importance of these surrogate problems. We illustrate the results in the context of composite problems, stochastic optimization, and Rockafellians constructed by augmentation.
Christoph
Sorger
CNRS Nantes
Holomorphic symplectic geometry
Vasily
Rogov
HU Berlin
$\vec{w}h\alpha\mathfrak{t}\;\; \forall\mathbb{R}\varepsilon\ldots$holomorphic symplectic varieties?
Abstract.
If you ask a specialist in holomorphically symplectic varieties what they are, and why these objects are interesting, you can get very different answers, depending on whether that person comes from algebraic geometry, or from differential geometry. Or maybe they come from complex analysis, theoretical physics, representation theory, or number theory. In all these areas holomorphically symplectic manifolds play their own exceptional role, and in order to study them one needs to combine all these different points of view. In my talk, I will discuss the main properties of holomorphically symplectic manifolds: some of them follow immediately from the definition, and some are deep and difficult theorems. In addition, I will try to explain why it is so important to construct new examples of holomorphically symplectic manifolds, and why this problem is incredibly difficult.
Hannes
Tarun
Konstruktion und Geometrische Repräsentationen dreiecksfreier Graphen mit hoher chromatischer Zahl (Wiederholung)
Todor
Antić
Star-Forest Decompositions of Complete (Geometric) Graphs
Nils
Detering
Universität Düsseldorf
Local Volatility Models for Commodity Forwards
Abstract.
We present a dynamic model for forward curves in commodity markets, which is defined as the solution to a stochastic partial differential equation (SPDE) with state-dependent coefficients, taking values in a Hilbert space H of real valued functions. The model can be seen as an infinite dimensional counterpart of the classical local volatility model frequently used in equity markets. We first investigate a class of point-wise operators on H, which we then use to define the coefficients of the SPDE. Next, we derive growth and Lipchitz conditions for coefficients resulting from this class of operators to establish existence and uniqueness of solutions. We also derive conditions that ensure positivity of the entire forward curve. Finally, we study the existence of an equivalent measure under which related traded, 1-dimensional projections of the forward curve are martingales. Our approach encompasses a wide range of specifications, including a Hilbert-space valued counterpart of a constant elasticity of variance (CEV) model, an exponential model, and a spline specification which can resemble the S shaped local volatility function that well reproduces the volatility smile in equity markets. A particularly pleasant property of our model class is that the one-dimensional projections of the curve can be expressed as one-dimensional stochastic differential equation. This provides a link to models for forwards with a fixed delivery time for which formulas and numerical techniques exist. In a first numerical case study we observe that a spline based, S shaped local volatility function can calibrate the volatility surface in electricity markets. Joint work with Silvia Lavagnini (BI Norwegian Business School)
Alexandre
Pannier
Université Paris Cité
A path-dependent PDE solver based on signature kernels
Abstract.
We develop a provably convergent kernel-based solver for path-dependent PDEs (PPDEs). Our numerical scheme leverages signature kernels, a recently introduced class of kernels on path-space. Specifically, we solve an optimal recovery problem by approximating the solution of a PPDE with an element of minimal norm in the signature reproducing kernel Hilbert space (RKHS) constrained to satisfy the PPDE at a finite collection of collocation paths. In the linear case, we show that the optimisation has a unique closed-form solution expressed in terms of signature kernel evaluations at the collocation paths. We prove consistency of the proposed scheme, guaranteeing convergence to the PPDE solution as the number of collocation points increases. Finally, several numerical examples are presented, in particular in the context of option pricing under rough volatility. Our numerical scheme constitutes a valid alternative to the ubiquitous Monte Carlo methods. Joint work with Cristopher Salvi (Imperial College London).
Janike
Oldekop
TU Berlin
Likelihood Geometry of Reflexive Polytopes
Abstract.
We study the problem of maximum likelihood (ML) estimation for statistical models defined by reflexive polytopes. Our focus is on the ML degree of these models as a way of measuring the algebraic complexity of the corresponding optimization problem. We compute the ML degrees of all 4319 classes of three-dimensional reflexive polytopes and prove formulas for several general families, which include the hypercube and the cross-polytope in any dimension. We find some surprising behavior in terms of the gaps between ML degrees and degrees of the associated toric varieties, and we encounter some models of ML degree one. This is joint work with Carlos Améndola.
Gregory
Pearlstein
Pisa
KSBA stable limits associated to quasi-homogeneous surface singularities
Abstract.
Smooth minimal surfaces of general type with \(K^2=1\), \(p_g=2\), and \(q=0\) constitute a fundamental example in the geography of algebraic surfaces. The moduli space of their canonical models admits a modular compactification \(M\) via the minimal model program. In previous work with Patricio Gallardo and Luca Schaffler we constructed eight new irreducible boundary divisors in \(M\) arising from unimodal singularities. In this talk, we will discuss extension of this work to quasi-homogeneous surface singularities.
Dmitrii
Pavlov
MPI Leipzig
Santaló Geometry of Convex Polytopes
Abstract.
The Santaló point of a convex polytope is the interior point which leads to a polar dual of minimal volume. This minimization problem is relevant in interior point methods for convex optimization, where the logarithm of the dual volume is known as the universal barrier function. When translating the facet hyperplanes, the Santaló point traces out a semi-algebraic set. In my talk I will describe this geometry and dive into connections with statistics, optimization and physics.
Tim
Stiebert
HU Berlin
Guaranteed lower eigenvalue bounds via a conforming FEM
Pierluigi
Colli
Università Pavia, Italy
Curvature effects in pattern formation: analysis and control of a sixth-order Cahn--Hilliard equation
Jobst
Heitzig
Potsdam Institute for Climate Research
Social Choice for AI Ethics and Safety
Abstract.
Many recent advances in the field of AI, including tools such as ChatGPT, rely on the training of foundation models. Foundation models are fine-tuned to avoid unsafe or otherwise problematic behavior, so that, for example, they refuse to comply with requests for help with committing crimes, or with producing racist text. One approach to fine-tuning, called reinforcement learning from human feedback, learns from humans’ expressed preferences over multiple outputs. Another approach is constitutional AI, in which the input from humans is a list of high-level principles. But which humans get to provide the feedback or principles? And how is their potentially diverging input aggregated into consistent data about “collective” preferences or otherwise used to make collective choices about model behavior? In this talk, I argue that the field of social choice is well positioned to address these questions, and discuss ways forward for this agenda, drawing on discussions in a recent workshop on Social Choice for AI Ethics and Safety held in Berkeley, CA, USA in December 2023.
Carsten
Carstensen
HU Berlin
Discussion on duality in the Poisson model problem
Gil
Kur
ETH Zürich
Connections between minimum norm interpolation and local theory of Banach spaces
Sebastian
Pokutta
ZIB
An Introduction to Conditional Gradients
Abstract.
Conditional Gradient methods are an important class of methods to minimize (non-)smooth convex functions over (combinatorial) polytopes. Recently these methods received a lot of attention as they allow for structured optimization and hence learning, incorporating the underlying polyhedral structure into solutions. In this talk I will give a broad overview of these methods, their applications, as well as present some recent results both in traditional optimization and learning as well as in deep learning.
Miriam
Goetze
Recognition Complexity of Subgraphs of 2- and 3-Connected Planar Cubic Graphs
Christa
Cuchiero
U of Vienna
Moritz
Egert
TU Darmstadt
Lucca
Tiemens
A balanced Transposition Grey Code for the Symmetric Group and on SCDs for the Permutahedron of Order 5 and below
March 2024
Hannes
Tarun
Konstruktion und Geometrische Repräsentationen dreiecksfreier Graphen mit hoher chromatischer Zahl
Yassine
El Maazouz
George
Balla
and
Marcel
Wack
and
Lena
Weis
Arxiv Seminar
Marina
Garrote-López
MPI MIS
Identifiability of level-1 species networks from gene tree quartets
Abstract.
Understanding evolutionary relationships, particularly in the context of hybridization and horizontal gene transfer, requires the inference of phylogenetic networks rather than traditional trees. While standard phylogenetic methods can infer gene trees from genetic data, these trees only indirectly reflect the species network topology due to horizontal inheritance and incomplete lineage sorting. Previous research has shown that certain network topologies and numerical parameters can be identified, but gaps remain in understanding the full topology of level-1 phylogenetic networks under the Network Multispecies Coalescent model. In this talk, we will give an overview of the inference of gene trees and address the identifiability problem of the topology of species networks, by investigating the ideals defined by quartet concordance factors for topological semi-directed networks. This is a joint work with Elizabeth S. Allman, Hector Baños and John A. Rhodes
Marina
Garrote
MPI MIS
Nikola
Sadovek
Convex equipartitions inspired by the little cubes operad
Abstract.
Nandakumar & Ramana Rap conjecture in the plane asks whether it is possible to divide a given convex polygon into n convex pieces such that the pieces have equal area and equal perimeter. A decade ago, two groups of authors (Karasev, Hubard & Aronov, and Blagojević & Ziegler) have shown that the regular convex partitions of a Euclidean space into n parts yield a solution to the (higher-dimensional analogue of the) Nandakumar & Ramana Rao conjecture when n is a prime power. This was obtained by parametrising the space of regular equipartitions of a given convex body by the classical configuration space. We repeat the process of regular convex partitions many times, first partitioning the Euclidean space into n_1 parts, then each part into n_2 parts, and so on. After doing this process k times, we obtain an ‘iterated' equipartion of a given convex body into n=n_1...n_k parts. We parametrise such iterated partitions by the (wreath) product of classical configuration spaces, and develop a new test-map scheme for solving the (higher dimensional analogue of) Nandakumar & Ramana Rao conjecture. The new scheme yields a solution to the conjecture if and only if all the n_i's are powers of the same prime number. Outside of this case, the conjecture remains open. This talk is based on the joint work with Pavle Blagojević.
Tom
ter Elst
Department of Mathematics, University of Auckland, New Zealand
The Dirichlet problem for elliptic equations without the maximum principle
Abstract
Sandro
Roch
Exact covering with unit disks
February 2024
Diane
Maclagan
Toric Bertini theorems in arbitrary characteristic
Abstract.
The classical Bertini theorem on irreducibility when intersecting by hyperplanes is a standard part of the algebraic geometry toolkit. This was generalised recently, in characteristic zero, by Fuchs, Mantova, and Zannier to a toric Bertini theorem for subvarieties of an algebraic torus, with hyperplanes replaced by subtori. I will discuss joint work with Gandini, Hering, Mohammadi, Rajchgot, Wheeler, and Yu in which we give a different proof of this theorem that removes the characteristic assumption. The proof surprisingly hinges on better understanding algebraically closed fields containing the field of rational functions in n variables, which involve polyhedral constructions. An application is a tropical Bertini theorem.
Javier
Cembrano
Impartial rank aggregation
Abstract.
We study functions that produce a ranking of n individuals from n such rankings and are impartial in the sense that the position of an individual in the output ranking does not depend on the input ranking submitted by that individual. When n ≥ 4, two properties concerning the quality of the output in relation to the input can be achieved in addition to impartiality: individual full rank, which requires that each individual can appear in any position of the output ranking; and monotonicity, which requires that an individual cannot move down in the output ranking if it moves up in an input ranking. When n ≥ 5, monotonicity can be dropped to strengthen individual full rank to weak unanimity, requiring that a ranking submitted by every individual must be chosen as the output ranking. Mechanisms achieving these results can be implemented in polynomial time. Both results are best possible in terms of their dependence on n. The second result cannot be strengthened further to a notion of unanimity that requires agreement on pairwise comparisons to be preserved. This is joint work with Felix Fischer and Max Klimm.
Alexandra
Wesolek
Reconfiguration of plane trees in convex geometric graphs
Diane
Maclagan
Warwick
Tropical vector bundles
Raluca
Vlad
Leipzig
Wonderful polytopes
Bernd
Sturmfels
Leipzig
Kinematics on \(\mathcal{M}_{0,n}\)
Annette
Werner
Frankfurt
Non-abelian p-adic Hodge theory
Saul
Schleimer
Solving the word problem in the mapping class group in quasi-linear time
Abstract.
The word problem for the mapping class group was first posed, and first solved, by Dehn [1922] in his Breslau lectures. His method was rediscovered, and greatly extended, by Thurston [1970-80's]. Mosher [1995] proved that the mapping class group is automatic and so found a quadratic-time algorithm for the word problem. Hamidi-Tehran [2000] and Dynnikov [2023] gave quadratic-time algorithms using train-tracks. We give the first sub-quadratic-time algorithm. We combine the work of Dynnikov with a generalisation of the half-GCD algorithm to obtain an algorithm running in time O(n log^3(n)). This is joint work with Mark Bell.
Computing multiple solutions of topology optimization problems
Abstract.
Topology optimization finds the optimal material distribution of a fluid or solid in a domain, subject to PDE and volume constraints. The models often result in a PDE, volume and inequality constrained, nonconvex, infinite-dimensional optimization problem that may support many local minima. In practice, heuristics are used to obtain the global minimum, but these can fail even in the simplest of cases. In this talk, we will introduce the deflated barrier method, a second-order algorithm that solves such problems, has local superlinear convergence, and can systematically discover many of these local minima. We will present examples which include finding 42 solutions of the topology optimization of a fluid satisfying the Navier-Stokes equations and more recent work involving the three-dimensional topology optimization of a fluid in Stokes flow.Underpinning the algorithm is the deflation mechanism. Deflation prevents a Newton-like solver from converging to a solution that has already been discovered. Deflation is computationally cheap, it does not affect the conditioning of the discretized systems, it may be coupled with a finite difference, finite volume or finite element discretization, and it is easy to implement.
Yassine
El Maazouz
Ekin
Ergen
Disbalance of Machines in Total Completion Time Scheduling Under Scenarios
Abstract.
We revisit the problem of scheduling unit-weight jobs onto parallel machines under scenarios. In our model, a scenario is defined as a subset of a predefined and fully specified set of jobs. The aim is to find a schedule of the whole set of jobs on parallel machines such that the schedule, obtained for the given scenarios by simply skipping the jobs not in the scenario, optimizes the average completion time over all scenarios (Bosman et al., 2023). In the first half of the talk, we recall total completion time scheduling under scenarios, painting an almost complete picture of its complexity landscape. We conjecture a structural property regarding load differences of machines of optimal schedules. This property implies a polynomial-time algorithm for a constant number of scenarios, settling the complexity of the problem. We discuss proven special cases of the conjecture, employing algorithmic ideas as well as tools from integer programming and polyhedral geometry. The second half of the talk is dedicated to a problem about the finiteness of a purely combinatorial process, which, depending on its solution, either proves another special case of the conjecture, or disproves the conjecture completely. We present examples, computational studies and observations. The first half of the talk is based on joint work with Thomas Bosman, Martijn van Ee, Csanad Imreh, Alberto Marchetti-Spaccamela, Martin Skutella and Leen Stougie. The second half is based on ongoing joint work with Martin Skutella and Maximilian Stahlberg.
Kyle
Huang
FU Berlin
Generating Smooth 3-Polytopes
Abstract.
Smooth polytopes are an important class of lattice polytope in combinatorial algebraic geometry, corresponding to smooth toric varieties. Open questions such as Oda's conjecture, Bøgvad's Conjecture, and Ewald's question guide research, but the lack of meaningful examples hinders our ability to disprove even dramatic strengthenings of such conjectures. We describe and implement a novel algorithm for classifying 3-polytopes which extends past classification results by Haase, Lorenzo, and Paffenholz as well as Lundman. We also present theoretical findings on smooth 3-polytopes--in fact our theoretical and computational findings are intertwined and inform one another.
Pedro
Maristany de las Casas
ZIB
Multiobjective Shortest Path Problems
Abstract.
In this talk we discuss new algorithms for the Multiobjective Shortest Path (MOSP) problem. The baseline algorithm, the Multiobjective Dijkstra Algorithm (MDA) has already been introduced in seminars at ZIB. New aspects discussed in this talk are its output-sensitive running time bound and how the bound compares to the one derived for previously existing MOSP algorithms, a version of the MDA for One-to-One MOSP instances, and the usage of the MDA as a subroutine. The discussed application in which the MDA acts as a subroutine are the Multiobjective Minimum Spanning Tree problem and the K-Shortest Simple Path problem.
Matthias
Schymura
TU Cottbus
Deep lattice points in lattice zonotopes
Abstract.
Given a polytope P and a point w in its interior one may want to measure the centrality (or the depth) of w within P. An established way to do so is via the so-called coefficient of asymmetry. This notion has been studied extensively in the realm of Hensley’s conjecture on the maximal volume of a d-dimensional lattice polytope that contains a fixed positive number of interior lattice points. Motivated by the Lonely Runner Conjecture from Diophantine approximation, we prove the existence of interior lattice points in lattice d-zonotopes, for which the coefficient of asymmetry is bounded above by an explicit function in O(d * log log d). In the general case of arbitrary lattice polytopes such a bound necessarily must be double exponential in the dimension. This is based on joint work with Matthias Beck.
Yannick
Mogge
TUHH
Creating a tree universal graph in positional games
Riccarda
Rossi
University of Brescia, Italy
Balanced Viscosity solutions for multi-rate systems in damage
with perfect plasticity
Rahul
Pandharipande
ETH Zürich
Tautological projections and the cohomology of the moduli space of abelian varieties
Abstract.
I will construct the projection operator on the Chow ring for the moduli of abelian varieties and compute many new examples (related to the geometry of the Lagrangian Grassmannian) elucidating the structure of this ring.
Kateryna
Buryachenko
HUB, Donetsk University, Ukraine
Regularity problems for anisotropic models
Abstract
Andrei
Bud
Goethe Universität Frankfurt
Degenerations of Prym-Brill-Noether loci
Abstract.
I will describe the Prym-Brill-Noether loci for curves in the boundary of the moduli of Prym curves. As consequences of this, I prove the irreducibility of the Universal Prym-Brill-Noether locus and compute the class of the Prym-Brill-Noether divisor.
Martin
Wahl
Universität Bielefeld
Heat kernel PCA with applications to Laplacian eigenmaps
Abstract.
Laplacian eigenmaps and diffusion maps are nonlinear dimensionality reduction methods that use the eigenvalues and eigenvectors of (un)normalized graph Laplacians. Both methods are applied when the data is sampled from a low-dimensional manifold, embedded in a high-dimensional Euclidean space. From a mathematical perspective, the main problem is to understand these empirical Laplacians as spectral approximations of the underlying Laplace-Beltrami operator. In this talk, we study Laplacian eigenmaps through the lens of kernel PCA, and consider the heat kernel as reproducing kernel feature map. This leads to novel points of view and allows to leverage results for empirical covariance operators in infinite dimensions.
Walter
Gubler
Regensburg
Abstract divisorial spaces and an extension of adelic
intersection numbers
Abstract.
This is joint work in progress with Yulin Cai. Yuan and Zhang defined an adelic intersection theory over number fields and Yuan used this to give a striking new approach to the uniform Mordell-Lang approach. Recently, Burgos and Kramer extended the arithmetic intersection pairing allowing more singular metrics on the archimedean side. We complete the picture on the non-archimedean side. Using the framework of so called abstract divisorial spaces, we show that Yuan-Zhang's construction is a completion process which works in various situations. In particular, we can extend arithmetic intersection numbers allowing more singular metrics working over any reasonable base field with product formula. In particular, we can do that for proper adelic base curves in the framework of Chen and Moriwaki.
John
Ball
Heriot Watt University, Edinburgh
Image comparison and scaling via nonlinear elasticity
Oleh
Omel'chenko
Uni Potsdam
Activity patterns in ring networks of theta neurons
A stochastic model for the growth of a filamentous fungus
Marta
dai Pra
HU Berlin
What is a (multi-type) branching process?
Abstract.
Branching processes are an important class of stochastic processes that models the growth of a population. They are widely used in biology and epidemiology to study the spread of infectious diseases and epidemics, and consist of a collection of independent random variables determining the number of children an individual will have. The subject has been actively developing since the pioneering works of Bienaymé, Galton and Watson. The purpose of the talk is to introduce some basic ideas about these processes. We begin by defining the simple Galton--Watson process and its properties. Of particular interest in this field is the study of the extinction probability; in fact, these processes either explode or become extinct with probability 1. We also state some simple limit theorems. The second part of the talk focuses on multi-type branching processes, generalizing the previous model by allowing individuals to have different 'types' with different probabilistic behaviors. We can think of types as the different genetic traits of a population. We carefully define this new setting and describe the new version of the main properties and limit theorems.
Michael
Krivelevich
Tel Aviv University
Improving graph's parameters through random perturbation
Vladimir
Shikhman
TU Chemnitz
Eva
Philippe
Sorbonne Université
Geometric realizations via triangulations of flow polytopes
Abstract.
Building upon the example of the s-permutahedron, a combinatorial complex introduced by Ceballos and Pons in 2019 and conjectured to be polytopal, I will present a pathway to obtain geometric realizations of combinatorial complexes thanks to nice triangulations of flow polytopes, Cayley trick and tropical dualization. This is based on joint work with Rafael S. González D’León, Alejandro H. Morales, Daniel Tamayo Jiménez, Yannic Vargas, Martha Yip.
Martin
Eigel
WIAS
Alexander
Kasprzyk
University of Nottingham
Machine learning detects terminal singularities
Abstract.
I shall explain how we recently used machine learning to accurately determine when certain Q-factorial Fano toric varieties have (at worst) terminal singularities. Inspired by the success of the machine, we were then able to prove an elegant new combinatorial characterisation, although this result is certainly not what the machine had learnt. This is joint work with Tom Coates and Sara Veneziale (and a machine)
Etienne
Jäckle
ZIB
/
TU Berlin
and
Yves
Formal Theorem Provers and Formal Proofs from THE BOOK
Abstract.
This talk introduces and illustrates the ITP Lean, that allows the user to write mathematical statements and their proofs in a way that can be mechanically checked for correctness by a computer. Lean has gained increased attention in the past few years, due to hosting formalization projects of two Fields medalists, and due to the rise of automated theorem proving via AI models as a field of research. In this talk, we will exemplify how Lean is used by discussing our Master thesis project and the experiences we gained from it. We will also survey some large and completed formalization projects and give an insight into existing AI models surrounding Lean.
Yves
Etienne Jäckle
ZIB
/
TU Berlin
Formal Theorem Provers and Formal Proofs from THE BOOK
Abstract.
This talk introduces and illustrates the ITP Lean, that allows the user to write mathematical statements and their proofs in a way that can be mechanically checked for correctness by a computer. Lean has gained increased attention in the past few years, due to hosting formalization projects of two Fields medalists, and due to the rise of automated theorem proving via AI models as a field of research. In this talk, we will exemplify how Lean is used by discussing our Master thesis project and the experiences we gained from it. We will also survey some large and completed formalization projects and give an insight into existing AI models surrounding Lean.
Sophie
Huiberts
Clermont Auvergne University
Open problems about the simplex method
Abstract.
The simplex method is a very efficient algorithm. In this talk we see a few of the state-of-the-art theories for explaining this observation. We will discuss what it takes for a mathematical model to explain an algorithm’s qualities, and whether existing theories meet this bar. Following this, we will question what the simplex method is and if the theoretician's simplex method is the same algorithm as the practitioner's simplex method.
Antonio
Tribuzio
Institute for Applied Mathematics, University of Bonn
Scaling laws for multi-well nucleation problems
Abstract
Evgenii
Chzhen
LMO Orsay, Paris
Anne
Spiering
HU Berlin
Elliptic Feynman integrals from a symbol bootstrap
Abstract.
A Feynman integral is a multi-dimensional integral that encodes the probability amplitude for particle interactions within the framework of quantum field theory. While Feynman integrals play a crucial role in connecting theoretical models with experimental data, their evaluation can pose significant challenges. The “symbol bootstrap” has proven to be a powerful tool for calculating specific (polylogarithmic) Feynman integrals that bypasses a direct integration. I will discuss a generalisation of this method to the elliptic case, mainly focusing on the so-called double-box integral where elliptic structures appear in the integration.
Lukas
Abel
HU
A scaling law for a model of epitaxially strained elastic films with dislocations
Hélène
Esnault
Universität Kopenhagen
Lokale Systeme in der Algebraischen und Arithmetischen Geometrie
Abstract.
Von Galois, zu Riemann, zu Poincaré, zu… Grothendieck, zu… Simpson, zu… Langlands… Die Fundamentalgruppen (die Galoisgruppen) sind zwar sehr klar definiert, über deren Eigenschaften weiß man aber extrem wenig. Deswegen studiert man deren (lineare) Darstellungen, modulo Isomorphismen, als erste Approximation. Es sind die lokalen Systeme. Wo findet man sie, wo kommen sie her? Kann man sie alle parametrisieren, gibt es spezielle Eigenschaften, wenn sie sich zum Beispiel nicht deformieren lassen? Ich werde versuchen, historisch und anschaulich einige Punkte vorzubringen.
Peter
Martin
FU Berlin
Lower bounds on Ramsey multiplicity of cliques.
Moritz
Link
Universität Konstanz
Multiobjective mixed-integer nonlinear optimization with application to energy supply networks
Abstract.
In light of the ongoing developments in the climate crisis, it is necessary to consider factors beyond the sole economic perspective in energy supply network planning. This gives rise to a classical multiobjective optimization problem involving conflicting objective functions. The presence of choices in the model, coupled with the consideration of stationary flow equations, results in an optimization problem with a mixed-integer nonlinear structure. Following a short introduction of some model-specific details and arising challenges, we present a novel method for computing an enclosure of the nondominated set of such problems. We prove finite convergence and demonstrate its effectiveness through numerical experiments. We conclude by offering a preview of ongoing extensions of this work.
Sophie
Huiberts
LIMOS
Open problems about the simplex method
Abstract.
The simplex method is a very efficient algorithm. In this talk we see a few of the state-of-the-art theories for explaining this observation. We will discuss what it takes for a mathematical model to explain an algorithm’s qualities, and whether existing theories meet this bar. Following this, we will question what the simplex method is and if the theoretician's simplex method is the same algorithm as the practioner's simplex method.
Daniel
Walter
HU Berlin
Towards optimal sensor placement for inverse problems in spaces of measures
Caroline
Geiersbach
WIAS
PDE-Constrained Optimization Problems with Probabilistic State Constraints
January 2024
Ben
Hollering
TU Munich
Computing Implicitizations of Multi-Graded Polynomial Maps
Abstract.
In this talk, we'll introduce a new method for computing the kernel of a polynomial map which is homogeneous with respect to a multigrading. We first demonstrate how to quickly compute a matrix of maximal rank for which the map has a positive multigrading. Then in each graded component we compute the minimal generators of the kernel in that multidegree with linear algebra. We have implemented our techniques in Macaulay2 and show that our implementation can compute many generators of low degree in examples where Gröbner basis techniques have failed. This includes several examples coming from phylogenetics where even a complete list of quadrics and cubics were unknown. When the multigrading refines total degree, our algorithm is embarassingly parallel. This is joint work with Joseph Cummings.
Moritz
Link
University of Konstanz
Multiobjective mixed-integer nonlinear optimization with application to energy supply networks
Abstract.
In light of the ongoing developments in the climate crisis, it is necessary to consider factors beyond the sole economic perspective in energy supply network planning. This gives rise to a classical multiobjective optimization problem involving conflicting objective functions. The presence of choices in the model, coupled with the consideration of stationary flow equations, results in an opti mization problem with a mixed-integer nonlinear structure. Following a short introduction of some model-specific details and arising challenges, we present a novel method for computing an enclosure of the nondominated set of such prob lems. We prove finite convergence and demonstrate its effectiveness through numerical experiments. We conclude by offering a preview of ongoing extensions of this work.
Gianluca
Finocchio
Universität Wien
An extended latent factor framework for ill-posed linear regression
Abstract.
The classical latent factor model for linear regression is extended by assuming that, up to an unknown orthogonal transformation, the features consist of subsets that are relevant and irrelevant for the response. Furthermore, a joint low-dimensionality is imposed only on the relevant features vector and the response variable. This framework allows for a comprehensive study of the partial-least-squares (PLS) algorithm under random design. In particular, a novel perturbation bound for PLS solutions is proven and the high-probability L²-estimation rate for the PLS estimator is obtained. This novel framework also sheds light on the performance of other regularisation methods for ill-posed linear regression that exploit sparsity or unsupervised projection. The theoretical findings are confirmed by numerical studies on both real and simulated data.
Annette
Werner
Frankfurt
The Hodge-Tate sequence for commutative rigid analytic groups
Abstract.
We consider generalizations of Scholze's Hodge-Tate sequence on smooth, proper rigid analytic varieties. These generalizations feature coefficients in commutative rigid groups, which are locally p-divisible. We will also discuss applications to p-adic versions of Simpson's correspondence with coefficients in commutative rigid groups. This is joint work with Ben Heuer and Mingjia Zhang.
Christoph
Hertrich
Johann-Wolfgang-Goethe-Universität Frankfurt
(Old and New) Facets of Neural Network Complexity
Abstract.
How to use discrete mathematics and theoretical computer science to understand neural networks? Guided by this question, I will focus on neural networks with rectified linear unit (ReLU) activations, a standard model and important building block in modern machine learning pipelines. The functions represented by such networks are continuous and piecewise linear. But how does the set of representable functions depend on the architecture? And how difficult is it to train such networks to optimality? In my talk I will answer fundamental questions like these using methods from polyhedral geometry, combinatorial optimization, and complexity theory. This stream of research was started during my doctorate within 'Facets of Complexity' and carried much further since then.
Matthias
Grützner
HU
Using complex analysis to study microswimmers in two dimensional Stokes flow
Rupert
Klein
FU Berlin
Thoughts on Machine Learning
Abstract.
Techniques of machine learning (ML) and what is called “artificial intelligence” (AI) today find a rapidly increasing range of applications touching upon social, economic, and technological aspects of everyday life. They are also being used increasingly and with great enthusiasm to fill in gaps in our scientific knowledge by data-based modelling approaches. I have followed these developments over the past almost 20 years with interest and concern, and with mounting disappointment. This leaves me sufficiently worried to raise here a couple of pointed remarks.
The K-Theory of Z/n
Vittorio
di Fraia
FU Berlin
What is Algebraic K-theory?
Abstract.
Algebraic K-theory originated in the 1950s from Grothendieck's studies on algebraic varieties. Since then, it has proven to be a powerful tool in various fields such as algebraic geometry, algebraic topology, and number theory. We will focus on defining the first K-group, K₀, with some examples, and extend the discussion to higher K-groups.
Malin
Heinacher
Trees and co-trees in planar graphs
Alexandra
Wesolek
TU Berlin
Cops and Robber Game on Surfaces
Andreas
Tillmann
TU Braunschweig
MIP and ML approaches to matrix sparsification
Abstract.
Sparsity of matrices can be exploited, in particular, to speed up algorithms in various applications, e.g., in linear system solvers or second-order optimization schemes. This motivates to consider the matrix sparsification problem, in which we seek an equivalent (column-space preserving) representation of a given matrix with as few nonzero entries as possible. We derive sequential solution approaches based on bilinear and linear mixed-integer programs, respectively, and point out connections to certain machine learning tasks. One particular problem appears as a subproblem or special case in both these approaches: Finding a sparsest nonzero vector in the nullspace of a matrix. We will outline how a dedicated branch-and-cut method for this problem can be utilized in the envisioned matrix sparsification algorithms, and touch upon some open questions and challenges of our ongoing work
Nils
Mosis
TU Darmstadt
A Unified Worst Case for Classical Simplex and Policy Iteration Pivot Rules
Abstract.
We construct a family of Markov decision processes for which the policy iteration algorithm needs an exponential number of improving switches with Dantzig’s rule, with Bland’s rule, and with the Largest Increase pivot rule. This immediately translates to a family of linear programs for which the simplex algorithm needs an exponential number of pivot steps with the same three pivot rules. Our results yield a unified construction that simultaneously reproduces well-known lower bounds for these classical pivot rules, and we are able to infer that any (deterministic or randomized) combination of them cannot avoid an exponential worst-case behavior. Regarding the policy iteration algorithm, pivot rules typically switch multiple edges simultaneously and our lower bound for Dantzig’s rule and the Largest Increase rule, which perform only single switches, seem novel. Regarding the simplex algorithm, the individual lower bounds were previously obtained separately via deformed hypercube constructions. In contrast to previous bounds for the simplex algorithm via Markov decision processes, our rigorous analysis is reasonably concise.
Victoria
Schleis
Universität Tübingen
Tropical Quiver Grassmannians
Abstract.
Grassmannians and flag varieties are important moduli spaces in algebraic geometry. Quiver Grassmannians are generalizations of these spaces arising in representation theory as the moduli spaces of quiver subrepresentations. These represent arrangements of vector subspaces satisfying linear relations provided by a directed graph.The methods of tropical geometry allow us to study these algebraic objects combinatorially and computationally. We introduce matroidal and tropical analoga of quivers and their Grassmannians obtained in joint work with Alessio Borzì and separate joint work with Giulia Iezzi; and describe them as affine morphisms of valuated matroids and linear maps of tropical linear spaces.
Gian Pietro
Pirola
Università di Pavia
Second fundamental form on \(\mathcal{M}_g\) associated to the period map and its asymptotic lines
Abstract.
The aim is to study the second fundamental form associated with the image period map for curves. We present some computational improvements that allow classifying the asymptotic low-rank complex line with respect to the infinitesimal variation of the Hodge structure map and its relation to the Clifford index. This is a joint work with Elisabetta Colombo and Paola Frediani.
Simon
Wood
University of Edinburgh
On neighbourhood cross validation
Abstract.
Cross validation comes in many varieties, but some of the more interesting flavours require multiple model fits with consequently high cost. This talk shows how the high cost can be side-stepped for a wide range of models estimated using a quadratically penalized smooth loss, with rather low approximation error. Once the computational cost has the same leading order as a single model fit, it becomes feasible to efficiently optimize the chosen cross-validation criterion with respect to multiple smoothing/precision parameters. Interesting applications include cross-validating smooth additive quantile regression models, and the use of leave-out-neighbourhood cross validation for dealing with nuisance short range autocorrelation. The link between cross validation and the jackknife can be exploited to obtain reasonably well calibrated uncertainty quantification in these cases
Norbert
Zeh
Dalhousie University
An FPT Algorithm for Scanwidth
Abstract.
Recently, Berry, Scornavacca, and Weller proposed scanwidth as a measure of how close a phylogenetic network is to being a phylogenetic tree. Much like the definition of the treewidth of a graph via the width of a tree decomposition of the graph, the scanwidth of a network is defined as the width of a rooted tree that reflects the structure of the network, called a tree extension of the network. In the same way that a low-width tree decomposition enables fast solutions to a wide range of NP-hard problems via dynamic programming over the tree decomposition, a low-width tree extension is expected to enable efficient dynamic programming solutions to various problems in phylogenetics (some examples of such problems exists already). This raises the question of how difficult it is to compute an optimal tree extension of a network. Berry, Scornavacca, and Weller proved that this problem is NP-hard. Holtgrefe proposed fast exponential-time algorithms, an XP-algorithm, and fast heuristics that appear to produce very good tree extensions in practice. In this talk, I will show how to reuse much of the machinery for computing an optimal tree decomposition of a graph to also compute an optimal tree extension of a phylogenetic network. The algorithm has running time O(f(k) * poly(n)), where k is the width of the computed tree extension (and f(.) is neither a completely crazy nor a particularly attractive function). Thus, our result proves that computing an optimal tree extension of a graph is fixed-parameter tractable when parameterized by the scanwidth of the network. I will also mention recent results on approximating the scanwidth of a network, as well as open problems. Joint work with: Niels Holtgrefe, Leo van Iersel, and Mark Jones
Fabien
Pazuki
Copenhagen
Explicit bounds on the coefficients of modular polynomials and
the size of \(X_0(N)\)
Abstract.
We give explicit upper and lower bounds on the size of the coefficients of the modular polynomials for the elliptic \(j\)-function. These bounds make explicit the best previously known asymptotic bounds. The proof relies on a careful study of the Mahler measure of a family of specializations of the modular polynomial. We also give an asymptotic comparison between the Faltings height of the modular curve \(X_0(N)\) and the height of this modular polynomial, giving a link between these two ways of measuring the "size" of the modular curve. The talk is based on joint work with Florian Breuer and Desirée Gijon Gomez.
Hugo
Parlier
Universite du Luxembourg
Crossing the line: from graphs to curves
Abstract.
The crossing lemma for simple graphs gives a lower bound on the necessary number of crossings of any planar drawing of a graph in terms of its number of edges and vertices. Viewed through the lens of topology, this leads to other questions about arcs and curves on surfaces. Here is one: how many crossings do a collection of m homotopically distinct curves on a surface of genus g induce? The talk will be about joint work with Alfredo Hubard where we explore some of these, using tools from the hyperbolic geometry of surfaces in the process.
Tim
Oosterwijk
VU Amsterdam
The Secretary Problem with Independent Sampling
Abstract.
The secretary problem is probably the most well-studied optimal stopping problem with many applications in economics and management. In the secretary problem, a decision-maker faces an unknown sequence of values, revealed one after the other, and has to make irrevocable take-it-or-leave-it decisions. Her goal is to select the maximum value in the sequence. While in the classic secretary problem, the values of upcoming elements are entirely unknown, in many realistic situations, the decision-maker still has access to some information, for example, in the form of past data. In this talk, I will take a sampling approach to the problem and assume that before starting the sequence, each element is independently sampled with probability p. This leads to what we call the random order and adversarial order secretary problems with p-sampling. In the former, the sequence is presented in random order, while in the latter, the order is adversarial. Our main result is to obtain the best possible algorithms for both problems and all values of p. As p grows to 1, the obtained guarantees converge to the optimal guarantees in the full information case, that is, when the values are i.i.d. random variables from a known distribution. Notably, we establish that the best possible algorithm in the adversarial order setting is a simple fixed threshold algorithm. In the random order setting, we characterize the best possible algorithm by a sequence of thresholds, dictating at which point in time we should start accepting a value. Surprisingly, this sequence is independent of p. I will then complement our theoretical results with practical insights obtained from numerical experiments on real life data obtained from Goldstein et al. (2020), who conducted a large-scale behavioral experiment in which people repeatedly played the secretary problem. The results help explain some behavioral issues they raised and indicate that people play in line with a strategy similar to our optimal algorithms from the first game onwards, albeit slightly suboptimally. This is joint work with José Correa, Andrés Cristi, Laurent Feuilloley, and Alexandros Tsigonias-Dimitriadis.
Jürg
Kramer
HU
Rationale Lösungen polynomialer Gleichungen
Abstract.
Ist f(X) ein Polynom in der Variablen X mit komplexen Koeffizienten, so besagt der Fundamentalsatz der Algebra, dass f mindestens eine komplexe Nullstelle besitzt, sobald der Grad von f positiv ist. In der Zahlentheorie interessiert die Frage, wie es um die Existenz rationaler Nullstellen von f steht, wenn f rationale Koeffizienten hat. Die Antwort darauf wird im Wesentlichen durch das Lemma von Gauß gegeben. In unserem Vortrag gehen wir nun der analogen Frage nach, wenn f = f(X,Y) ein Polynom mit rationalen Koeffizienten in zwei Variablen ist. Wir werden dabei finden, dass – generisch gesehen – solche Polynome (unendlich) viele rationale Nullstellen haben, wenn deren Grad „klein“ ist, wogegen sie höchstens endlich viele rationale Nullstellen besitzen, falls deren Grad „groß“ ist. Um diese qualitativen Sachverhalte quantitativ genauer zu verstehen, werden wir uns vom Irrationalitätsbeweis von √2 mit Hilfe von Fermats unendlichem Abstieg inspirieren lassen. Diese Überlegungen werden uns auch zu interessanten zahlentheoretischen Fragestellungen führen, wie beispielsweise dem Kongruenzzahlproblem, die Gegenstand aktueller mathematischer Forschung sind.
Bennet
Gebken
Universität Paderborn
First- and second-order descent methods for nonsmooth optimization based on deterministic gradient sampling
Abstract.
In nonsmooth optimization, it is well known that the classical gradient is not a suitable object for describing the local behavior of a function. Instead, generalized derivatives from nonsmooth analysis, like the Clarke subdifferential, have to be employed. While in theory, the Clarke subdifferential inherits many useful properties from the classical gradient, there is a large discrepancy in practice: It is unstable, and for a general locally Lipschitz continuous function, it is impossible to compute. Thus, in practice, the Clarke subdifferential has to be approximated. A simple strategy to achieve this, known as gradient sampling, is based on approximating it by taking the convex hull of classical gradients evaluated at smooth points from a small neighborhood of a given point. In this talk, I will present two descent methods for nonsmooth optimization that are based on this idea. However, in contrast to the standard gradient sampling approach, where the gradients are sampled randomly, both methods will be deterministic. I will begin with a first-order method, where new gradients are computed using a bisection subroutine. Afterwards, I will demonstrate how the gradient sampling methodology can be generalized to second-order derivates by sampling Hessian matrices in addition to gradients. This will lead to a second-order descent method that, at least in numerical experiments, shows a high speed of convergence.
Matthias
Schymura
BTU Cottbus-Senftenberg
On the size of integer programs with bounded subdeterminants
Abstract.
We study a combinatorial question related to the recent and ongoing interest in understanding the complexity of integer linear programming with bounded subdeterminants: Given a number Delta and a full-rank integer matrix A with n rows such that the absolute value of every n-by-n minor of A is bounded by Delta, at most how many pairwise distinct columns can A have? The case Delta = 1 is the classical result of Heller (1957) saying that the maximal number of pairwise distinct columns of a totally unimodular integer matrix with n rows equals n^2 + n + 1. We investigate the problem in two settings: First, in the general case we obtain an upper bound of order O(Delta * n^4), that is, a linear bound in the parameter Delta. Secondly, under the additional assumption that no n-by-n minor of A vanishes, we prove an optimal linear bound for n=2 rows, and a sublinear bound for any n >= 3 rows. In the talk I will focus on the second setting and describe how we use tools from the theory of finite fields and from the geometry of numbers to get our results. This is based on joint work with Gennadiy Averkov, Björn Kriepke and Gohar Kyureghyan.
Nikola
Sadovek
FU Berlin
Convex equipartitions inspired by the little cubes operad
Abstract.
Nandakumar & Ramana Rap conjecture in the plane asks whether it is possible to divide a given convex polygon into n convex pieces such that the pieces have equal area and equal perimeter. A decade ago, two groups of authors (Karasev, Hubard & Aronov, and Blagojević & Ziegler) have shown that the regular convex partitions of a Euclidean space into n parts yield a solution to the (higher-dimensional analogue of the) Nandakumar & Ramana Rao conjecture when n is a prime power. This was obtained by parametrising the space of regular equipartitions of a given convex body by the classical configuration space. We repeat the process of regular convex partitions many times, first partitioning the Euclidean space into n_1 parts, then each part into n_2 parts, and so on. After doing this process k times, we obtain an ‘iterated' equipartion of a given convex body into n=n_1...n_k parts. We parametrise such iterated partitions by the (wreath) product of classical configuration spaces, and develop a new test-map scheme for solving the (higher dimensional analogue of) Nandakumar & Ramana Rao conjecture. The new scheme yields a solution to the conjecture if and only if all the n_i's are powers of the same prime number. Outside of this case, the conjecture remains open.This talk is based on the joint work with Pavle Blagojević.
Michael
Krivelevich
Tel Aviv University
Percolation through isoperimetry
Jona
Rörig
HU Berlin
Gromov's Compactness Theorem for Alexandrov Spaces with Lower Curvature Bounds
RDM – More than a Chore!
Matteo
Giordano
Università degli Studi di Torino
Likelihood methods for low frequency diffusion data
Abstract.
The talk will consider the problem of nonparametric inference in multi-dimensional diffusion models from low-frequency data. Implementation of likelihood-based procedures in such settings is a notoriously delicate task, due to the computational intractability of the likelihood. For the nonlinear inverse problem of inferring the diffusivity in a stochastic differential equation, we propose to exploit the underlying PDE characterisation of the transition densities, which allows the numerical evaluation of the likelihood via standard numerical methods for elliptic eigenvalue problems. A simple Metropolis-Hastings-type MCMC algorithm for Bayesian inference on the diffusivity is then constructed, based on Gaussian process priors. Furthermore, the PDE approach also yields a convenient characterisation of the gradient of the likelihood via perturbation techniques for parabolic PDEs, allowing the construction of gradient-based inference methods including MLE and Langevin-type MCMC. The performance of the algorithms is illustrated via the results of numerical experiments. Joint work with Sven Wang.
Pietro
Longhi
Uppsala University
Open topological strings and symplectic cuts
Abstract.
The study of A-branes as boundary conditions for open topological strings has extensive ramifications across physics and mathematics. Yet, from a mathematical perspective a generally valid definition of open Gromov-Witten invariants is still lacking, while on the physics side computations rely heavily on the use of large N dualities and mirror symmetry. In this talk I will present a novel approach to the computation of genus-zero open topological string amplitudes on toric branes based on a worldsheet description. We consider an equivariant gauged linear sigma model whose target is a certain modification of the Calabi-Yau threefold, known as symplectic cut and determined by the toric brane data. This leads to equivariant generating functions of open and closed genus-zero string amplitudes that extend smoothly across the entire moduli space, and which provide a unifying description of standard Gromov-Witten potentials.
Tropical geometry
Kamillo
Ferry
TU Berlin
What is a tropical plane curve?
Abstract.
"Tropical geometry is a combinatorial shadow of algebraic geometry." This is a very popular slogan among tropical geometers. We will take our time to study a very simple tropical plane curve, have a first look at the connection to algebraic geometry and how polyhedral geometry shows up.
Raphael
Steiner
Chromatic number is not tournament-local
Raphael
Steiner
ETH Zürich
Hadwiger's conjecture and topological bounds.
Ansgar
Freyer
TU Wien
Unimodular valuations beyond Ehrhart
Abstract.
The Betke-Kneser theorem characterizes the Ehrhart coefficients as the unique scalar-valued valuations on the set of lattice polytopes that are invariant under affine unimodular transformations. For tensor-valued valuations, a classification result in a similar spirit was pursued by Ludwig and Silverstein. If the tensor degree is at most 8, they showed that any unimodular equivariant and translation covariant tensor valuation is a linear combination of the so-called Ehrhart tensor coefficients. Moreover, they gave an example of a valuation from lattice polygons into tensors of degree 9 that does not arise as a linear combination of Ehrhart tensor coefficients. In the talk we complete the classification of tensor valued valuations in the planar case. To this end, we draw a connection between tensor valuations on lattice polygons and invariants with respect to a specific finite group. This connection will explain the particular significance of the number 8 in the work of Ludwig and Silverstein. Our approach also yields a classification of power series-valued valuations on lattice polygons, as well as a partial classification of tensor-valuations on 3-dimensional lattice polytopes. This is joint work with Monika Ludwig and Martin Rubey.
Andreas
Kretschmer
Magdeburg
Characteristic Numbers for Cubic Hypersurfaces
Abstract.
Given an \(N\)-dimensional family \(F\) of subvarieties of some projective space, the number of members of \(F\) tangent to \(N\) general linear spaces is called a characteristic number for \(F\). More general contact problems can often be reduced to the computation of these characteristic numbers. However, determining the characteristic numbers themselves can be very difficult, and very little is known even for the family of smooth degree \(d\) hypersurfaces of projective \(n\)-space as soon as both \(n\) and \(d\) are greater than 2. In this talk I will survey the construction of a so-called 1-complete variety of cubic hypersurfaces, generalizing Aluffi's variety of complete plane cubic curves to higher dimensions. This allows us to explicitly compute, for instance, the numbers of smooth cubic surfaces, threefolds and fourfolds tangent to 19, 34 and 55 lines, respectively.
Andreas
Kreschmer
Magdeburg
Characteristic Numbers for Cubic Hypersurfaces
Abstract.
Given an \(N\)-dimensional family \(F\) of subvarieties of some projective space, the number of members of \(F\) tangent to \(N\) general linear spaces is called a characteristic number for \(F\). More general contact problems can often be reduced to the computation of these characteristic numbers. However, determining the characteristic numbers themselves can be very difficult, and very little is known even for the family of smooth degree \(d\) hypersurfaces of projective \(n\)-space as soon as both \(n\) and \(d\) are greater than 2. In this talk I will survey the construction of a so-called 1-complete variety of cubic hypersurfaces, generalizing Aluffi's variety of complete plane cubic curves to higher dimensions. This allows us to explicitly compute, for instance, the numbers of smooth cubic surfaces, threefolds and fourfolds tangent to 19, 34 and 55 lines, respectively.
Vicky
Holfeld
ITWM
Amos
Onn
Single Cell Lineage Reconstruction using Short Tandem Repeats
Abstract.
Inferring the lineage relationships of single cells is a useful tool for an answer to a wide variety of fundamental questions. Current sequencing-based approaches rely on either genetic editing - not applicable for living humans - or on extensive coverage, which is non-scalable and might be affected by functional bias. Here we will present a scalable, low-cost method, which is based on targeted sequencing of Short Tandem Repeats (STRs), genomic regions known to be highly mutable. The unique mutation pattern of STRs and their high mutation rates present us with challenges specific to our method. In addition, since these regions are difficult to analyze and have no known biological function, they have not been studied very extensively. From read alignment, through mutations occurring in-vitro, to separate and compare alleles - we have developed various custom solutions, including ones using signal processing approaches, in order to analyze STRs, and were able to successfully reconstruct lineage trees, demonstrated by testing on data with a known reference. In this lecture I will describe the challenges of the analysis, our solutions to them, and the current state of the method, including open questions and directions for further research. I also want to go in depth into a specific application of the method in getting insight into the development of cancer metastasis.
Thomas
Walpuski
HU Berlin
The Gopakumar–Vafa finiteness conjecture
Abstract.
The purpose of this talk is to illustrate an application of the powerful machinery of geometric measure theory to a conjecture in Gromov–Witten theory arising from physics. Very roughly speaking, the Gromov–Witten invariants of a symplectic manifold \((X,\omega)\) equipped with a tamed almost complex structure \(J\) are obtained by counting pseudo-holomorphic maps from mildly singular Riemann surfaces into \((X,J)\). It turns out that Gromov–Witten invariants are quite complicated (or “have a rich internal structure”). This is true especially for if \((X,\omega)\) is a symplectic Calabi–Yau 3–fold (that is: \(\mathrm{dim}X=6\), \(c_1(X,\omega) = 0\)). In 1998, using arguments from M–theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi–Yau 3–folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov–Witten invariants by a transformation of the generating series. The Gopakumar–Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition. The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that Gromov’s compactness theorem can (and should!) be replaced with the work of Federer–Flemming, Allard, and De Lellis–Spadaro–Spolaor. This upgrade of Ionel and Parker’s cluster formalism proves both the integrality and finiteness conjecture. This talk is based on joint work with Eleny Ionel and Aleksander Doan.
Nonlinear Electrokinetics in Anisotropic Microfluids – Analysis, Simulation, and Optimal Control
Abstract
Thorsten
Koch
ZIB
On the state of QUBO solving
Abstract.
It is regularly claimed that quantum computers will bring breakthrough progress in solving challenging combinatorial optimization problems relevant in practice. In particular, Quadratic Unconstrained Binary Optimization (QUBO) problems are said to be the model of choice for use in (adiabatic) quantum systems during the NISQ era. Even the first commercial quantum-based systems are advertised to solve such problems and QUBOs are certainly an interesting way of modeling combinatorial optimization problems. Theoretically, any Integer Program can be converted into a QUBO. In practice, however, there are some caveats. Furthermore, even for problems that can be nicely modeled as a QUBO, this might not be the most effective way to solve them. We review the state of QUBO solving on digital and Quantum computers and give some insights regarding current benchmark instances and modeling.
Eric
Moulines
Ecole Polytechnique
Score-based diffusion models and applications
Abstract.
Deep generative models represent an advanced frontier in machine learning. These models are adept at fitting complex data sets, whether they consist of images, text or other forms of high-dimensional data. What makes them particularly noteworthy is their ability to provide independent samples from these complicated distributions at a cost that is both computationally efficient and resource efficient. However, the task of accurately sampling a target distribution presents significant challenges. These challenges often arise from the high dimensionality, multimodality or a combination of these factors. This complexity can compromise the effectiveness of traditional sampling methods and make the process either computationally prohibitive or less accurate. In my talk, I will address recent efforts in this area that aim to improve traditional inference and sampling algorithms. My major focus will be on score-based diffusion models. By utilizing the concept of score matching and time-reversal of stochastic differential equations, they offer a novel and powerful approach to generating high-quality samples. I will discuss how these models work, their underlying principles and how they are used to overcome the limitations of conventional methods. The talk will also cover practical applications, demonstrating their versatility and effectiveness in solving complex real-world problems.
Michele
Pernice
KTH Stockholm
A stacky Castelnuovo’s contraction theorem
Abstract.
In this talk, we are going to discuss a generalization to weighted blow-ups of the classical Castelnuovo's contraction theorem. Moreover, we will show as a corollary that the moduli stack of n-pointed stable curves of genus 1 is a weighted blow-up. This is a joint work with Arena, Di Lorenzo, Inchiostro, Mathur, Obinna.
Kento
Osuga
Tokyo University
Recent progress in refined topological recursion
Abstract.
I will first present recent progress in the formulation of refined topological recursion with a brief overview of previous attempts. I will then show its interesting properties such as refined quantum curves, the refined variational formula, and refined BPS structures. I will also discuss an intriguing relation between refined topological recursion, W-algebras, and b-Hurwitz numbers. Finally, I will conclude with open questions and future directions. This talk is partly based on joint work with Kidwai, and also partly joint work in progress with Chidambaram and Dolega.
Raphael
Steiner
ETH Zürich
Shortest paths on combinatorial polytopes: Hardness and approximation
Abstract.
I will present some of my joint work with Jean Cardinal on the complexity of computing and approximating shortest paths in the skeleton of a combinatorially defined polytope. In particular, I will discuss proofs for the inapproximability of finding shortest paths on the skeleton of perfect matching polytopes, and of polymatroids, and discuss various related context and problems in which our work is embedded.
December 2023
Robert
Lauff
Facet hamiltonian paths in graph associahedra of complete bipartite graphs (and maybe trees)
Hannah
Tillmann-Morris
University College London
Detecting blow-ups of Fano varieties via their Laurent polynomial mirrors
Abstract.
Mirror symmetry gives a correspondence between certain Fano varieties and Laurent polynomials, translating the classification of Fano varieties up to deformation into a combinatorial problem. I will present a set of combinatorial conditions $\Phi$ on pairs of Laurent polynomials $(f,g)$ which imply the existence of mirror Fano varieties $X_f$ and $X_g$ related by a blow-up map $X_g \to X_f$. These criteria generalise the relationship between fans of toric varieties related by toric blowup; I will explain how in some key examples. Time permitting, I will discuss a new approach to constructing mirrors to Laurent polynomials, which is the main idea in the proof that Laurent polynomials in two variables satisfying the conditions $\Phi$ have mirrors related by blowing up in one point. This is based on upcoming joint work with Mark Gross.
Hanno
Gottschalk
TU Berlin
From Probabilistic Models of Mechanical Failure to Multi-Objective Shape Optimization
Zoe
Geiselmann
TU Berlin
Two ways of constructing graph associahedra
Abstract.
A tube of a connected graph G is a subset of its vertices, inducing a connected subgraph, whereas a tubing of G is a non-intersecting, non-adjacent set of tubes. The poset of tubings, ordered by reverse inclusion can be realized by the face lattice of a polytope, called the graph associahedron. We will take a look at two ways to construct graph associahedra. The first consists of consecutively truncating particular faces of a simplex and yields a whole class of polytopes. For the complete graph, one instance of this class is the permutahedron. The second way uses the nice hyperplane description of the permutahedron and removes some of the hyperplanes - however this will not always yield the desired polytope. We will see under which condition we can employ this method.
Hélène
Esnault
FU Berlin/Harvard/Copenhagen
Survey on some arithmetic properties of rigid local systems
Abstract.
A central conjecture of Simpson predicts that complex rigid local systems on a smooth complex variety come from geometry. In the last couple of years, we proved some arithmetic consequences of it: integrality (using the arithmetic Langlands program), F-isocrystal properties, crystallinity of the underlying p-adic representation (using the Cartier operator over the Witt vectors and the Higgs-de Rham flow) (for Shimura varieties of real rank at least 2, this is the corner piece of Pila-Shankar-Tsimerman's proof of the André-Oort conjecture), weak integrality of the character variety (using de Jong's conjecture proved with the geometric Langlands program) (yielding a new obstruction for a finitely presented group to be the topological fundamental group of a smooth complex variety). We'll survey some aspects of this (please ask if there is something on which you would like me to focus on). The talk is based mostly on joint work with Michael Groechenig, also, even if less, with Johan de Jong.
Simon
Felten
Columbia University
Global logarithmic deformation theory
Abstract.
A well-known problem in algebraic geometry is to construct smooth projective Calabi-Yau varieties \(X\). In the smoothing approach, one constructs first a degenerate (reducible) Calabi-Yau variety \(V\) by gluing pieces. Then one aims to find a family with special fiber \(V\) and smooth general fiber \(X\). In this talk, we see how infinitesimal logarithmic deformation theory solves the second step of this approach: the construction of a family out of a degenerate fiber \(V\). This is achieved via the logarithmic Bogomolov-Tian-Todorov theorem as well as its variant for pairs of a log Calabi-Yau space \(f_0: X_0 \to S_0\) and a line bundle \(\mathcal{L}_0\) on \(X_0\). Both theorems are a consequence of the abstract unobstructedness theorem for curved Batalin-Vilkovisky algebras.
Maximal parabolic regularity for the treatment of real world problems
Abstract.
This is a series of three lectures on non-smooth problems, the first dedicated to elliptic ones and the third to parabolic ones. We start by explainig what means 'nonsmooth' and describe effects which cancel classical regularity results. Then we introduce a general geometric setting which allows in consequence to prove elliptic regularity results sufficient for attacking two- and three dimensional real world problems.Having this at hand, elliptic operators on the scale of (negatively indexed) Sobolev spaces and Lp spaces are introduced and their regularity properties are investigated, among them L∞ estimates, Hölder estimates and W 1,q estimates for the solution. After these preparations we pass to parabolic equations in the third and last lecture. Here the notion of 'maximal parabolic regularity' for an elliptic operator A is the central one - being now for some decades the ultimative instrument also for the investigation in particular of nonlinar problems. Since into this notion the domain of A enters explicitly, it becomes clear that an exact knowledge of dom(A), prepared in the first two lectures, is highly desirable. So, after having introduced maximal parabolic regularity and explained some of its properties, we show that second order divergence operators satisfy this property even if the domain is highly non-smooth, the coefficient function is only bounded, measurable and elliptic and the boundary conditions are mixed
Bruno
Kahn
IMJ Paris
Universal Weil cohomology
Abstract.
In this joint work with Luca Barbieri-Viale, we show that a universal Weil cohomology exists over any field k. The story is actually a bit more complicated: to a suitable class of smooth projective k-varieties (all varieties is the default) we associate 4 universal Weil cohomologies, depending on whether the universal problem concerns targets which are additive or abelian categories, and whether the axioms for the Weil cohomology are plain or if one adds requirements in the style of Weak and Strong Lefschetz. In the latter case, the universal additive category obtained can be used to recover André’s category of motives for ''motivated'' cycles. If time permits, I will explain how the construction extends over a base, and some open problems.
Jakob
Sudau
On the Variational Analysis of Spin Models on Triangular Lattices
Super-resolved Lasso
Abstract.
Super-resolution of pointwise sources is of utmost importance in various areas of imaging sciences. Specific instances of this problem arise in single molecule fluorescence, spike sorting in neuroscience, astrophysical imaging, radar imaging, and nuclear resonance imaging. In all these applications, the Lasso method (also known as Basis Pursuit or l1-regularization) is the de facto baseline method for recovering sparse vectors from low-resolution measurements. This approach requires discretization of the domain, which leads to quantization artifacts and consequently, an overestimation of the number of sources. While grid-less methods, such as Prony-type methods or non-convex optimization over the source position, can mitigate this, the Lasso remains a strong baseline due to its versatility and simplicity. In this work, we introduce a simple extension of the Lasso, termed ``super-resolved Lasso" (SR-Lasso). Inspired by the Continuous Basis Pursuit (C-BP) method, our approach introduces an extra parameter to account for the shift of the sources between grid locations. Our method is more comprehensive than C-BP, accommodating both arbitrary real-valued or complex-valued sources. Furthermore, it can be solved similarly to the Lasso as it boils down to solving a group-Lasso problem. A notable advantage of SR-Lasso is its theoretical properties, akin to grid-less methods. Given a separation condition on the sources and a restriction on the shift magnitude outside the grid, SR-Lasso precisely estimates the correct number of sources.
The symplectic topology of singularities
Muhammed E.
Guelen
HU Berlin
What is a Fukaya category?
Abstract.
A finite collection of points of the complex plane lying in general position determines a polygonal shape. By extending each edge of the polygon into a line, these vertices describe certain intersections points of these lines. In general, the intersecting objects do not need to be lines nor need this to happen in the complex plane. In this talk, we want to give a taste of intersection problems in symplectic manifolds where the intersecting objects are a class of half-dimensional spaces, called Lagrangians. We will discuss (with examples) what this has to do with the method of Lagrangian multipliers and how such Lagrangian intersection problems reveal a path towards categorification.
Laura
Sangalli
MOX Milano, Italien
Physics-informed spatial and functional data analysis
Abstract.
Recent years have seen an explosive growth in the recording of increasingly complex and high-dimensional data, whose analysis calls for the definition of new methods, merging ideas and approaches from statistics and applied mathematics. My talk will focus on spatial and functional data observed over non-Euclidean domains, such as linear networks, two-dimensional manifolds and non-convex volumes. I will present an innovative class of methods, based on regularizing terms involving Partial Differential Equations (PDEs), defined over the complex domains being considered. These Physics-Informed statistical learning methods enable the inclusion of the available problem specific information, suitably encoded in the regularizing PDE. Illustrative applications from environmental and life sciences will be presented.
Stefan
Felsner
On digons in arrangements of (pseudo)circles
Gabriele
Eichfelder
Fachgebiet Mathematische Methoden des Operations Research, TU Ilmenau
Multiobjective Mixed Integer Programming
Abstract.
Multiobjective mixed integer nonlinear optimization refers to mathematical programming problems where more than one nonlinear objective function needs to be optimized simultaneously and some of the variables are constrained to take integer values. In this talk, we give a short introduction to the basic concepts of multiobjective optimization. We give insights why the famous approach of scalarization might not be an appropriate method to solve these problems. Instead, we present two procedures to solve the problems directly. The first is a branch-and-bound method based on the use of properly defined lower bounds. We do not simply rely on convex relaxations, but we built linear outer approximations of the image set in an adaptive way. The second method is tailored for convex objective functions and is purely based on the criterion space. It uses ingredients from the well-known outer approximation algorithm from single-objective mixed-integer optimization and combines them with strategies to generate enclosures of nondominated sets by iteratively improving approximations. For both algorithms, we are able to guarantee correctness in terms of detecting the nondominated set of multiobjective mixed integer problems according to a prescribed precision.
Rosa
Preiß
U Potsdam
An algebraic geometry of paths via the iterated-integrals signature
Abstract.
Contrary to previous approaches bringing together algebraic geometry and signatures of paths, we introduce a Zariski topology on the space of paths itself, and study path varieties consisting of all paths whose signature satisfies certain polynomial equations. Particular emphasis lies on the role of the non-associative halfshuffle, which makes it possible to describe varieties of paths that satisfy certain relations all along their trajectory. Specifically, we may understand the set of paths on a given classical algebraic variety in R^d starting from a fixed point as a path variety. While halfshuffle varieties are stable under stopping paths at an earlier time, we furthermore study varieties that are stable under concatenation of paths. We point out how the notion of dimension for path varieties crucially depends on the fact that they may be reducible into countably infinitely many subvarieties. Finally, we see that studying halfshuffle varieties naturally leads to a generalization of classical algebraic curves, surfaces and affine varieties in finite dimensional space, where these generalized algebraic sets are now described through iterated-integral equations. Keywords. path variety, shuffle ideal, halfshuffle, deconcatenation coproduct, tensor algebra, Zariski topology, concatenation of paths, Chen's identity, tree-like equivalence, regular map, generalized variety
Gergely
Berczi
Aarhus University
Geometry of the Hilbert scheme of points on manifolds, part II
Abstract.
While the Hilbert scheme of points on surfaces is well-explored and extensively studied, the Hilbert scheme of points on higher-dimensional manifolds proves to be exceptionally intricate, exhibiting wild and pathological characteristics. Our understanding of their topology and geometry is very limited. In these talks I will present recent results on various aspects of their geometry. I will discuss i) the distribution of torus fixed points within the components of the Hilbert scheme (work with Jonas Svendsen) ii) intersection theory of Hilbert schemes: a new formula for tautological integrals over geometric components, and applications in enumerative geometry iii) new link invariants of plane curve (and hypersurface) singularities coming from p-adic integration and Igusa zeta functions (work with Ilaria Rossinelli). This talk will be relatively independent from part I on 12th December at the Arithmetic Geometry Seminar
Boris
Buchmann
ANU Canberra, Australia
Weak subordination of multivariate Levy processes
Abstract.
Subordination is the operation which evaluates a Levy process at a subordinator, giving rise to a pathwise construction of a "time-changed" process. In probability semigroups, subordination was applied to create the variance gamma process, which is prominently used in financial modelling. However, subordination may not produce a levy process unless the subordinate has independent components or the subordinate has indistinguishable components. We introduce a new operation known as weak subordination that always produces a Levy process by assigning the distribution of the subordinate conditional on the value of the subordinator, and matches traditional subordination in law in the cases above. Weak subordination is applied to extend the class of variance-generalised gamma convolutions and to construct the weak variance-alpha-gamma process. The latter process exhibits a wider range of dependence than using traditional subordination. Joint work with Kevin W. LU - Australian National University (Australia) & Dilip B. Madan - University of Maryland (USA)
László
Kozma
FU
Algorithms: from sorting to saddle points
Abstract.
Algorithmic problems typically ask to transform each given input according to some well-defined mathematical function. For example, in the sorting problem, given a sequence of (comparable) items, we want to put them in increasing order.When can we say that we fully understand the complexity of an algorithmic problem? Ideally, we should find an algorithm that solves the task in a certain number of elementary steps, and prove that no algorithm can achieve this in fewer steps. But how can we argue about all possible inputs and all possible algorithms, including those not yet invented? This basic question is behind some of the great mysteries of theoretical computer science; we have satisfactory answers only for relatively simple problems in restricted models of computation.As a case study we will look at the problem of finding a saddle point, a task that arises both in optimization and game theory. Seemingly related to sorting, the problem allows for some surprising algorithmic improvements, with its precise complexity not yet settled.
Gergely
Berczi
Aarhus
Geometry of the Hilbert scheme of points on manifolds, part I
Abstract.
While the Hilbert scheme of points on surfaces is well-explored and extensively studied, the Hilbert scheme of points on higher-dimensional manifolds proves to be exceptionally intricate, exhibiting wild and pathological characteristics. Our understanding of their topology and geometry is very limited. In this series of two talks I will present recent results on various aspects of their geometry. I will discuss i) the distribution of torus fixed points within the components of the Hilbert scheme (work with Jonas Svendsen), ii) intersection theory of Hilbert schemes: a new formula for tautological integrals over geometric components, and applications in enumerative geometry, iii) new link invariants of plane curve (and hypersurface) singularities coming from p-adic integration and Igusa zeta functions (work with Ilaria Rossinelli). Part II of the talk will be relatively independent from part I and takes place on Wednesday 13 December in the Algebraic Geometry Seminar.
Xavier
Coulter
University of Auckland
A one-parameter deformation of the monotone Hurwitz numbers
Abstract.
The monotone Hurwitz numbers are involved in a wide array of mathematical connections, linking topics such as integration on unitary groups, representation theory of the symmetric group, and topological recursion. In recent work, we introduce a one-parameter deformation of the monotone Hurwitz numbers and show that the resulting family of polynomials admits a similarly broad network of connections. We will discuss these results and some non-trivial conjectures on the roots of these polynomials.
Olexandr
Burylko
Institute of Mathematics, Kyiv, Ukraine
Coexistence of conservative and dissipative dynamics in rings of coupled phase oscillators
Christof
Schütte
ZIB
The few and the many
Abstract.
The talk will give a short introduction to complex dynamics of interacting systems of individual units that can be particles (molecules, …), or agents (individual humans, media agents, …). We are interested in systems with at least two types of such units, one type of which just a “few” individual units are present and another type of which there are “many”. For such systems we will review mathematical models on different levels: from the micro-level in which all particles/agents are described individually to the macro-level where the “many” are modelled in an aggregated way. The effective dynamics given by these models will be illustrated by examples from cellular systems (neurotransmission processes) and opinion dynamics in social networks. You will be able to follow the talk even if you do not have any detailed knowledge about particles/agents or cellular/social processes (at least I hope).
Carsten
Carstensen
HU Berlin
Hs(Ω) for 0<s<1 (Fortsetzung)
Neil
Olver
LSE
A strongly polynomial algorithm for linear programs with at most two non-zero entries per row or column
Abstract.
We give a strongly polynomial algorithm for minimum cost generalized flow, and hence for optimizing any linear program with at most two non-zero entries per row, or at most two non-zero entries per column. Our result can be viewed as progress towards understanding whether all linear programs can be solved in strongly polynomial time, also referred to as Smale's 9th problem. Our approach is based on the recent primal-dual interior point method (IPM) due to Allamigeon, Dadush, Loho, Natura and Végh (FOCS '22). They show that the number of iterations needed by the IPM can be bounded in terms of the straight line complexity of the central path; roughly speaking, this is the minimum number of pieces of any piecewise linear curve that multiplicatively approximates the central path. As our main contribution, we show that the straight line complexity of any minimum cost generalized flow instance is polynomial in the number of arcs and vertices. By a reduction of Hochbaum, the same bound applies to any linear program with at most two non-zeros per column or per row. Further, we demonstrate how to handle initialization, and how to ensure that the bit complexity of each iterate remains bounded during the execution of the algorithm. Joint work with Daniel Dadush, Zhuan Khye Koh, Bento Natura and László Végh.
Igor
Burban
Universität Padelborn
Algebraic geometry of the torus model of the fractional quantum
Hall effect
Abstract.
The experimental discovery of the quantum Hall effect is widely considered to be a one of the major events in the condensed matter physics in the second half of the twentieth century. Both experimental and theoretical aspects of this phenomenon still continue to attract an enormous attention. In 1993 Keski-Vakkuri and Wen introduced a model for the quantum Hall effect based on multilayer two-dimensional electron systems satisfying quasi-periodic boundary conditions. Such a model is specified by a choice of a complex torus \(E\) and a symmetric positively definite matrix \(K\) of size \(g\) with integer coefficients. The space of the corresponding wave functions turns out to be \(d\)-dimensional, where \(d\) is the determinant of \(K\). I am going to explain a construction of a hermitian holomorphic bundle of rank \(d\) on the abelian variety \(A\) (which is the \(g\)-fold product of the torus \(E\) with itself), whose fibres can be identified with the space of wave function of Keski-Vakkuri and Wen. A rigorous construction of this "magnetic bundle" involves the technique of Fourier-Mukai transforms on abelian varieties. This bundle turns out to be simple and semi-homogeneous. Moreover, for special classes of the matrix \(K\), the canonical Chern-Weil connection of the magnetic bundle is shown to be projectively flat. This talk is based on a joint work with Semyon Klevtsov (arXiv:2309.04866).
Viktor
Levandovskyy
U Kassel
Gröbner bases over free associative algebras: Algorithmics,
Implementation, and Applications
Abstract.
In this talk we will make a journey to the constructive methods for ideals of free associative algebra. These methods, especially those based on Gröbner bases are important constituents in a vast number of applications. However, there are numerous intrinsic complications to be treated: for example, a typical computation of a Gröbner basis will not terminate after finitely many steps. Balancing on the edge of decidability we will show, what is possible to compute and how these computations are implemented. Our implementation, providing a lof of functionality at a decent speed, is called Singular:Letterplace and it is OSCAR-aware.Slides
Deborah
Hendrych
ZIB
/
TU Berlin
Solving the Optimal Experiment Design Problems with Mixed-Integer Frank-Wolfe-based methods
Abstract.
We tackle the Optimal Experiment Design Problem, which consists in choosing experiments to run or observations to select from a finite set to estimate the parameters of a system. The objective is to maximize some measure of information gained on the system from the observations, leading to a convex integer optimization problem. We leverage Boscia, a recent algorithmic framework, which is based on a nonlinear branch-and-bound with node relaxations solved to approximate optimality using Frank-Wolfe algorithms. One particular advantage of the method is its efficient utilization of the polytope formed by the original constraints which remains preserved by the method, unlike in those relying on epigraph-based formulations. We assess our method against both generic and specialized convex mixed-integer approaches. Computational results highlight the performance of the proposed method, especially on large and challenging instances.
PDAEs with Uncertainties for the Analysis, Simulation and Optimization of Energy Networks
Abstract
Gabriel
Dill
Bonn
The modular support problem over number fields and over function
fields
Abstract.
In 1988, Erdős asked: let \(a\) and \(b\) be positive integers such that for all \(n\), the set of primes dividing \(a^n - 1\) is equal to the set of primes dividing \( b^n - 1\). Is \(a = b\)? Corrales and Schoof answered this question in the affirmative and showed more generally that, if every prime dividing \(a^n - 1\) also divides \(b^n - 1\), then \(b\) is a power of \(a\). In joint work with Francesco Campagna, we have studied this so-called support problem with the Hilbert class polynomials \( H_D(T)\) instead of the polynomials \(T^n - 1\), replacing roots of unity by singular moduli. I will present the results we obtained both in the number field case, where \(a\) and \(b\) lie in some ring of \(S\)-integers in a number field \(K\), as well as in the function field case, where \(a\) and \(b\) are regular functions on a smooth irreducible affine curve over an algebraic closure of a finite field.
Johannes
Storn
Universität Bielefeld
Adaptive Mesh Refinement for arbitrary initial Triangulations
Abstract.
This talk introduces a simple initialization of the Maubach/Traxler bisection routine for adaptive mesh refinements. This initialization applies to any conforming initial triangulation. It preserves shape-regularity, satisfies the closure estimate needed for optimal convergence of adaptive schemes, and allows for the intrinsic use of existing implementations. This talk results from joint work with Lars Diening (Bielefeld University) and Lukas Gehring (Friedrich-Schiller-Universität Jena).
Thomas
Buc-d'Alché
ENS Lyon
Fay-like identities for hyperelliptic curves
Abstract.
Fay's identity is a determinantal formula between Riemann theta functions associated to the period matrix of a Riemann surface. In random matrix theory, the theta function appears in the asymptotic expansion of the partition function of the β-model. Using Pfaffian formulae for averages of characteristic polynomials when β = 1 or β =4, we derive Pfaffian identities involving the theta function associated to half or twice the period matrix of a hyperelliptic curve. This is joint work with Gaëtan Borot.
Equations over groups
Jeremie
Houssineau
University of Warwick
Ensemble Kalman filtering for epistemic uncertainty
Carsten
Carstensen
HU Berlin
Hs(Ω) for 0<s<1
Manfred
Scheucher
On the empty hexagon theorem
November 2023
Sonja
Steffensen
Institut für Geometrie und Praktische Mathematik, RWTH Aachen
On Multilevel Game Theory and its Applications
Abstract.
Hierarchical Nash game models are an important modelling tool in various applications to study a strategic non-cooperative decision process of individuals, where the individuals can be split into a hierarchy of at least two different groups. Such models are in general mathematically described by multilevel games. In this talk we will give a short introduction into standard and in particular hierarchical game theory and present some important mathematical structures and properties of Single- and Multi-Leader-Follower Games. Moreover, we will have a look at suitable numerical methods for such games. In the second part of the talk, we first study a discrete-dynamic multilevel game that is used to model the optimal control of a gas transmission network. The players of this game are given by the controller of the network, i.e. the so-called technical system operator (TSO) on the one hand and the users of the network, namely the gas buyers and sellers on the other hand. Here, we consider a fully dynamic version of the TSO’s optimal control problem using a coupled system of semi-linear isothermal Euler equations to describe the time dependent gas dynamics. We will analyse the original four-level problem, which can be reduced to a bilevel discrete-dynamic optimal control problem by means of the underlying potential game structure. Finally, we briefly present a dynamic Stackelberg game, i.e. a Single-Leader-Follower game, where the number of followers becomes (infinitely) large giving rise to a so-called Meanfield Stackelberg game.
Myfanwy
Evans
Uni Potsdam
Geometry-based simulation of self assembly
Abstract.
The morphometric approach to solvation free energy is a geometry-based theory that incorporates a weighted combination of geometric measures over the solvent accessible surface for solute configurations in a solvent. In this talk, I will demonstrate that employing this geometric technique in simulating the self assembly of sphere clusters and small loops results in an assortment of interesting geometric configurations. This gives insight into the role of shape in the physical process of self assembly, potentially relevant to proteins, viruses and other complex systems.
André-Alexander
Zepernick
FU Berlin
An introduction to TorchPhysics: Deep Learning for partial differential equations
Giancarlo
Urzúa
Pontificia Universidad Católica de Chile
The birational geometry of Markov numbers
Abstract.
The projective plane is rigid. However, it may degenerate to surfaces with quotient singularities. After the work of Bădescu and Manetti, Hacking and Prokhorov 2010 classified these degenerations completely. They are \(\mathbb{Q}\)-Gorenstein partial smoothings of \(\mathbb{P}(a^2,b^2,c^2)\), where \(a,b,c\) satisfy the Markov equation \(x^2+y^2+z^2=3xyz\). Let us call them Markovian planes. They are part of a bigger picture of degenerations with Wahl singularities, where there is an explicit MMP whose final results are either \(K\) nef, smooth deformations of ruled surfaces, or Markovian planes. Although it is a final result of MMP, we can nevertheless run MMP on small modifications of Markovian planes to obtain new numerical/combinatorial data for Markov numbers via birational geometry. New connections with Markov conjecture (i.e. Frobenius Uniqueness Conjecture) are byproducts. This is joint work with Juan Pablo Zúñiga (Ph.D. student at UC Chile), the pre-print can be found here.
Martin
Spindler
Universität Hamburg
High-dimensional L2-boosting: Rate of convergence (hybrid talk)
Yannik
Schüler
University of Sheffield
Gromov-Witten theory from the 5-fold perspective
Abstract.
The observation that the Gromov-Witten theory of a Calabi-Yau threefold X may be viewed as a mathematical realisation of the A-model topological string on this target is the corner stone of some of the most exciting developments in Enumerative Geometry in the last decades. Despite this, the so called refined topological string so far lacked a mathematical description. In this talk I will make a proposal for a rigorous formulation in terms of equivariant Gromov-Witten theory on the fivefold X x C^2. To convince you of our construction I will mention several precision checks our proposal passes. Most of these results were expected by physics but some are new.
Julia
Schaeffer
HU Berlin
Contour integration methods for nonlinear eigenvalue problems in nanooptics
Jonas
Lautenschläger
HU
L^p-Regularity of the Poisson equation
Jonas
Charfreitag
University of Bonn, Germany
Integer Programming for the Maximum Cut Problem Models, Branching and Solvers
Abstract.
While the Maximum Cut Problem and Unconstrained Binary Quadratic Optimization are of high interest in the scientific community and gain increasing importance, no state-of-the-art solvers for these problems were publicly available before 2022. This changed with the development of our solver McSparse, which is available online: http://mcsparse.uni-bonn.de/. We discuss the relevant building blocks that lead to McSparse and present recent results on improved ilp-models, separation and branching rules for the Maximum Cut problem, as well as further directions for improvement.
Maximilian
Wittmann
The Dimension of Products of Orders
Marie
Brandenburg
MPI MiS Leipzig
How to slice a polytope
Abstract.
Given a 3-dimensional cube, the intersection with an affine hyperplane is always a polygon with 3,4,5, or 6 vertices. But how can one understand the slices of a general polytope? And which slice is “the best”, e.g. is the slice of maximal volume? In this talk, we consider the structure of all possible affine hyperplane sections of a convex polytope, and we craft algorithms that compute optimal sections for various combinatorial and metric criteria. Along the way, we will encounter several famous hyperplane arrangements which will guide our algorithms. This is based on joint work with Jesus De Loera and Chiara Meroni.
Marco
Cuturi
Apple / CREST-ENSAE
On the optimization of Monge Maps: Structured Priors and Neural Networks
Abstract.
After providing a short self-contained introduction on the Monge problem, its potential applications and its computational challenges, I will present in this talk two recent contributions that offer practical solutions. In the [first part](https://proceedings.mlr.press/v202/cuturi23a.html) I will show how changing the so-called ground cost function of optimal transport problems directly influences the structure of such maps in theory, and how this can be turned into a practical tool. In the [second part](https://proceedings.mlr.press/v202/uscidda23a.html) I present a simple approach to estimate Monge maps using a simple regularizer (both works were presented at the ICML'23 conference.
Quanling
Deng
Australian National University
Particle-Continuum Multiscale Modeling of Sea Ice Floes
Abstract.
In this talk, I will start by presenting some quick facts about Arctic and Antarctic sea ice floes followed by a quick overview of the major sea ice continuum and particle models. I will then present our main contribution to its multiscale modelling. The recent Lagrangian particle model based on the discrete element method (DEM) has shown improved model performance and started to gain more attention from groups that are working on Global Climate Models (GCMs). We adopt the DEM model for sea ice dynamical simulation. The major challenges are 1) model coupling in different frames of reference (Lagrangian for sea ice while Eulerian for the ocean and atmosphere dynamics); 2) the heavy computational cost when the number of the floes is large; and 3) inaccurate floe parameterisation when the floe distribution has multiscale features. To overcome these challenges, I will present a superfloe parameterisation to reduce the computational cost and a superparameterisation method to capture the multiscale features. In particular, the superfloe parameterisation facilitates noise inflation in data assimilation that recovers the unobserved ocean field underneath the sea ice. To capture the multiscale features, we adopt the Boltzmann equation for particles and superparameterise the sea ice floes as continuity equations governing the statistical moments. This leads to a particle-continuum coupled multiscale model. I will present several numerical experiments to demonstrate the success of the proposed method. This is joint work with Sam Stechmann (UW-Madison) and Nan Chen (UW-Madison).
Jarkko
Kari
University of Turku
From Wang Tiles to the Domino Problem: A Tale of Aperiodicity
Abstract.
This presentation delves into the remarkable history of aperiodic tilings and the domino problem. Aperiodic tile sets refer to collections of tiles that can only tile the plane in a non-repeating, or non-periodic, manner. Such sets were not believed to exist until 1966 when R. Berger introduced the first aperiodic set consisting of an astonishing 20,426 Wang tiles. Over the years, ongoing research led to significant advancements, culminating in 2015 with the discovery of a mere 11 Wang tiles by E. Jeandel and M. Rao, alongside a computer-assisted proof of their minimality. Simultaneously, researchers found even smaller aperiodic sets composed of polygon-shaped tiles. Notably, Penrose's kite and dart tiles emerged as early examples, and most recently, a groundbreaking discovery was made - a solitary aperiodic tile known as the "hat" that can tile the plane exclusively in a non-periodic manner. Aperiodic tile sets are intimately connected with the domino problem that asserts how certain tile sets can tile the plane without us ever being able to establish their tiling nature with absolute certainty. Moreover, aperiodic tilings hold a distinct visual aesthetic allure. In today's musical presentation, their artistic appeal transcends the visual domain and extends into the realm of music.
Daniel
Dadush
CWI Amsterdam
Interior point methods are not worse than Simplex
Abstract.
https://arxiv.org/abs/2206.08810
Lukas
Schürmann
University of Bonn, Germany
Gap-based Optimality Cuts: A new Paradigm for Branch-and-Cut Approaches for Binary Linear Programming
Abstract.
Cutting plane algorithms are a key tool for solving (Mixed) Integer Linear programs. The constraints defining the cutting planes can be divided into the classes of feasibility cuts and optimality cuts. Both classes cut off certain parts of the linear relaxation. However, the optimality cuts, in contrast to the feasibility cuts, can also cut off feasible solutions by using information provided by the objective function. In this work, we even go one step further and present new gap-based optimality (GO) cuts for Binary Linear Programs (BLP) that may cut off optimal integer solutions. Given a feasible solution providing a primal bound U for our given (BLP), our newly developed GO constraints are designed to cut off solutions that are not better than U. We show that they improve a certain class of integer rounding cuts as well as Dantzig cuts and their strengthened version by Charnes and Cooper. Our computational results show that our new constraints can have a positive impact on the solution process.
Sparse Personalized PageRank:
New results on the 25 billion dollar eigenvector problem
Abstract.
This talk will go over the basics of the PageRank problem, studied initially by the founders of Google, which allowed them to create their search engine by applying it to the internet graph with links defining edges. Then, we will explain some of our results on the problem for undirected graphs, whose main application is finding local clusters in networks, and is used in many branches of science. We can now find local clusters fast in a time that does not depend on the whole graph but on the local cluster itself. <p>This is joint work with and Sebastian Pokutta.</p>
Lara
Theallier
HU Berlin
Hybrid-high-order methods for linear elasticity
Marc
Hoffmann
Université Paris-Dauphine
On estimating multidimensional diffusions from discrete data
Georgi
Mitsov
HU Berlin
Time-space variational formulations for the heat equation
Murad
Alim
Hamburg Universität
Resurgence, BPS structures and topological string S-duality
Abstract.
The partition function of topological string theory is an asymptotic series in the topological string coupling and provides in a certain limit a generating function of Gromov-Witten (GW) invariants of a Calabi-Yau threefold. I will discuss how the resurgence analysis of the partition function allows one to extract Donaldson-Thomas (DT) or BPS invariants of the same underlying geometry. I will further discuss how the analytic functions in the topological string coupling obtained by Borel summation admit a dual expansion in the inverse of the topological string coupling leading to another asymptotic series at strong coupling and to the notion of topological string S-duality. This S-duality leads to a new modular structure in the topological string coupling. I will also discuss relations to difference equations and the exact WKB analysis of the mirror geometry. This is based on various joint works with Lotte Hollands, Arpan Saha, Iván Tulli and Jörg Teschner as well as on work in progress.
Regularity for non-smooth elliptic problems II
Abstract.
This is a series of three lectures on non-smooth problems, the first dedicated to elliptic ones and the third to parabolic ones. We start by explainig what means 'nonsmooth' and describe effects which cancel classical regularity results. Then we introduce a general geometric setting which allows in consequence to prove elliptic regularity results sufficient for attacking two- and three dimensional real world problems.Having this at hand, elliptic operators on the scale of (negatively indexed) Sobolev spaces and Lp spaces are introduced and their regularity properties are investigated, among them L∞ estimates, Hölder estimates and W 1,q estimates for the solution. After these preparations we pass to parabolic equations in the third and last lecture. Here the notion of 'maximal parabolic regularity' for an elliptic operator A is the central one - being now for some decades the ultimative instrument also for the investigation in particular of nonlinar problems. Since into this notion the domain of A enters explicitly, it becomes clear that an exact knowledge of dom(A), prepared in the first two lectures, is highly desirable. So, after having introduced maximal parabolic regularity and explained some of its properties, we show that second order divergence operators satisfy this property even if the domain is highly non-smooth, the coefficient function is only bounded, measurable and elliptic and the boundary conditions are mixed
Jan
Volec
Czech Technical University in Prague
On the minimum eigenvalue of regular triangle-free graphs
Low complexity colorings of the two-dimensional grid
Manuel
Staiger
FU Berlin
What is a sofic group?
Abstract.
In view of the MATH+ Friday lecture by Andreas Thom we introduce sofic groups and discuss some examples.
Jonas
Köppl
WIAS
What is a low-complexity coloring?
Abstract.
Informally, a tiling is a covering of the plane with tiles of various shapes, arranged to avoid any overlapping. Usually, these tiles have simple shapes (e.g. polygons), and one only allows a small number of different shapes to be used for a tiling. One particularly interesting class of tiles are the so-called Wang tiles which can alternatively be represented via finite colorings of \Z^d. Given a set of such tiles, one might ask whether one can actually use them to cover the plane and whether that is possible without ever repeating oneself, i.e., without becoming periodic. The goal of this talk is to introduce (finite) colorings of Z^d, discuss their relation to Wang tiles and the domino problem, and then speak about low complexity colorings and Nivat’s conjecture.
Helena
Bergold
Plane Hamiltonian cycles and paths in convex drawings
Paul
Strang
EDF, Paris, France
Learning in hyperbolic space: an introduction for combinatorial optimization practitioners
Abstract.
Over the last decade, many contributions have sought to leverage machine learning algorithms to improve the performances of combinatorial optimization solvers. In particular, due to the ubiquitous nature of graphs in combinatorial tasks, the use of graph neural networks has flourished in the literature. Parallel to these developments, due to their capacity to embed hierarchically structured data such as trees and scale-free graphs with arbitrary low distortion, hyperbolic spaces have gained an increasing amount of attention in the machine learning community. This talk first introduces the necessary conceptual tools in both Riemannian geometry and deep learning to provide a clear understanding of hyperbolic neural networks. Then, it motivates the use of hyperbolic representations in combinatorial applications and focuses on its potential in mixed integer programming.
Christoph
Hertrich
Goethe-Universität Frankfurt
Understanding Neural Network Expressivity via Polyhedral Geometry
Abstract.
Neural networks with rectified linear unit (ReLU) activations are one of the standard models in modern machine learning. Despite their practical importance, fundamental theoretical questions concerning ReLU networks remain open until today. For instance, what is the precise set of (piecewise linear) functions representable by ReLU networks with a given depth? Even the special case asking for the number of layers to compute a function as simple as max{0, x_1, x_2, x_3, x_4} has not been solved yet. In this talk I will convince you that polyhedral geometry is a useful perspective to study such questions and I will report about recent progress using mixed-integer programming and subdivisions of lattice polytopes. This is based on joint works with Amitabh Basu, Marco Di Summa, and Martin Skutella (NeurIPS 2021), as well as Christian Haase and Georg Loho (ICLR 2023).
Lorenz
Richter
Zuse Institute Berlin
An optimal control perspective on diffusion-based generative modeling leading to robust numerical methods
Giulia
Codenotti
FU Berlin
Lattice-reduced and complete convex bodies
Abstract.
Reduced and complete convex bodies are classical objects in convex geometry. We introduce and study discrete analogues of these bodies in the presence of a lattice, which we call lattice-reduced and lattice-complete bodies. As we will see, these convex bodies are necessarily polytopes and have strong restrictions on the number of vertices and facets. We will in particular discuss the case of bodies that are both lattice-reduced and lattice-complete, exploring their characteristics and giving conditions on their existence.To conclude, we will see one of the main motivations for these definitions, namely the connection with the celebrated flatness theorem. This theorem establishes, in fixed dimension, a maximum lattice width of convex bodies that do not contain lattice points in the interior. In particular, we will explore how the study of lattice-reduced bodies can help determine the maximum width constants of the theorem.
Frank-Olaf
Schreyer
Universität des Saarlandes
Tate resolutions of Gorenstein Rings and a construction from Clifford modules of complete intersection of two quadrics
Abstract.
The concept of MCM approximations of Auslander-Buchweitz is a beautiful concept which builds on Tate resolution. I will decribe the complexes explicitly in case for the case of the coordinate ring a complete intersection \((x_1,\ldots,x_n)\) as a module over a coordinate ring of a further complete intersection \((q_1,\ldots,q_r)\). In the second part I will explain how one can directly construct Tate resolution from a module over the Clifford algebra of a complete intersection of two quadrics \(X \subset \mathbb{P}^{2g+1}\) and their relation to Ulrich bundles on \(X\).
Tobias
Kreutz
Bonn
On Simpson's Standard Conjecture for unipotent local systems
Abstract.
Simpson's Standard Conjecture predicts that a local system which is defined over \( \overline{\mathbb{Q}}\) on both sides of the Riemann-Hilbert correspondence is motivic. In this talk, I want to discuss this conjecture for unipotent local systems. Conditional on classical transcendence conjectures for mixed Tate motives over number fields, we show that a unipotent local system over \( X = \mathbb{P}^1 \setminus \{s_1,...,s_n\}\) which is defined over \(\overline{\mathbb{Q}}\) on both sides of the Riemann-Hilbert correspondence is the monodromy of a mixed Tate motive over \(X\). This uses the construction of the motivic fundamental group due to Deligne-Goncharov, Borel's computation of the (rational) algebraic \(K\)-theory of number fields, and a homotopy exact sequence for the motivic fundamental group due to Esnault-Levine. For unipotent local systems of small index, we obtain some unconditional results due to transcendence/irrationality results of Baker and Apéry. We also prove a version of the Standard Conjecture over smooth projective curves of genus one, using transcendence results of Chudnovsky and Wüstholz.
Paolo
Gregori
IPhT - CEA Saclay
New results in non-perturbative topological recursion
Abstract.
I will present recent techniques which combine topological recursion with ideas from the theory of resurgence. In this framework, one can compute non-perturbative contributions to the formal power series one usually obtains from topological recursion, upgrading them to resurgent 'transseries'. The computation of such contributions serves two main purposes: on the one hand, it allows for an in-depth study of instanton effects in 2d gravitational theories such as Jackiw-Teitelboim gravity. On the other hand, it leads to new formulas for the large genus asymptotics of a large class of enumerative invariants, such as Weil-Petersson volumes and intersection numbers.
Neela
Nataraj
IIT Bombay
Adaptive Computation of Fourth-Order Problems
Helene
Frankowska
Sorbonne Université, Paris
Differential Inclusions and Optimal Control on Wasserstein spaces
Abstract.
Optimal Control on Wasserstein spaces addresses control of systems with large number of agents. Recently many models arising in social sciences use these metric spaces of Borel probability measures. The aim of this talk is to demonstrate that for Lipschitz kind dynamics, some corner stone results of classical control theory known in the Euclidean framework have their analogues in Wasserstein spaces. In this talk I will first discuss an extension of the theory of differential inclusions to the setting of general Wasserstein spaces. Indeed, it is well known that for optimal control of ODEs, the differential inclisions theory provides useful tools to investigate existence of optimal controls, necessary optimality conditions and Hamilton-Jacobi-Bellman equations. Same happens for Wasserstein spaces. In particular, I will present necessary and sufficient conditions for the existence of solutions to state-constrained continuity inclusions from [2] building on a suitable notion of contingent cones in Wasserstein spaces leading to viability and invariance theorems. They were already applied in [5], [6] to investigate stability of controlled continuity equations and uniqueness of solutions to HJB equations and will be recalled by the end of the talk. References [1] BONNET B. & FRANKOWSKA H., Carathéodory Theory and a Priori Estimates for Continuity Inclusions in the Space of Probability Measures, preprint https://arxiv.org/pdf/2302.00963.pdf, 2023. [2] BONNET B. & FRANKOWSKA H., On the Viability and Invariance of Proper Sets under Continuity Inclusions in Wasserstein Spaces, SIAM Journal on Mathematical Analysis, to appear. [3] BONNET B. & FRANKOWSKA H., Differential inclusions in Wasserstein spaces: the Cauchy-Lipschitz framework, Journal of Diff. Eqs. 271: 594 - 637, 2021. [4] BONNET B. & FRANKOWSKA H., Mean-field optimal control of continuity equations and differential inclusions, Proceedings of 59th IEEE Conference on Decision and Control, Republic of Korea, December 8-11, 2020: 470 - 475, 2020. [5] BONNET B. & FRANKOWSKA H., Viability and exponentially stable trajectories for differential inclusions in Wasserstein spaces, Proceedings of 61st IEEE Conference on Decision and Control, Mexico, December 6-9, 2022: 5086 - 5091, 2022. [6] BADREDDINE Z. & FRANKOWSKA H., Solutions to Hamilton-Jacobi equation on a Wasserstein space, Calculus of Variations and PDEs 81: 9, 2022.
Norbert
Heuer
Pontificia Universidad Católica de Chile
Normal-normal continuous symmetric stresses in finite element elasticity
Vesna
Iršič
Domination of subcubic planar graphs with large girth
Olaf
Parczyk
FU Berlin
New Ramsey Multiplicity Bounds and Search Heuristics
Abstract.
We study two related problems concerning the number of monochromatic cliques in two-colorings of the complete graph that go back to questions of Erdős. Most notably, we provide the first substantial improvement on the 25-year-old upper bounds of Thomason on the Ramsey multiplicity of K_4 and K_5 and we settle the minimum number of independent sets of size 4 in graphs with clique number at most 4. Motivated by the elusiveness of the symmetric Ramsey multiplicity problem, we also introduce an off-diagonal variant and obtain tight results when counting monochromatic K_4 or K_5 in only one of the colors and triangles in the other. The extremal constructions for each problem turn out to be blow-ups of a finite graph and were found through search heuristics. They are complemented by lower bounds and stability results established using Flag Algebras, resulting in a fully computer-assisted approach. More broadly, these problems lead us to the study of the region of possible pairs of clique and independent set densities that can be realized as the limit of some sequence of graphs. Joint work with Sebastian Pokutta, Christoph Spiegel and Tibor Szabó.
Tomáš
Roubíček
Charles University Prague and Inst. of Thermomechanics, Czech Acad. Sci.
Some gradient theories in linear visco-elastodynamics
towards dispersion and attenuation of waves in relation
to large-strain models
Abstract
Alessandro
Verra
Università Roma Tre
From Enriques surfaces to the Artin-Mumford counterexample
Abstract.
The talk deals with the multiple relations between Enriques surfaces and rationality problems. Artin-Mumford's counterexample to Lueroth's problem is revisited: the role of Enriques surfaces, the family of Reye congruences is emphasized and the 2-torsion cohomology of the threefold is geometrically reconstructed from that of these surfaces. The same construction extends to higher dimensions.
Sven
Wang
Humboldt-Universität zu Berlin
Rupert
Klein
FU
Wie die Mathematik zur Klimadebatte beitragen kann
Abstract.
Im Kontext der Klimadebatte wird die Mathematik oftmals lediglich als Lieferantin von Rechenmethoden gesehen. In Wirklichkeit ist sie deutlich breiter aufgestellt, wie drei Beispiele zeigen sollen: Die Klimaforschung arbeitet vielfach mit vereinfachten Gleichungen für Atmosphären- und Ozeansimulationen. Die Mathematik liefert rigorose Aussagen zur Gültigkeit solcher reduzierter Modelle und hilft so, die Klimaforschungsergebnisse abzusichern. Das Klima wird verkürzt als "30 jährige Wetterstatistik" definiert. Da aber die entsprechend eingesetzten statistischen Methoden zeitunabhängige Zufallsverteilungen annehmen, stellt sich die Frage, was dann unter "Klimawandel" überhaupt zu verstehen ist. Die mathematische Zeitreihenanalyse liefert hier ganz neue Ansatzpunkte. Im Schulterschluss suchen Sozial-, Wirtschafts- und Klimawissenschaften nach gemeinsamen Grundlagen für die Politikberatung. Dabei kommt es oft zu Missverständnissen aufgrund ihrer doch sehr unterschiedlichen Fachsprachen. Das Beispiel einer mathematischen Formalisierung des Begriffs der "Vulnerabilität bezüglich des Klimawandels" zeigt, wie Mathematik helfen kann, interdisziplinäre Diskurse zu strukturieren.
Luigi
Lombardi
Milano
On the invariance of Hodge numbers of irregular varieties under derived equivalence
Abstract.
A conjecture of Orlov predicts the invariance of the Hodge numbers of a smooth projective complex variety under derived equivalence. For instance this has been verified to the case of varieties of general type. In this talk, I will examine the case of varieties that are fibered by varieties of general type through the Albanese map. For this class of varieties I will prove the derived invariance of Hodge numbers of type \( h^{0,p}\), together with a few other invariants arising from the Albanese map. This talk is based on a joint work with F. Caucci and G. Pareschi.
Sophie
Puttkammer
HU Berlin
Notes on Morley FEM in 3D
Martin
Markl
Czech Academy of Sciences
Transfers of strongly homotopy structures as Grothendieck bifibrations
Abstract.
It is well-known that strongly homotopy structures can be transferred over chain homotopy equivalences. Using the uniqueness results of Markl & Rogers we show that the transfers could be organized into a discrete Grothendieck bifibration. An immediate application is e.g. functoriality up to isotopy.
Approximation Algorithms for Network Design Problems
Ekin
Ergen
and
Lizaveta
Manzhulina
TU Berlin
What is (integer) linear programming?
Abstract.
Below is the abstract initially proposed by Lizaveta Manzhulina. Due to illness, however, Ekin Ergen did kindly and on our very short notice take over the talk. A plentitude of highly relevant real-world problems can be modeled by means of linear programs (LP): optimization problems with a linear objective and constraints. Luckily, there are efficient algorithms for solving them. However, once we additionally require the variables to be integers --- which unsurprisingly is oftentimes so in applications --- things go downhill. Integer linear programming (ILP) is NP-hard, and thus to deal with it in practice heuristic methods and approximation algorithms are used. After convincing ourselves that (I)LP arises everywhere around us, we will explore the intricate relationship between LP and ILP. In particular, we are going to discuss how LP can be of help in tackling NP-hard ILP. Along the way, we will be accompanied by classical examples coming from network problems --- and from combinatorial optimization in general.
Hannes
Meinlschmidt
FAU Erlangen-Nürnberg
Renormalized solutions for optimal control of the drift in Fokker-Planck equations
Sandro
Roch
Coloring problems on arrangements of pseudolines
Yamaan
Attwa
FU Berlin
Erdős-Hajnal is true for an infinite family of prime graphs. Part II
Justin
Lanier
University of Sydney
Section problems for configuration spaces
Abstract.
Is there a continuous rule for adding a new distinct point to configurations of n distinct points in a given space X? For instance, when X is a closed ball and n is 1, the answer is no by Brouwer’s fixed-point theorem. This “add a point” problem can be phrased as a question about the existence of a continuous section of the forgetful map from the configuration space on X of n+1 points to the configuration space on X of n points. I will discuss the cases where the X is the plane, a closed ball, or a graph, treated respectively in work by Lei Chen, in my joint work with Lei Chen and Nir Gadish, and in work by Alexander Bauman.
Ali
El-Rahal
FU Berlin
Effiziente Synergien durch integrierte Prozessoptimierung – Bedarfsgerechter Einsatz von Produktionskapazitäten unter Berücksichtigung der partiellen Produktionssysteme
Ruben
Harris
FU Berlin
Ensemble Kalman Inversion for time-dependent forward operators
Mara
Belotti
TU Berlin
Tangency and point constraints in computational geometry
Abstract.
Wissenschaftliche Aussprache
Fredi
Tröltzsch
TU Berlin
Honorary Langenbach Seminar on the occasion of Jürgen Sprekels's 75th birthday
(flyer) :
Optimal control of partial differential equations - selected results and recent trends
Abstract
Kristian
Ranestad
University of Oslo
Quaternary quartic forms and Gorenstein rings
Abstract.
The Betti tables of their apolar rings give rise to a stratification of the space of quartic forms. The strata may be characterized by possible power sum decompositions and liftings to Calabi-Yau 3-folds. I shall report on work with G. And M. Kapustka, H. Schenk, M. Stillman and B. Yuan.
Tien
Tran Ngoc
Universität Augsburg
Lower eigenvalue bounds with hybrid high-order methods
Victor
Panaretos
EPFL Lausanne
Optimal transport for covariance operators
Abstract.
Covariance operators are fundamental in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen-Loève expansion. These operators may themselves be subject to variation, for instance in contexts where multiple functional populations are to be compared. Statistical techniques to analyse such variation are intimately linked with the choice of metric on covariance operators, and the intrinsic infinite-dimensionality and of these operators. I will describe how the geometry and tools of optimal transportation can be leveraged to construct natural and effective statistical summaries and inference tools for covariance operators, taking full advantage of the nature of their ambient space. Based on joint work with Valentina Masarotto (Leiden), Leonardo Santoro (EPFL), and Yoav Zemel (EPFL).
October 2023
Erwan
Rousseau
Brest
An Albanese construction for Campana's C-pairs
Abstract.
We will explain a construction of Albanese maps for orbifolds (or C-pairs), with applications to hyperbolicity such as a generalization of the Bloch-Ochiai theorem. (Joint with Stefan Kebekus).
Davide
Scazzuso
HU Berlin
Topological gravity, volumes and matrices
Abstract.
Jackiw-Teitelboim (JT) gravity is a simple model of two-dimensional quantum gravity that describes the low-energy dynamics of any near-extremal black hole and provides an example of AdS_2/CFT_1. In 2016 Saad, Shenker and Stanford showed that the path integral of JT gravity is computed by a Hermitian matrix model, by reinterpreting Mirzakhani's results on the volumes of moduli spaces of Riemann surfaces through the lenses of Eynard and Orantin's topological recursion. Thus, a beautiful threefold story connecting quantum gravity in two dimensions, random matrices and intersection theory emerged. In this talk I will review such connection from the point of view of physics and touch upon its generalization to N=1 JT supergravity and super Riemann surfaces.
Benedikt
Gräßle
HU Berlin
On nonconforming approximations for a class of semilinear problems
Alexandra
Wesolek
Cops and robber game on surfaces
Yamaan
Attwa
FU Berlin
Erdős-Hajnal is true for an infinite family of prime graphs. Part I
Moritz
Grillo
TU Berlin
Topological expressive power of ReLU neural networks
Abstract.
We study the expressivity of ReLU neural networks in the setting of a binary classification problem from a topological perspective. Recently, empirical studies showed that neural networks operate by changing topology, transforming a topologically complicated data set into a topologically simpler one as it passes through the layers. This topological simplification has been measured by Betti numbers. We use the same measure to establish lower and upper bounds on the topological simplification a ReLU neural network can achieve with a given architecture. We therefore contribute to a better understanding of the expressivity of ReLU neural networks in the context of binary classification problems by shedding light on their ability to capture the underlying topological structure of the data. In particular the results show that deep ReLU neural networks are exponentially more powerful than shallow ones in terms of topological simplification.
Sven
Wang
HU Berlin
On polynomial-time mixing for high-dimensional MCMC in inverse problems
Felix
Goldmann
HU Berlin
Unstetige Galerkinverfahren für das parabolische Hindernisproblem
Justin
Lanier
U Sydney
Mapping class groups and dense conjugacy classes
Abstract.
I’ll start by introducing mapping class groups and infinite-type surfaces—those with infinite genus or infinitely many punctures. One difference from the finite-type setting is that these mapping class groups come with natural non-discrete topologies. I’ll discuss joint work with Nick Vlamis where we fully characterize which surfaces have mapping class groups with dense conjugacy classes, so that there exists an element that well approximates every mapping class, up to conjugacy.
Volker
Kaibel
Otto-von-Guericke-Universität Magdeburg
Linear Programming and Diameters of Polytopes
Abstract.
We describe constructions of extended formulations that establish a certain relaxed version of the Hirsch-conjecture. Those constructions can be used to show that if there is a pivot rule for the simplex algorithm for which one can bound the number of steps by the diameters of the bases-exchange graphs of the polyhedra of feasible solutions then the general linear programming problem can be solved in strongly polynomial time. The talk is based on joint work with Kirill Kukharenko.
The Impact of Dormancy on the Evolutionary, Ecological and Pathogenic Properties of Microbial Populations
Abstract
Denis
Belomestny
Universität Duisburg-Essen
Provable benefits of policy learning from human preferences
Abstract.
A crucial task in reinforcement learning (RL) is a reward construction. It is common in practice that no obvious choice of reward function exists. Thus, a popular approach is to introduce human feedback during training and leverage such feedback to learn a reward function. Among all policy learning methods that use human feedback, preference-based methods have demonstrated substantial success in recent empirical applications such as InstructGPT. In this work, we develop a theory that provably shows the benefits of preference-based methods in tabular and linear MDPs. The main idea of our method is to use KL-regularization with respect to the learned policy to ensure more stable learning.
Sophie
Puttkammer
HU Berlin
An enriched Crouzeix-Raviart FEM for guaranteed lower eigenvalue bounds
Emanuel
Malek
HU Berlin
Exceptional generalised geometry and Kaluza-Klein spectra of string theory compactifications
Abstract.
Most interesting solutions of string theory are of the form M x C, where M is some D-dimensional non-compact space (e.g. Minkowski or Anti-de Sitter), and C is some (10-D)- or (11-D)-dimensional compact space, known as a compactification. Many interesting questions about string theory then reduce about understanding the properties of the 'Kaluza-Klein spectra' of certain differential operators on C. Because these operators often involve a complicated interplay between the p-forms arising in string theory and the metric on C, few general results are known. Generalised geometry is the study of structures on TM + T*M and similar extensions of TM, and naturally 'geometrises' the interaction between p-forms and metric in string theory. I will review generalised geometry and show how it allows us to study the Kaluza-Klein spectra for a large class of string theory compactifications.
Analysis of a variational contact problem arising in thermoelasticity
Abstract.
We study a model of a thermoforming process involving a membrane and a mould as implicit obstacle problems. Mathematically, the model consists of parabolic and elliptic PDEs coupled to a (quasi-)variational inequality. We study the related stationary (elliptic) model. We look at the existence of weak solutions, and by exploring the fine properties of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions. Under certain contraction conditions, we also show a uniqueness result. This is based on a joint paper with Jose-Francisco Rodrigues (Lisbon, Portugal) and Carlos N. Rautenberg (Virginia, USA).
Nadiia
Derevianko
Georg-August-Universität Göttingen
On Prony-type methods
Stefan
Felsner
Triangulations and partial triangulations: flips and counting
Angelina
Senchukova
LUT University
Edge-preserving inversion with heavy-tailed Bayesian neural networks priors
Regularity for non-smooth elliptic problems I
Abstract.
This is a series of three lectures on non-smooth problems, the first dedicated to elliptic ones and the third to parabolic ones. We start by explainig what means 'nonsmooth' and describe effects which cancel classical regularity results. Then we introduce a general geometric setting which allows in consequence to prove elliptic regularity results sufficient for attacking two- and three dimensional real world problems.Having this at hand, elliptic operators on the scale of (negatively indexed) Sobolev spaces and Lp spaces are introduced and their regularity properties are investigated, among them L∞ estimates, Hölder estimates and W 1,q estimates for the solution. After these preparations we pass to parabolic equations in the third and last lecture. Here the notion of 'maximal parabolic regularity' for an elliptic operator A is the central one - being now for some decades the ultimative instrument also for the investigation in particular of nonlinar problems. Since into this notion the domain of A enters explicitly, it becomes clear that an exact knowledge of dom(A), prepared in the first two lectures, is highly desirable. So, after having introduced maximal parabolic regularity and explained some of its properties, we show that second order divergence operators satisfy this property even if the domain is highly non-smooth, the coefficient function is only bounded, measurable and elliptic and the boundary conditions are mixed
Christian
Merdon
WIAS Berlin
Gradient-robust hybrid discontinuous Galerkin discretizations for the compressible Stokes equations
Abstract.
The talk introduces the concept of gradient-robustness for the velocity-density formulation of the compressible Stokes and its connection to the preservation of certain well-balanced states. Gradient-robust hybrid discontinuous Galerkin (HDG) discretisations of arbitrary order are discussed. The lowest-order scheme is shown to be non-negativity preserving and, at least in the isothermal case with linear equation of state, to be stable and provably convergent. The gradient-robustness property dramatically enhances the accuracy in well-balanced situations, such as the hydrostatic balance where the pressure gradient balances the gravity force, but also in non-hydrostatic cases for low Mach numbers and small viscosities. This is demonstrated in some numerical examples. (joint work with Philip Lederer)
Pedro
Tamaroff
HU Berlin
Differential operators of higher order and their homotopy trivializations
Abstract.
In the classical Batalin–Vilkovisky formalism, the BV operator is a differential operator of order two with respect to a commutative product; in the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a genus zero level cohomological field theory induced on homology. In this talk, we will explore generalisations of non-commutative Batalin-Vilkovisky algebras for differential operators of arbitrary order, showing that homotopically trivial operators of higher order also lead to interesting algebraic structures on the homology. This is joint work with V. Dotsenko and S. Shadrin.
Mazen
Ali
Fraunhofer-Institut ITWM, Kaiserslautern
Quantum Computing for Differential Equations and Surrogate Modeling
Abstract.
Quantum computing has transitioned from theoretical promise to practical reality, with multiple devices now accessible to the public. This technological evolution has catalyzed a multidisciplinary race to achieve the first 'quantum advantage,' drawing experts from fields as diverse as physics, computer science, finance, mathematics, and chemistry. However, despite the immense potential, the practical utility of current quantum computing implementations remains modest. Much of the research is concentrated on similar, easily-attainable goals, often accompanied by overstated claims and unwarranted optimism. Consequently, pivotal questions about the true nature of 'quantum advantage,' the roadmap to achieving it, and its fundamental relevance remain Our focus is on harnessing the capabilities of quantum computing for material simulations at the macroscopic scale. In this presentation, I will offer an overview of the current state of quantum computing, discuss methodologies for solving differential equations directly on quantum platforms, and explore the use of quantum machine learning to create surrogate models for complex systems.
Matthew
Maat
U Twente
The connection between linear programs and games
Abstract.
The simplex algorithm is a well-known combinatorial algorithm that is commonly used to solve linear programs (LPs). It requires a pivot rule to decide which steps it takes. Despite the algorithm being fast in practice, for many different pivot rules it has been shown that in the worst case the number of steps may be exponential, and it is not known whether there exists a pivot rule with better worst-case bounds. In order to gain more understanding of the worst case behaviour, we consider a class of LPs that come from games, specifically mean payoff games and parity games. We show that strategy improvement (a well-known combinatorial algorithm for solving these games) is equivalent to the simplex method in the LP formulation. We use this result to derive lower bounds for pivot rules that only use certain combinatorial information. Finally, we shortly discuss combinatorial types for these types of games.
Rimma
Hämäläinen
Few Slopes Without Collinearity
Sayan
Mukherjee
MPI-MiS Leipzig
Modeling shapes and surfaces - Geometry meets machine learning
Abstract.
We will consider modeling shapes and fields via topological and lifted-topological transforms. Specifically, we show how the Euler Characteristic Transform and the Lifted Euler Characteristic Transform can be used in practice for statistical analysis of shape and field data. We also state a moduli space of shapes for which we can provide a complexity metric for the shapes. We also provide a sheaf theoretic construction of shape space that does not require diffeomorphisms or correspondence. A direct result of this sheaf theoretic construction is that in three dimensions for meshes, 0-dimensional homology is enough to characterize the shape. We will also discuss Gaussian processes on fiber bundles and applications to evolutionary questions about shapes. Applications in biomedical imaging and evolutionary anthropology will be stated throughout the talk.
Reinhard
Scholz
Leibniz-Institut für Polymer-Forschung Dresden
Simulation amphiphiler Polymernetzwerke
Abstract
Rahul
Pandharipande
ETH Zürich
The Gromov-Witten/Donaldson-Thomas correspondence, Hilbert schemes of the affine plane and the moduli of abelian varieties
Abstract.
I will explain how these three directions of study are fundamentally linked.
September 2023
Felix
Schröder
Edge density of RAC graphs
Ariel
Liebman
Monash University
Why have even less time than you think on Climate Action, and why Optimisation and Artificial Intelligence are critical to the solution?
Abstract.
Computer science, software engineering and artificial intelligence are essential for climate change mitigation, and no hallucinations with GPT (generative pre-trained transformer) needed. With the world needing to halve greenhouse gas emissions by 2030, and reduce these emissions by a further 80% by 2040, the climate science based 1.5°C Paris Agreement trajectory seems to become more and more difficult to achieve. Despite the challenges, there’s hope on the horizon with mass-scale renewable technologies becoming competitive against fossil sources, even in places such as Australia, where coal and natural gas can be produced cheaper than in most other parts of the world. When combined with technologies such as batteries, rooftop solar photovoltaic and electric vehicles, we have a complete set of “solution blocks” for mass decarbonisation. However, while we have many forms of renewable technologies to address the climate situation, integrating all these “solution blocks” together and into the existing grid infrastructure is challenging. This is where artificial intelligence and its full range of forms comes into play.
Akiko
Takeda
University of Tokyo
Applying random projection techniques to nonconvex optimization problems
Abstract.
Random projection techniques based on the Johnson-Lindenstrauss lemma are used for randomly aggregating the constraints or variables of optimization problems while approximately preserving their optimal values, which leads to smaller-scale optimization problems. In this talk, we show the following three applications of random matrix techniques for constructing smaller-scale optimization problems or for constructing random subspace algorithms that iteratively solve smaller-scale subproblems. 1. We use this technique for nonconvex quadratic optimization problems (NQOPs) to find approximate global optimal solutions by solving convex optimization problems of smaller dimensions. This takes advantage of the fact that the nonconvexity of an NQOP is mitigated by random projection. 2. In order to obtain approximate stationary points to more general nonconvex optimization problems, we propose an algorithm that iteratively solves smaller dimensional subproblems and evaluate the convergence speed of the algorithm. 3. We will also report our research results on random subspace algorithms for constrained optimization problems. This talk is based on joint works with Terunari Fuji, Ryota Nozawa, and Pierre-Louis Poirion.
Ievgen
Makedonskyi
BIMSA
Semi-infinite Pluecker relations and the snake formula
Abstract.
We study a certain infinitization of the Pluecker algebra. This is the so-called arc ring of the classical algebra generated by minors of a matrix and it also has a natural representation-theoretical meaning. We prove that this ring is defined by relations of degree 2 and give a combinatorial formula of the graded components of this algebra. We call it the ''snake formula''. I will explain the method used to study this ring. It is an arc analogue of the standard monomial theory and can be applied to a wide family of arc rings.
Julian
Sampels
Zählen von Elimination Trees
Ji Hoon
Chun
TU Berlin
Finite Sphere Packings and the Sausage Conjecture
Abstract.
The Sausage Conjecture of L. Fejes Tóth (1975) states that for all dimensions $d \geq 5$, the densest packing of any finite number of spheres in $\mathbb{R}^{d}$ occurs if and only if the sphere centers are all placed as closely as possible on one line, i.e., a 'sausage.' We discuss the progress made in the literature, including the result of Betke and Henk (1998) that the Sausage Conjecture is true for all $d \geq 42$. Our work builds upon the methods of Betke and Henk to improve the lower bound to $d \geq 36$ with the aid of interval arithmetic for certain complicated portions. Joint work with Martin Henk.
David
Steurer
ETH Zurich
Planted cliques, robust inference, and sum-of-squares polynomials
Abstract.
We design new polynomial-time algorithms for recovering planted cliques in the semi-random graph model introduced by Feige and Kilian as a proxy for robust inference. The previous best algorithms for this model succeed if the planted clique has size at least n<sup>2/3</sup> in a graph with n vertices. Our algorithms work for planted-clique sizes approaching n<sup>1/2</sup> — the information-theoretic threshold in the semi-random model (Steinhard 2017) and a conjectured computational threshold even in the easier fully-random model. This result comes close to resolving open questions by Feige and Steinhardt. Our algorithms rely on a new conceptual connection between planted cliques in the semi-random graph model and certificates of upper bounds on unbalanced biclique numbers in Erdős–Rényi random graphs. We show that higher-degree sum-of-squares polynomials allow us to obtain almost tight certificates of this kind. Based on a joint work with Rares-Darius Buhai and Pravesh K. Kothari.
Julian
Pfeifle
Different embeddings of the Grassmannian, applied to realization problems
August 2023
Likhit
Ganedi
RWTH Aachen University
A Vector Field Model for Disclinations in Nematic Liquid Crystals
Nicholas
Nelsen
Caltech
Analysis of vector-valued random features
A proximal trust-region method for nonsmooth optimization with inexact function and gradient evaluations
Abstract.
We develop a novel trust-region method to minimize the sum of a smooth nonconvex function and a nonsmooth convex function. This class of problems that is ubiquitous in data science, learning, optimal control, and inverse problems. Our method is unique in that it permits and systematically controls the use of inexact objective function and derivative evaluations inherent in large-scale system solves and compression techniques, e.g. randomized sketching. When using a quadratic Taylor model for the trust-region subproblem, our algorithm is an inexact, matrix-free proximal Newton-type method that permits indefinite Hessians. We prove global convergence of our method in Hilbert space and elaborate on potential nonsmooth subproblem solvers based on ideas taken from their smooth counter-parts. Under additional assumptions, we can also prove superlinear, or even quadratic convergence to local minima. We demonstrate its efficacy on examples from data science and PDE-constrained optimization.
Amir
Maghoul
Investigation of continuum limits in adaptive dynamical networks
Robert
Lauff
Facet guarding cycles in graph associahedra
Fabian
Lenzen
TU München
Computation of two-parameter persistent (co)homology
Abstract.
Persistent homology is a central tool in topological data analysis (TDA) that seeks to understand the topological structure of data. Given a filtered topological space, persistent homology tracks the changes in the homology of that space along the filtration, and assigns to the filtration an invariant called the barcode. It can be efficiently computed, and many different implementations are available. However, persistent homology has some shortcomings. Most notably, it is susceptible to outliers. A possible remedy is seen in multi-parameter persistent homology, which tracks the changes in homology of a space filtered in several parameters. However, multi-parameter persistent homology is much harder to compute. In this talk, I explain a common approach for the computation of a minimal free resolution of persistent homology in the special case of two-parameter persistence. Motivated by observations from (ordinary) one-parameter persistence, I will show how computation of cohomology (instead of homology) can be used to obtain other efficient algorithms for the computation of minimal free resolutions of one-parameter persistence.
Chaoliang
Tang
Fudan University
Turán number of the linear 3-graph Crown
Abstract.
Let the crown C13 be the linear 3-graph on 9 vertices {a,b,c,d,e,f,g,h,i} with edges E = {{a,b,c}, {a, d,e}, {b, f, g}, {c, h,i}}. Proving a conjecture of Gyárfás et. al., we show that for any crown-free linear 3-graph G on n vertices, its number of edges satisfy |E(G)| ≤ 3(n - s)/2, where s is the number of vertices in G with degree at least 6. This result, combined with previous work, essentially completes the determination of linear Turán number for linear 3-graphs with at most 4 edges.The result is the joint work with my tutor Hehui Wu and my fellow Zeyu Zheng. It is also simutaneously proved by Shengtong Zhang.
Antony
Della Vecchia
TU Berlin
A FAIR File Format for Mathematical Software
Abstract.
Due to the complexity of mathematical objects, storing data so it is FAIR has its challenges. We describe a generic \texttt{JSON} based file format which is suitable for computations in computer algebra, highlighting some design decisions and demonstrating some of the benefits with an emphasis on interoperability and reproducibility. This is implemented in the computer algebra system \texttt{OSCAR}, but we also show how it can be used in a different context. This is joint work with Michael Joswig and Benjamin Lorenz.
Michael
Joswig
TU Berlin
Smooth Fano Polytopes and Their Triangulations
Abstract.
The classification of the d-dimensional simplicial, terminal, and reflexive polytopes with at least 3d-2 vertices is presented. It turns out that all of them are smooth Fano polytopes. This builds on and improves previous results of Casagrande (2006) and Obro (2008). The second half of the presentation is concerned with triangulations of such polytopes. Joint work with Benjamin Assarf, Andreas Paffenholz and Julian Pfeifle.
July 2023
Ekin
Ergen
Total completion time scheduling under scenarios
Claudia
Yun
MPI Leipzig
Amalgamating groups via linear programming
Abstract.
A compact group $A$ is called an amalgamation basis if, for every way of embedding $A$ into compact groups $B$ and $C$, there exist a compact group $D$ and embeddings $B\to D$ and $C\to D$ that agree on the image of $A$. Bergman in a 1987 paper studied the question of which groups can be amalgamation bases. A fundamental question that is still open is whether the circle group $S^1$ is an amalgamation basis in the category of compact Lie groups. Further reduction shows that it suffices to take $B$ and $C$ to be the special unitary groups. In our work, we focus on the case when $B$ and $C$ are the special unitary group in dimension three. We reformulate the amalgamation question into an algebraic question of constructing specific Schur-positive symmetric polynomials and use integer linear programming to compute the amalgamation. We conjecture that $S^1$ is an amalgamation basis based on our data. This is joint work with Michael Joswig, Mario Kummer, and Andreas Thom.
Esther
Müller
Universal tree-based phylogenetic networks with minimal number of reticulations
Felix
Schröder
A new Lemma for proving density bounds of beyond-planar graphs
Uncertainty quantification for models involving hysteresis operators
Abstract.
Parameters within models involving hysteresis operators that are supposed to describe with real world objects like, e.g. magneto mechanical devices, have to be identified from measurements. Hence, they are subject to corresponding errors. The methods of Uncertainty Quantification (UQ) are applied to investigate the influence of these errors. As an example, results of forward UQ for a play operator with uncertain yield limit will be presented. Afterwards, the model for a magneto mechanical devices involving a generalized Prandtl-Ishlinskiĭ operator considered in Sec. 5 in Davino-Krejčí--Visone-2013, Fully coupled modeling of magneto-mechanical hysteresis through `thermodynamic' compatibility. Smart Mater. Struct. https://doi.org/10.1088/0964-1726/22/9/095009 will be considered. Starting from data used to generated a First-Order-Reversal-Curves (FORC)-diagram inverse UQ is performed by formulating appropriate Bayesian Inverse Problems (BIPs) and applying Bayes' Theorem. The density of the resulting posterior density is represented by samples resulting from MCMC-computations using UQLab, the “The Framework for Uncertainty Quantification”, see https://www.uqlab.com/. Afterwards, forward UQ is performed and the results are compared to measurements. These are results of a joined work with Carmine Stefano Clemente and Daniele Davino of the Università degli Studi del Sannio, Benevento, Italy and Ciro Visone of Università di Napoli Federico II, Napoli, Italy, see also: K.-Davino-Visone-2020, On forward and inverse uncertainty quantification for models involving hysteresis operators, Math. Model. Nat. Phenom 15, https://doi.org/10.1051/mmnp/2020009 and Clemente-Davino-K.-Visone-2023, Forward and Inverse Uncertainty Quantification for a model for a magneto mechanical device involving a hysteresis operator, WIAS Preprint 3009
Tatiana
Levinson
FU Berlin
Transversal Nandakumar & Ramana Rao problem with the glimpse of algebraic topology side
Abstract.
In 2006 Nandakumar asked whether, for a given polygon on a plane and an integer m, it is always possible to find a partition of this polygon into m convex pieces of equal area and perimeter. The problem turned out to be surprisingly hard - almost two decades later the full answer is not known. On the other hand, this question allows many interesting generalisations, for example by asking the same question for a d-polytope, and partitions that equalise volume and some d-1 continuous functions on the space of full-dimensional compact convex bodies in R^d. In this talk, we will discuss the original Nandakumar & Ramano Rao problem and will see how an application of algebraic topology methods allowed Karasev, Hubard & Aronov and Blagojevic & Ziegler to find a partial solution. Then we will consider its transversal generalisation, in which we aim to equipart more than d-1 functions by choosing a suitable polytope from some family of d-polytopes. We will examine the new challenges it brings to the algebraic topology side of the problem and some ways to overcome them. We will also discuss the inherent limitations of the algebraic approach to this problem and the questions about the polytopes it gives rise to. This talk is based on my PhD thesis.
Jonathan
Spreer
University of Sydney
An algorithm to compute the crosscapnumber of a knot
Abstract.
The crosscap number of a knot is the non-orientable counterpart of its genus. It is defined as the minimum of one minus the Euler characteristic of S, taken over all non-orientable surfaces S bounding the knot. Computing thecrosscap number of a knot is tricky, since normal surface theory - the usualtool to prove computability of problems in 3-manifold topology, does notdeliver the answer 'out-of-the-box'.In this talk, I will review the strengths and weaknesses of normal surfacetheory, focusing on why we need to work to obtain an algorithm to computethe crosscap number. I will then explain the theorem stating that an algorithm due to Burton and Ozlen can be used to give us the answer.This is joint work with Jaco, Rubinstein, and Tillmann.
Vsevolod
Shevchishin
U Olsztyn
Lagrangian tori and exceptional symplectic spheres in symplectic 4-manifolds
Abstract
Guillermo
Olicón Méndez
FU Berlin
A random dynamical system perspective on chemical reaction networks
Lassi
Roininen
LUT University
Bayesian inversion with alpha-stable priors
Iris
Rammelmüller
Universität Klagenfurt
Ensemble-based Data Assimilation for high-dimensional nonlinear dynamical systems
Silas
Rathke
FU Berlin
The asymptotics of R(4,t)
Abstract.
The Ramsey number R(4,t) is the minimum number n of vertices such that each 2-colouring of K_n either contains a red K_4 or a blue K_t. Erdős conjectured that this should essentially grow with the order t^3. For forty years, there has not been any progress on that problem. Until last month, when Mattheus and Verstraete proved this conjecture.In the talk, I will present their proof. It combines a concrete algebraic construction with some probabilistic arguments.
V.
Krizhanovski
Vasyl' Stus Donetsk National University
Some mathematical problems in the practice of radio physics and electronics
Giacomo
Nannicini
University of Southern California
Solving the semidefinite relaxation of QUBOs in matrix multiplication time, and faster with a quantum computer
Abstract.
Recent works on quantum algorithms for solving semidefinite optimization (SDO) problems have leveraged a quantum-mechanical interpretation of positive semidefinite matrices to develop methods that obtain quantum speedups with respect to the dimension n and number of constraints m. While their dependence on other parameters suggests no overall speedup over classical methodologies, some quantum SDO solvers provide speedups in the low-precision regime. We exploit this fact to our advantage, and present an iterative refinement scheme for the Hamiltonian Updates algorithm of Brandão et al. [Faster quantum and classical SDP approximations for quadratic binary optimization, Quantum 6, 625 (2022)] to exponentially improve the dependence of their algorithm on the precision 𝜖, defined as the absolute gap between primal and dual solution. As a result, we obtain a classical algorithm to solve the semidefinite relaxation of Quadratic Unconstrained Binary Optimization problems (QUBOs) in matrix multiplication time. Provided access to a quantum read/classical write random access memory (QRAM), a quantum implementation of our algorithm exhibits O(ns + n<sup>1.5</sup> polylog(n, ‖C‖<sub>F</sub>, 1/𝜖)) running time, where C is the cost matrix, ‖C‖<sub>F</sub> is its Frobenius norm, and s is its sparsity parameter (maximum number of nonzero elements per row).
Sandro
Roch
A general lower bound on the mixing time of Glauber dynamics
Anton
Dochtermann
Texas State University
Parking function ideals on generalized permutohedra
Abstract.
Suppose G is a finite graph with specified sink vertex q. The `G-parking functions' (relative to q) are combinatorial objects with connections to algebra (eg power ideals) and combinatorics (eg chip-firing on G). As studied by Postinov and Shapiro, one way to define a G-PF is as a standard monomial of a certain monomial ideal M_G associated with G. It turns out that the generators of M_G are described by the facets of the corresponding zonotope Z_G that are visible when one views Z_G `from the direction q'. Furthermore the facial structure of Z_G describes a minimal cellular resolution of M_G. We extend these constructions to generalized permutohedra P given by sums of simplices. Viewing P from a sink direction, we again obtain a monomial ideal generated by the visible facets. The algebraic results seem to generalize nicely, and can even be extended to mixed subdivisions of P. On the combinatorial side, the standard monomials of this ideal lead to notions of chip-firing on hypergraphs, where many questions remain. This is joint work with Ayah Almousa and Ben Smith, as well as an REU program co-advised with Suho Oh.
Sai Ganesh
Nagarajan
EPFL
The Advantages of Parameterized Seeding in k-means++
Abstract.
Clustering using k-means is a classic problem with significant practical implications. The elegant k-means++ algorithm, proposed by Arthur and Vassilvitskii [k-means++: the advantages of careful seeding SODA 2007,1027–1035], is one of the most popular approaches for solving it and is a O(log k)-approximation. However, some limitations of this algorithm have been identified, specifically in cases where the data is highly clusterable. Recently, Balcan et al. [Data-Driven Clustering via Parameterized Lloyd's Families NeurIPS 2018, 31, 10641–10651] explored a new data-driven approach to overcome these limitations. They proposed to learn a parameter 𝛼 in order to parameterize the seeding as follows: a point is selected as a cluster center with probability proportional to the 𝛼-powered distance from the point to its closest center selected thus far. The standard k-means++ is then the particular case of 𝛼=2. While it is known that selecting 𝛼≠2 may lead to a worse guarantee in the worst case, they qualitatively and experimentally show the advantage of this parameterized seeding in several settings. In this talk, I will describe the analysis of (standard) k-means++ and how this leads to a general analysis of parameterized seeding in k-means++, through a carefully constructed potential function. Using this potential, I will describe how the approximation guarantee (for any 𝛼 > 2) depends on parameters that captures the concentration of points of any optimal cluster, the ratio of the maximum number of points to the minimum number of points in the optimal clusters, and the ratio of the maximum and the minimum variance of the set of optimal clusters. As a corollary, we obtain that seeding with 𝛼 > 2 results in a constant factor approximation guarantee on a wide class of instances. Finally, I will provide examples that necessitates the dependence on some of the aforementioned parameters. This is joint work with Etienne Bamas (EPFL->ETHZ) and Ola Svensson (EPFL).
Norbert
Heuer
PUC Chile
Conforming Galerkin schemes via traces and applications to plate bending -- Teil 2
Jan Hendrik
Bruinier
TU Darmstadt
Special cycles on toroidal compactifications of orthogonal Shimura
varieties
Abstract.
A famous theorem of Gross-Kohnen-Zagier states that the generating series of Heegner divisors on a modular curve is a weight 3/2 modular form with values in the first Chow group. An analogous result for special divisors on orthogonal Shimura varieties was proved by Borcherds, and for higher codimension special cycles by Zhang, Raum and myself. We report on joint work with Shaul Zemel generalizing the modularity result to special divisors on toroidal compactifications of orthogonal Shimura varieties.
Anand
Srivastav
Christian-Albrechts-Universität zu Kiel
A new Bound for the Maker-Breaker Triangle Game
Anand
Srivastav
Kiel
Recent Advances in the Maker Breaker Subgraph Game
Abstract.
The triangle game introduced by Chvátal and Erdős (1978) is one of the old and famous combinatorial games. For n, q ∈ N, the (n,q)-triangle game is played by two players, called Maker and Breaker, on the complete graph K_n .Alternately Maker claims one edge and thereafter Breaker claims q edges of the graph. Maker wins the game if he can claim all three edges of a triangle. Otherwise Breaker wins. Chvátal and Erdős (1978) proved that for q < sqrt(n/2), Maker has a winning strategy, while for q > 2 sqrt(n), Breaker wins. So, the threshold bias must be in the interval [sqrt(1/2)sqrt(n) , 2 sqrt(n)].Since then, the problem of finding the exact constant (and an associated Breaker strategy) for the threshold bias of the triangle game has been one of the interesting open problems in combinatorial game theory. In fact, the constant is not known for any graph with a cycle and we do not even know if such a constant exists. Balogh and Samotij (2011) slightly improved the Chvátal-Erdős constant for Breaker’s winning strategy from 2 to 1.935 with a randomized approach. Thereafter, no progress was made. In this work, we present a new deterministic strategy for Breaker leading to his win if q > sqrt(8/3) sqrt(n), for sufficiently large n. This almost matches the Chvátal-Erdős bound of sqrt(1/2)sqrt(n) for Maker's win (Glazik, Srivastav, Europ. J. Comb. 2022).In contrast to previous (greedy) strategies we introduce a suitable non-linear potential function on the set of nodes. By keeping the potential small, Breaker picks edges that neutralize the most ‘dangerous’ nodes with incident Maker edges blocking Maker triangles. A characteristic property of the dynamics of the game is that the total potential is not monotone decreasing. In fact, the total potential of the game may increase, even for several turns, but finally Breaker’s strategy prevents the total potential of the game from exceeding a critical level, which results in Breaker’s win. We further survey recent results for cycles of length k, and a general potential function theorem (Sowa, Srivastav 2023). This is joint work with Christian Glazik, Christian Schielke and Mathias Sowa, Kiel University.
A semismooth Newton solver with automatic differentiation written in C++
Abstract.
In this talk we consider problems of the form F(x)=0 where F is a nonlinear Newton differentiable mapping between Solbolev spaces. It is well-known that a semismooth Newton method ensures local superlinear convergence towards a solution. The function spaces are discretized by suitable finite elements over a given grid. A major difficulty is the practical implementation of generalized Jacobians. To this end, we present automatic differentiation techniques to obtain discrete subgradients of F. The resulting sparse linear problems are solved by efficient linear solvers. The framework is easy to use and to implement: The user only needs to implement a local evaluation of F in the weak form for a given set of test functions. An example implementation is given for a thermoforming model from a recent paper. To verify the solver, the results of this model are reproduced.
Dongho
Chae
Chung-Ang University, Seoul
On the Liouville type problem in the fluid mechanics
Dennis
Chemnitz
FU Berlin
What is the blow-up method?
Abstract.
Many dynamical systems have interesting dynamics at several time scales (think weather vs climate). The resulting phenomena can be difficult to capture using classical ODE-theory and thus new tools are required. The goal of this talk is to introduce fast/slow systems, geometric singular perturbation theory and the blow-up method. The Van-der-Pol oscillator will serve as a guiding example.
Benedikt
Gräßle
HU Berlin
Stabilization-free a posteriori error analysis for hybrid-high order methods
Ilja
Klebanov
FU Berlin
On definitions of modes and MAP estimators
Jonas
Ketteler
Uni Leipzig
Some ideas for the quasi-orthogonality for the Fortin-Soulie FEM
Lara
Theallier
HU Berlin
HHO for linear elasticity
Christian
Kuehn
TUM
Multiscale Dynamics: From Finite to Infinite
Carsten
Carstensen
HU Berlin
Lower eigenvalue bounds of the Laplacian
Mira
Schedensack
Uni Leipzig
Discrete Helmholtz decompositions
Helena
Bergold
FU Berlin
An Extension Theorem for Signotopes
Abstract.
In 1926, Levi showed that, for every pseudoline arrangement A and two points in the plane, A can be extended by a pseudoline which contains the two prescribed points. Later extendability was studied for arrangements of pseudohyperplanes in higher dimensions. While the extendability of an arrangement of proper hyperplanes in R^d with a hyperplane containing d prescribed points is trivial, Richter-Gebert found an arrangement of pseudoplanes in R^3 which cannot be extended with a pseudoplane containing two particular prescribed points. In this talk, we investigate the extendability of signotopes, which are a combinatorial structure encoding a rich subclass of pseudohyperplane arrangements. We show that signotopes of odd rank are extendable in the sense that for two prescribed crossing points we can add an element containing them.
Patrick
Morris
Universitat Politècnica de Catalunya
The canonical Ramsey theorem in random graphs: even cycles and list constraints
Abstract.
The celebrated canonical Ramsey theorem of Erdős and Rado implies that for a given graph H, if n is sufficiently large then any colouring of the edges of Kn gives rise to copies of H that exhibit certain colour patterns, namely monochromatic, rainbow or lexicographic. We are interested in sparse random versions of this result and the threshold at which the random graph G(n,p) inherits the canonical Ramsey properties of Kn. We will discuss a theorem that pins down this threshold when we focus on colourings that are constrained by some prefixed lists and also a related result on the threshold for the canonical Ramsey property (with no list constraints) in the case that H is an even cycle.This represents joint work with José D. Alvarado, Yoshiharu Kohayakawa and Guilherme O. Mota.
Stephanie
Mui
NYU
The $L^{p}$ dual Minkowski problem: an overview and new results for
$p<0$
Abstract.
The $L^{p}$ dual curvature measure was introduced by Lutwak, Yang, and Zhang in 2018. The associated Minkowski problem, known as the $L^{p}$ dual Minkowski problem, asks about existence of a convex body with prescribed $L^{p}$ dual curvature measure. This question unifies the previously disjoint $L^{p}$ Minkowski problem with the dual Minkowski problem, two open questions in convex geometry. Among its important cases are the classical Minkowski problem, the Aleksandrov problem, and the log Minkowski problem. The case of negative $p$ is still largely unsolved, with the centro-affine Minkowski problem included in this region. We have obtained existence results assuming origin-symmetry for $-1 < p < 0$, $q < 1+p$, $p\neq q$. In this talk, we will also discuss the extensive history of the problem, critical cases, and more existence results in the negative $p$ region assuming bounded absolute continuity of the given data.
Rahul
Pandharipande
ETH Zürich
Beyond the tautological ring of the moduli of curves
Yuri
Tschinkel
New York University
Birational types
Abstract.
I will discuss joint work with Chambert-Loir, Kontsevich, and Kresch on new invariants in birational geometry.
Günter M.
Ziegler
FU
Das 24-Zell
Abstract.
Der Würfel und das Oktaeder sind einfache und gewöhnliche 3-dimensionale Objekte. Es gibt auch entsprechende Objekte in der 4-dimensionalen Geometrie, die man geometrisch, kombinatorisch und algebraisch beschreiben und untersuchen kann.In diesem Vortrag soll es darum gehen, ein weiteres, außergewöhnliches und in verschiedenster Weise einzigartiges Objekt kennenzulernen, das wohl zunächst von Ludwig Schläfli in der Mitte des 19. Jahrhunderts entdeckte „24-Zell“: Wir beschreiben es ebenfalls geometrisch, kombinatorisch und algebraisch, stellen dann aber außergewöhnliche Eigenschaften fest (etwa dass das 24-Zell „selbst-dual“ ist und mit bemerkenswerten Kugelanordnungen zusammenhängt) und stoßen dann schnell auf auch unbeantwortete Fragen (etwa nach Deformationen des 24-Zells, die die Kombinatorik aber nicht die Geometrie erhalten).
Hsueh-Yung
Lin
National Taiwan University
Motivic invariants of birational maps and Cremona groups
Abstract.
(Joint with E. Shinder) In characteristic zero, birational maps of projective varieties factorize through a sequence of blow-ups and blow-downs along smooth centers. We study to which extent these factorization centers are unique, and construct invariants of motivic nature which account for the non-uniqueness of centers. For surfaces over a perfect field, we prove (with E. Shinder and S. Zimmermann) the uniqueness of centers in the strongest possible sense. In higher dimension, we construct examples showing that the centers fail to be unique. Relying on the non-uniqueness, we provide new explanations of the non-simplicity of Cremona groups.
Dante
Luber
Technische Universität Berlin
Initial degenerations of Grassmannian via matroid subdivisions of hypersimplices
Abstract.
Nonempty initial degenerations of the Grassmannian are induced by weight functions in its tropicalization. On the other hand, the same weight function induces a regular matroidal subdivision of the hypersimplex. Hence, we study initial degenerations of the (3,8) Grassmannian via matroidal subdivisions of the (3,8) hypersimplex. Our techniques employ tropical, algebraic, and polyhedral geometry, as well as matroid theory, commutative algebra, and computation.
Minh Ha
Quang
RIKEN
An optimal transport and information geometric framework for Gaussian processes
Abstract.
Information geometry (IG) and Optimal transport (OT) have been attracting much research attention in various fields, in particular machine learning and statistics. In this talk, we present results on the generalization of IG and OT distances for finite-dimensional Gaussian measures to the setting of infinite-dimensional Gaussian measures and Gaussian processes. Our focus is on the Entropic Regularization of the 2-Wasserstein distance and the generalization of the Fisher-Rao distance and related quantities. In both settings, regularization leads to many desirable theoretical properties, including in particular dimension-independent convergence and sample complexity. The mathematical formulation involves the interplay of IG and OT with Gaussian processes and the methodology of reproducing kernel Hilbert spaces (RKHS). All of the presented formulations admit closed form expressions that can be efficiently computed and applied practically. The mathematical formulations will be illustrated with numerical experiments on Gaussian processes.
June 2023
Ruben
Harris
FU Berlin
Improving Ensemble Kalman Filter performance by adaptively controlling the ensemble
Robert
Lauff
Facet guarding cycles in graph associahedra
Eric
Larson
Brown University
Interpolation for Brill-Noether Curves
Abstract.
In this talk, we determine when there is a Brill-Noether curve of given degree and given genus that passes through a given number of general points in any projective space.
Per
Salberger
U Gothenburg
On n-torsion in class groups of number fields
Abstract.
It is well known that the class group of a number field is of size bounded above by roughly the square root of its discriminant. But one expects by conjectures of Cohen-Lenstra that the the n-torsion part of this group should be much smaller and there have recently been several papers on this by prominent mathematicians. We present in our talk some of these results and a partial improvement of an estimate of Bhargava, Shankar, Taniguchi, Thorne, Zimmerman and Zhao.
Tibor
Szabó
Freie Universität Berlin
Topology at the North Pole
Abstract.
In the max-min allocation problem a set P of players are to be allocated disjoint subsets of a set R of indivisible resources, such that the minimum utility among all players is maximized. We study the restricted variant, also known as the Santa Claus problem, where each resource has an intrinsic positive value, and each player covets a subset of the resources. Bezakova and Dani showed that this problem is NP-hard to approximate within a factor less than 2, consequently a great deal of work has focused on approximate solutions. The principal approach for obtaining approximation algorithms has been via the Configuration LP (CLP) of Bansal and Sviridenko. Accordingly, there has been much interest in bounding the integrality gap of this CLP. The existing algorithms and integrality gap estimations are all based one way or another on the combinatorial augmenting tree argument of Haxell for finding perfect matchings in certain hypergraphs. Here we introduce the use of topological tools for the restricted max-min allocation problem. This approach yields substantial improvements in the integrality gap of the CLP. In particular we improve the previously best known bound of 3.808 to 3.534. The talk represents joint work with Penny Haxell.
Jean-Michel
Coron
U Pierre-et-Marie-Curie
Boundary stabilization of 1-D hyperbolic systems
Guillermo
Pineda-Villavicencio
Deakin University
Some results and conjectures on convex polytopes: from edge-connectivity of their graphs to lower bound conjectures on their faces
Abstract.
In the talk, we will first discuss a recent theorem on edge cuts of minimal cardinality in the graph of a simplicial d-polytope of dimension d≥3: such a graph has no nontrivial minimum edge cut with fewer than d(d+1)/2 edges, hence the graph is min{δ,d(d+1)/2}-edge-connected where δ denotes the minimum degree. When d=3, this implies that every minimum edge cut in a plane triangulation is trivial. When d≥4, we construct a simplicial d-polytope whose graph has a nontrivial minimum edge cut of cardinality d(d+1)/2, proving that the aforementioned result is best possible. This is a joint work with Julien Ugon (Deakin University) and Vincent Pilaud (CNRS & LIX, École Polytechnique) In the second part of the talk, we will propose two Lower bound conjectures: one for d-polytopes with at most 3d-1 vertices and another for d-polytopes whose facets are all combinatorially isomorphic to a pyramid over a (d-2)-cube. Finally, if time permits, I will explore the notion of neighbourly d-polytopes beyond the simplicial and cubical case.
Philipp
Guth
RICAM
Optimal Control and Feedback Stabilization Under Uncertainty
Sebastian
Manecke
Goethe Universität Frankfurt
The twisted-incidence algebra of angles
Abstract.
The study of valuations emerged from Hilberts 3rd problem and has since been a useful tool to understand various dissection problems. In this talk we consider valuations on polyhedral cones, examples of which arise in convex and polyhedral geometry as well as algebraic combinatorics. Such valuations can be used to measure angles between faces of polyhedra and thus angles are naturally an element in the incidence algebra of face lattices. Adding a twist to the convolution product turns the set of angles into an subalgebra of the (twisted-)incidence algebra. This algebra provides a very general and unified framework for dissection statements on cones and can be used to recover old and obtain new dissections with simple proofs.
Theo
Grundhöfer
U Würzburg
Groups and nearfields
Abstract.
Nearfields are easy to define: just omit one of the two distributive laws in the definition of skew fields. We explain the connection of nearfields to sharply 2-transitive permutation groups. Every nearfield has an intrinsic vector space structure, hence one can study nearfields in terms of groups of linear transformations. All finite nearfields have been classified by Zassenhaus in 1936. We report on classification results for infinite nearfields, and we construct some `wild´ nearfields which are far away from skew fields.
Michael
McBreen
The Chinese University of Hong Kong
Introduction to microlocal sheaves (Lecture 2 of the minicourse)
Abstract.
Given a manifold \(M\) and an open subset \(U\) of the cotangent bundle \(T^*M\), we define the category \(\mu Sh(U)\) of microlocal sheaves on \(U\), as well as the category \(\mu Sh_Z(U)\) of microlocal sheaves supported on any given subset \(Z\) of \(U\). We sketch the basic features of this category, and describe it as explicitly as possible using the constructions from Lecture 1.
Leon
Bungert
TU Berlin
The Geometry of Adversarial Machine Learning
Abstract.
It is well-known that despite their aptness for complicated tasks like image classification, modern neural networks are prone to insusceptible input perturbations (a.k.a. adversarial attacks) which can lead to severe misclassifications. Adversarial training is a state-of-the-art method to train classifiers which are more robust against these adversarial attacks. The method features minimization of a robust risk and has interpretations as game-theoretic problem, distributionally robust optimization problem, dual of an optimal transport problem, or nonlocal geometric regularization problem. In this talk I will focus on the last interpretation which allows for the application of tools from calculus of variations and geometric measure theory to study existence, regularity, and asymptotic behavior of minimizers. In particular, I will show that adversarial training of binary agnostic classifiers is equivalent to a nonlocal and weighted perimeter regularization of the decision boundary. Furthermore, I will show Gamma-convergence of this perimeter to a local anisotropic perimeter as the strength of the adversary tends to zero, thereby establishing an asymptotic regularization effect of adversarial training. Lastly, I will discuss probabilistic relaxations of adversarial training which exhibit better clean accuracies and also have a perimeter-regularization interpretation.
Olaf
Parczyk
FU Berlin
Graphs with large minimum degree and no small odd cycles are 3-colourable
Abstract.
Let Ƒ be a fixed family of graphs. The homomorphism threshold of Ƒ is the infimum of those α for which there exists an Ƒ-free graph H=H(Ƒ, α), such that every Ƒ-free graph on n vertices of minimum degree αn is homomorphic to H. Letzter and Snyder showed that the homomorphism threshold of {C3,C5} is 1/5. They also found explicit graphs H({C3,C5}, α), for α > 1/5, which were in addition 3-colourable. Thus, their result also implies that {C3,C5}-free graphs of minimum degree at least (1/5+ ε)n are 3-colourable. For longer cycles, Ebsen and Schacht showed that the homomorphism threshold of {C3,C5,…,C2l-1} is 1/(2l-1). However, their proof does not imply a good bound on the chromatic number of{C3,C5,…,C2l-1}-free graphs of minimum degree n/(2l-1). Answering a question of Letzter and Snyder, we prove that such graphs are 3-colourable. In fact, we prove a stronger result that works with a slightly smaller minimum degree.This is joint work with Julia Böttcher, Nóra Frankl, Domenico Mergoni Cecchelli, and Jozef Skokan.
Deep Learning with variable time stepping
Abstract.
Feature propagation in Deep Neural Networks (DNNs) can be associated to nonlinear discrete dynamical systems. The novelty, in this talk, lies in letting the discretization parameter (time step-size) vary from layer to layer, which needs to be learned, in an optimization framework. The proposed framework can be applied to any of the existing networks such as ResNet, DenseNet or Fractional-DNN. This framework is shown to help overcome the vanishing and exploding gradient issues. Stability of some of the existing continuous DNNs such as Fractional-DNN is also studied. The proposed approach is applied to an ill-posed 3D-Maxwell's equation.
Michael
McBreen
U Hong Kong
Introduction to Microlocal Sheaves
Abstract.
This is the first of a series of lectures about microlocal sheaves. Details can be found here.
Benedikt
Gräßle
HU Berlin
The pressure-wired Stokes element
Abstract.
The conforming Scott-Vogelius pair for the stationary Stokes equation in 2D is a popular finite element which is inf-sup stable for any fixed regular triangulation. However, the inf-sup constant deteriorates if the "singular distance" (measured by some geometric mesh quantity Θₘᵢₙ > 0) of the finite element mesh to certain "singular" mesh configurations is small. In this paper we present a modification of the classical Scott-Vogelius element of arbitrary polynomial order k ≥ 4 for the velocity where a constraint on the pressure space is imposed if locally the singular distance is smaller than some control parameter η > 0. We establish a lower bound on the inf-sup constant in terms of Θₘᵢₙ+η > 0 independent of the maximal mesh width and the polynomial degree that does not deteriorate for small Θₘᵢₙ≪1. The divergence of the discrete velocity is at most of size O(η) and very small in practical examples. In the limit η→0 we recover and improve estimates for the classical Scott-Vogelius Stokes element. This talk presents joint work with Nis-Erik Bohne and Stefan Sauter, University of Zurich.
Raman
Sanyal
FU
From linear programming to colliding particles
Abstract.
The simplex algorithm is the method of choice for solving linear optimization problems in practice. However, it is a famous open problem to show that the simplex algorithm also performs well in theory. From a discrete-geometric perspective, the simplex algorithm follows a path in the graph of a convex polytope and the path is determined by a so-called pivot rule. The challenge is to find a pivot rule that always takes a “short” path.
Weiwen
Mo
KU Leuven
Approximating Multivariate Functions with Embedded Lattice-based Algorithms
Laurence
Wilkes
KU Leuven
A randomised lattice algorithm for integration using a fixed generating vector
Ryan
Matzke
Vanderbilt University
Energy, Discrepancy, and Polarization of Greedy Sequences on the Sphere
Stefan
Felsner
Geometric representations of signotopes
Matthias
Beck
San Francisco State University
Integer-point transforms, symmetric functions, and q-Ehrhart polynomials
Abstract.
The integer-point transform of a rational polyhedron P encodes all integer lattice points of P as a multivariate rational generating function. It has applications in combinatorics, commutative algebra, number theory, and many other fields. We will discuss some connections with work of Chapoton (2016) on the q-Ehrhart polynomial of P, a 2-variable generalization of the integer-point counting function of dilates of P. Motivated by Stanley's chromatic symmetric function (1995) generalization of the chromatic polynomial of a graph, we propose a study of symmetric functions attached to certain rational polytopes.
Phil
Engel
University of Georgia
The non-abelian Hodge locus
Abstract.
Given a family of smooth projective varieties, one can consider the relative de Rham space, of flat vector bundles of rank \(n\) on the fibers. The flat vector bundles which underlie a polarized \(\mathbb{Z}\)-variation of Hodge structure form the "non-abelian Hodge locus". Simpson proved that this locus is analytic, and he conjectured it is algebraic. This would imply a conjecture of Deligne that only finitely many representations of the fundamental group of a fiber appear. I will discuss a proof of Deligne's and Simpson's conjectures, under the additional hypothesis that the \(\mathbb{Z}\)-zariski closure of monodromy is a cocompact arithmetic group. This is joint work with Salim Tayou.
Cécile
Gachet
HU Berlin
Smooth projective surfaces with infinitely many real forms
Abstract.
A common undergraduate exercise is to classify quadratic forms over the real and complex numbers. Its conclusion could be that the two non-isomorphic real conics \(x^2 + y^2 + z^2 = 0\) and \(x^2 + y^2 - z^2 = 0\) are isomorphic as complex curves. In fact, the corresponding complex curve is the rational line, and it admits only the afore-mentioned two non-isomorphic real forms. Although it is quite common to find complex projective varieties admitting several real forms, the first example of a variety with infinitely many non-isomorphic real forms can be found in a 2018 paper by Lesieutre. More examples of varieties with infinitely many real forms have been found later, for instance as rational surfaces and as surfaces birational to K3 surfaces, see the 2022 paper by Dinh, Oguiso, and Yu. This talk, reporting on joint work with Tien-Cuong Dinh, Hsueh-Yung Lin, Keiji Oguiso, Long Wang, and Xun Yu, completes the picture sketched by theses examples in the case of smooth projective surfaces. It features the following two results: First, if a smooth projective surface admits infinitely many real forms, then it is rational, or birational to a K3 surface and non-minimal, or birational to an Enriques surface and non-minimal. Second, there are surfaces obtained by blowing-up one point in an Enriques surface, which admit infinitely many non-isomorphic real forms. In this talk, I will explain the key ideas involved in the proofs of the two results, and try to give an idea of the construction used for the second one. Interestingly, we will encounter a fair share of group theory, group actions, and dynamics.
Moritz
Reintjes
City University of Hong Kong
Optimal regularity and Uhlenbeck compactness in Lorentzian geometry and beyond
Abstract
Omri
Weinstein
Hebrew University
Approximate Matrix Multiplication via Spherical Convolutions
Abstract.
We introduce a new algorithmic framework, which we call Polyform, for fast approximate matrix multiplication through sums of spherical convolutions (sparse polynomial multiplications). This bilinear operation leads to several new (worst-case and data-dependent) improvements on the speed-vs-accuracy tradeoffs for approximate matrix multiplication. Polyform can also be viewed as a cheap practical alternative to matrix multiplication in deep neural networks (DNNs), which is the main bottleneck in large-scale training and inference. The algorithm involves unexpected connections to Additive Combinatorics, sparse Fourier transforms, and spherical harmonics. The core of the algorithm is optimizing the polynomial's coefficients, which is a *low-rank* SDP problem generalizing Spherical Codes. Meanwhile, our experiments demonstrate that, when using SGD to optimize these coefficients in DNNs, Polyform provides a major (3×-5×) speedup on state-of-art DL models with minor accuracy loss. This suggests replacing matrix multiplication with (variants of) polynomial multiplication in large-scale deep neural networks.
Emilie
Pirch
Uni Jena
Guaranteed lower eigenvalue bounds with three skeletal methods
Wolfgang
Mulzer
Freie Universität Berlin
The Multiplicative-Weights-Update Method
Abstract.
The multiplicative weights update method is a design paradigm for algorithms that is used in many different areas of theoretical computer science and machine learning.A famous survey by Arora, Hazan, and Kale provides an excellent overview over the method and its applications. Together with Nabil Mustafa, we are currently working on a monograph that explores the method in more detail. I will give an overview of the method and share some nuggets that we encountered. Based on joint work with Nabil Mustafa.
Sanaz
Pooya
U Potsdam
What is a group C*-algebra?
Abstract.
In this talk I will start by introducing the notion of C*-algebras, focusing on some basic examples. Subsequently, I will introduce the important example of group C*-algebras. They form a bridge between group theory and operator algebras. For some concrete examples of groups I will describe their associated C*-algebras.
Sven
Raum
U Potsdam
Operator algebraic group theory
Yan
Alves Radke
On the connectivity of the flipgraph of pseudoline arrangements
Esme
Bajo
UC Berkeley
q-analog chromatic polynomials
Abstract.
Stanley (1995) introduced a symmetric function generalization of the chromatic polynomial, and conjectured that it distinguishes non-isomorphic trees. We discuss a geometric approach to this problem using Chapoton’s q-analog weighted Ehrhart theory, wherein we express the principal specialization of the chromatic symmetric function as a polynomial in the q-integers and study its coefficients in different bases. This is joint work with Matthias Beck and Andrés R. Vindas-Meléndez.
Jacob
Matherne
Universität Bonn, MPI
Polynomials in combinatorics and representation theory
Abstract.
Many polynomials in combinatorics (and in other areas of mathematics) have nice properties such as having all of their roots being real numbers, or having all of their coefficients being nonnegative. By surveying recent advances in the Hodge theory of matroids (namely, the nonnegativity of Kazhdan-Lusztig polynomials of matroids and Dowling and Wilson's top-heavy conjecture for the lattice of flats of a matroid), I will give several examples of well-behaved polynomials, and I will indicate some connections of these properties to geometry and representation theory. The talk should be understandable to everyone, and should appeal to those with interests in at least one of the following topics: hyperplane arrangements, matroids, log-concavity, real-rooted polynomials, lattice theory, Coxeter groups, and the representation theory of Lie algebras. It will contain results that are joint work with Tom Braden, Luis Ferroni, June Huh, Nicholas Proudfoot, Matthew Stevens, Lorenzo Vecchi, and Botong Wang.
Physics-informed neural control of partial differential equations with applications to numerical homogenisation
Abstract.
In this talk we discuss a model for numerical homogenisation based on the combination of physics-informed neural networks and standard numerical approximation techniques. From a continuous viewpoint, the formulation corresponds to a non-standard PDE-constrained optimisation problem with a neural network objective. From a discrete viewpoint, the formulation represents a hybrid neural network numerical solver. We discuss physics-informed neural networks, the numerical homogenisation modelling framework and its aforementioned optimisation interpretation, as well as related discrete concepts and routes towards applications in materials science and fluid flows in porous media.
Haohao
Liu
IMJ Paris
Fourier-Mukai transform on complex tori
Abstract.
Classical Fourier transform occupies a major part of the analysis. An analog on abelian varieties is introduced by S. Mukai in 1981, which is now known as Fourier-Mukai transform. Similar to the Fourier inversion formula, Mukai proved a duality theorem for his transform. This result reveals the phenomenon that, the derived category of coherent modules of two non-isomorphic projective varieties can be equivalent. In this talk, I will present the work of O. Ben-Bassat, J. Block and T. Pantev about the analytic version of Fourier-Mukai transform on complex tori.
Carsten
Carstensen
HU Berlin
Smoother
Günter
Rote
Freie Universität Berlin
Grid Peeling and the Affine Curve-Shortening Flow
Abstract.
Grid Peeling is the process of taking the integer grid points inside a convex region and repeatedly removing the convex hull vertices. On the other hand, the Affine Curve-Shortening Flow (ACSF) is defined as a particular deformation of a smooth curve. It has been observed in 2017 by Eppstein, Har-Peled, and Nivasch, that, as the grid is refined, Grid Peeling converges to the Affine Curve-Shortening Flow. As part of the M.Ed. thesis of Moritz Rüber, we have investigated the grid peeling process for special parabolas, and we could observe some striking phenomena. This has lead to the precise value of the constant that relates the two processes. With Morteza Saghafian from IST Austria, we could prove the convergence of grid peeling for the class of parabolas with vertical axis.
Lutz
Recke
A common approach to singular perturbation and homogenization
Mattes
Mollenhauer
FU Berlin
Subgaussian concentration in Hilbert spaces and inference in inverse problems
Oleg
Butkovsky
Weierstrass Institute
Approximation of SDEs with irregular drift: stochastic sewing approach
Helena
Bergold
Strong Erdős-Hajnal properties in chordal graphs
May 2023
Martin
Winter
Warwick
Rigidity, Tensegrity and Reconstruction of Polytopes under Metric Constraints
Abstract.
In how far is it possible to reconstruct a convex polytope (up to isometry, affine equivalence or combinatorial equivalence) from its edge-graph and some 'graph-compatible metric data', such as its edge-lengths? In this talk we explore this question and pose the following conjecture: a convex polytope P is uniquely determined by its edge-graph, its edge length and the distance of each vertex from some common interior point of P, across all dimensions and combinatorial types. We conjecture even stronger, that two polytopes P and Q are already isometric if they have the same edge-graph, edges in P are at most as long as in Q, and vertex-point distances in P are at least as long as in Q. In other word, a convex polytope cannot become 'bigger' while its edges become shorter. We verify the conjectures in three special cases: if P and Q are combinatorically equivalent, if P and Q are centrally symmetric, and if Q is a slight perturbation of P. These results were obtained using techniques from the intersection of convex geometry and spectral graph theory.
Antoine
Deza
McMaster University
Kissing Polytopes
Abstract.
We investigate the following question: how close can two disjoint lattice polytopes be contained in a fixed hypercube? This question occurs in various contexts where this minimal distance appears in complexity bounds of optimization algorithms. We provide nearly matching lower and upper bounds on this distance and discuss its exact computation. Similar bounds are given in the case of disjoint rational polytopes whose binary encoding length is prescribed. Joint work with Shmuel Onn, Sebastian Pokutta, and Lionel Pournin.
Josh
Lam
HU Berlin
Properties of non-abelian Hodge theory mod p: Periodicity
Abstract.
This is part of the reading group on flat connections in positive and mixed characteristic. For details see here.
Joscha
Gedicke
Uni Bonn
P₁ finite element methods for an elliptic optimal control problem with pointwise state constraints
Abstract.
We present theoretical and numerical results for two P₁ finite element methods for an elliptic distributed optimal control problem on general polygonal/polyhedral domains with pointwise state constraints.
Musxhu (Julian)
Kern
WIAS Berlin
What is an interacting particle system?
Abstract.
Most parts of the real world can be described via interacting particles. Whether you look at the microscopic level and see molecules bumping together or try to understand how electrical impulses in your brain form thoughts, one question is common to all particle systems: how can local interactions induce a macroscopic behaviour? In this week's "What is... ?" seminar, we will see how particle systems model natural phenomena, where randomness comes into play and what behaviour we might expect to see at different scales (=zoom levels).
Anja
Sturm
U Göttingen
Interacting particle systems and the contact process on random graphs
Rimma
Hämäläinen
Area Proportionality of Euler diagrams
Proximal Galerkin: Structure-preserving finite element analysis for free boundary problems, maximum principles, and optimal design
Abstract.
One of the longest-standing challenges in finite element analysis is to develop a stable, scalable, high-order Galerkin method that strictly enforces pointwise bound constraints. The latent variable proximal Galerkin finite element method is a nonlinear, structure-preserving method with these properties. This talk will introduce proximal Galerkin and describe its capability for treating free-boundary problems, enforcing discrete maximum principles, and designing scalable, mesh-independent algorithms for optimal design. The talk begins with a derivation of the latent variable proximal point (LVPP) method: an unconditionally stable alternative to the interior point method. LVPP is an infinite-dimensional optimization algorithm that may be viewed as having an adaptive (Bayesian) barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Thereupon, we find that the original variational inequality can be replaced by a sequence of semilinear partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout this talk, we arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between positivity-preserving discretizations and infinite-dimensional Lie groups; and (3) a gradient-based, structure-preserving algorithm for two-field density-based topology optimization. The overall latent variable proximal Galerkin combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis.
Kris
Shaw
University of Oslo
The birational geometry of matroids
Abstract.
In this talk, I will consider isomorphisms of Bergman fans of matroids. Motivated by algebraic geometry, these isomorphisms can be considered as matroid analogs of birational maps. I will introduce Cremona automorphisms of the coarsest fan structure. These produce a class of automorphisms which do not come from automorphisms of the underlying matroid. I will then explain that the automorphism group of the coarse fan structure is generated by matroid automorphisms and Cremona maps when the fan is 2-dimensional and also for modularly complemented matroids. I will assume no background on matroid theory for this talk. This is based on joint work with Annette Werner.
Kris
Shaw
University of Oslo
Real phase structures on matroid fans and matroid orientations
Abstract.
In this talk, I will propose a first definition of real structures on polyhedral complexes. I’ll explain that in the case of matroid fans, specifying such a structure is cryptomorphic to providing an orientation of the underlying matroid. Real phase structures are used to patchwork the real part of a tropical variety in any codimension and this gives a closed chain in the real part of a toric variety. This provides a link between oriented matroids and real toric varieties and an obstruction to the orientability of matroids via the homology groups of toric varieties. This is based on joint work with Johannes Rau and Arthur Renaudineau.
Degenerate hysteresis in partially saturated porous media
Abstract.
We propose a model for fluid diffusion in partially saturated porous media taking into account hysteresis effects in the pressure-saturation relation. The resulting mathematical problem leads to a diffusion equation with Robin boundary condition for the pressure in an N-dimensional domain with a Preisach hysteresis operator under the time derivative. The problem is doubly degenerate in the sense that the saturation range is bounded, and no a priori control of the time derivative of the pressure is available. A bootstrapping argument based on particular geometric properties of the hysteresis operator makes it possible to prove the existence and uniqueness of a strong solution to the problem for arbitrarily large data. This is a joint work with Chiara Gavioli from TU Wien.
Johannes
Storn
Uni Bielefeld
Advantages of Dual Formulations in Computational Calculus of Variations
Abstract.
Duality theory is a very useful tool in the calculus of variations. In this talk we exploit this tool to overcome the Lavrentiev gap phenomenon and to design an iterative scheme for the computation of the p-Laplace problem with large exponents.
Alexandre
Minets
U Edinburgh
A proof of the P=W conjecture
Abstract.
Let \( C \) be a smooth projective curve. The non-abelian Hodge theory of Simpson is a homeomorphism between the character variety \( M_B \) of \( C \) and the moduli of (semi)stable Higgs bundles \( M_D \) on \( C \). Since this homeomorphism is not algebraic, it induces an isomorphism of cohomology rings, but does not preserve finer information, such as the weight filtration. Based on computations in small rank, de Cataldo-Hausel-Migliorini conjectured that the weight filtration on \( H^*(M_B) \) gets sent to the perverse filtration on \( H^*(M_D) \), associated to the Hitchin map. In this talk, I will explain a recent proof of this conjecture, which crucially uses the action of Hecke correspondences on \( H^*(M_D) \). Based on joint work with T. Hausel, A. Mellit, O. Schiffmann.
Jens
Vygen
University of Bonn
Resource Sharing, Routing, Chip Design
Abstract.
We survey the state of the art in VLSI routing. Interconnecting millions of sets of pins by wires is challenging because of the huge instance sizes and limited resources. We present our general approach as well as algorithms for fractional min-max resource sharing and goal-oriented shortest path search sped up by geometric distance queries. These are key components of BonnRoute, which is used for routing some of the most complex microprocessors in industry.
Felix
Schöder
Gons and Holes in Projective Point Sets (Part 2)
Aryaman
Jal
KTH
Polyhedral geometry of bisectors and bisection fans
Abstract.
Every symmetric convex body induces a norm on its affine hull. The object of our study is the bisector of two points with respect to this norm. A topological description of bisectors is known in the 2 and 3-dimensional cases and recent work of Criado, Joswig and Santos (2022) expanded this to a fuller characterisation of the geometric, combinatorial and topological properties of the bisector. A key object introduced was the bisection fan of a polytope which they were able to explicitly describe in the case of the tropical norm. We discuss the bisector as a polyhedral complex, introduce the notion of bisection cones and describe the bisection fan corresponding to other polyhedral norms. This is joint work with Katharina Jochemko.
Model Reduction and Uncertainty Quantification of Multiscale Diffusion with Parameter Uncertainties Using Nonlinear Expectations
Abstract
Dingyuan
Liu
FU Berlin
On the number of sets with bounded sumset
Abstract.
We will study the counting problem of sets with bounded sumset. More precisely, let m,n be natural numbers, for an n-element subset Y of an arbitrary abelian group, we give upper bounds on the number of sets A ⊆Y with |A+A| ≤ m according to the range of m. Furthermore, we also give an upper bound for the refined version of this problem, improving the previous result of Campos and confirming an earlier conjecture of Alon, Balogh, Morris, and Samotij in a certain range.This is joint work with Leticia Mattos and Tibor Szabó.
Ngoc Tien
Tran
Uni Jena
Minimal residual methods for PDE of second order in nondivergence form
Emil
Jacobsen
U Zürich / Stockholm University
On (generic) motivic fundamental groups
Abstract.
I will introduce the (generic) motivic fundamental group of a smooth variety, and explain its relation to the usual (generic) fundamental group. The main result can also be phrased as follows: generic local systems of motivic origin are stable under extension in the category of generic local systems. I will also present a (generic) motivic version of a classical theorem of Hain, on Malcev completions of monodromy representations. At the end, I'll explain some of the group/representation theoretic tools that go into these result. If there's time, I can explain some analogous Hodge theoretic results, as well as some future directions.
Felix
Goldmann
HU Berlin
Discontinuous Galerkin method for the Parabolic Obstacle Problem
Günter
Rote
Freie Universität Berlin
The Generalized Combinatorial Lasoń-Alon-Zippel-Schwartz Nullstellensatz Lemma
Abstract.
We survey strengthenings and generalizations of the Schwartz-Zippel Lemma and Alon’s Combinatorial Nullstellensatz. Both lemmas guarantee the existence of (a certain number of) nonzeros of a multivariate polynomial when the variables run independently through sufficiently large ranges.
Lukas
Abel
HU Berlin
What is important in mathematical modeling of materials?
Abstract.
Real world problems of material sciences are often mathematically modelled in the framework of calculus of variations. In this week’s „What is“ seminar starting with $\Gamma$-convergence we will go on a journey through different ideas, concepts and methods which are highly used and of great importance in this special area of applied analysis.
Anja
Schlömerkemper
U Würzburg
Various Structures in Mathematics of Materials Science
Christian
Haase
FU
Gitterpunkte und Hodge-Zahlen
Abstract.
„Das Studium torischer Varietäten ist ein wunderbarer Teil der algebraischen Geometrie mit tiefen Verbindungen zur polyedrischen Geometrie. Es gibt elegante Theoreme, unerwartete Anwendungen und phantastische Beispiele.“ (Frei nach Cox, Little, Schenck: Toric Varieties; AMS 2011). Dieser Vortrag ist eine Einladung in dieses Gebiet. Als ein Beispiel wie „wunderbar“ es hier ist, schauen wir uns Verallgemeinerungen des berühmten (und natürlich eleganten) Satzes von Bernstein-Kushnirenko an. Dieser Satz drückt die Anzahl der Lösungen eines Systems von n Polynomgleichungen in n Unbekannten durch das gemischte Volumen von n Polytopen aus. Wenn wir nur noch k < n Gleichungen haben, wird die Lösungsmenge nicht mehr endlich sein. Nichtsdestotrotz kann man Formeln beweisen, die (algebraisch) geometrische Invarianten der Lösungsmenge in Beziehung zu Gitterpunktzahlen in Minkowski-Summen von Polytopen setzen. Der Schwerpunkt des Vortrags liegt darin, eine Idee des Wechselspiels zwischen algebraischer und polyedrischer Geometrie zu vermitteln. Bei den aktuelleren Resultaten berichte ich über gemeinsame Arbeiten mit S. Di Rocco, M. Juhnke-Kubitzke, B. Nill, R. Sanyal und T. Theobald.
Bernd
Sturmfels
MPI Leipzig
Positive del Pezzo geometry
Abstract.
Positive geometry is an emerging field at the interface of combinatorics, algebraic geometry and physics. This lecture explores the positive geometries associated with del Pezzo surfaces and their moduli spaces. Scattering amplitudes offer new vistas on configurations of six and seven points in the plane, their Weyl group symmetries of types E6 and E7, and on 27 lines on real and tropical cubic surfaces. This is work in progress with Nick Early, Alheydis Geiger, Marta Panizzut and Claudia Yun.
Andreas
Löhne
Universität Jena
Stable algorithms for the 3-dimensional vertex enumeration problem
Abstract.
Vertex enumeration is the fundamental computational problem to determine the vertices of a convex polytope given by affine inequalities. The converse problem, called convex hull problem, is equivalent to polarity.Large instances of vertex enumeration problems frequently appear as subproblems in various applications. In practice, they are often treated by algorithms implemented in floating point arithmetic. Typically, correctness of these algorithms is only known for exact (rational) arithmetic, which can be too slow for practical requirements. Imprecise computations caused by floating point arithmetic can result in wrong results which are not even an approximation of the correct results in some suitable sense. Therefore, one is interested in stable algorithms where such kind of bad behavior can be excluded. Surprisingly, stability seems to be established so far only for the 2-dimensional vertex enumeration problem.We present two algorithms for the 3-dimensional vertex enumeration problem which are stable in some sense which needs to be explained further. We also discuss possible extensions to dimension 4.
Neela
Nataraj
IIT Bombay
Discussion on WOPSIP
Nicola
Tarasca
Virginia Commonwealth University
Coinvariants of vertex algebras and abelian varieties
Abstract.
Spaces of coinvariants have classically been constructed by assigning representations of affine Lie algebras, and more generally, vertex operator algebras, to pointed algebraic curves. Removing curves out of the picture, I will construct spaces of coinvariants at abelian varieties with respect to the action of an infinite-dimensional Lie algebra. I will show how these spaces globalize to twisted D-modules on moduli of abelian varieties, extending the classical picture from moduli of curves. This is based on my recent preprint arXiv:2301.13227.
Ákos
Magyar
University of Georgia
On a conjecture of Graham in geometric Ramsey theory
Abstract.
Let's call a finite point configuration F in a Euclidean space "Ramsey" if any finite coloring of a sufficiently high dimensional Euclidean space contains a monochromatic isometric copy of F. After some initial investigations of Erdos, Graham at al., Graham conjectured that a set F is Ramsey if and only if it is spherical, i.e. if it can be inscribed into a sphere.We outline a new approach and some new results based on some modern tools of additive combinatorics, such as the use of (geometric) uniformity norms and arithmetic regularity lemmas both in the context of Euclidean spaces as well as in the model case of vector spaces over finite fields.
Lara
Theallier
HU Berlin
Implementing HHO for linear elasticity
Marco
Flores
HU Berlin
A cohomological approach to formal Fourier-Jacobi series
Abstract.
Siegel modular forms admit various expansions, one of the most important being the Fourier-Jacobi expansion. Algebraically, these expansions take the form of a series whose coefficients are Jacobi forms satisfying a certain symmetry condition. One poses the following modularity question: does every formal series of this shape arise from a Siegel modular form? Bruinier and Raum answered the question affirmatively, over the complex numbers, in 2014. In this talk I will consider this question over the ring of integers, and reformulate it as a matter of cohomological vanishing on a toroidal compactification of the moduli space A_g of principally polarized abelian varieties. I will present a proof that said vanishing is equivalent to our modularity question, and explore its relationship with the singularities of the minimal compactification of A_g.
Avi
Wigderson
Institute for Advanced Study, Princeton
The Value of Errors in Proofs
Abstract.
Last year, a group of theoretical computer scientists posted a paper on the Arxiv with the strange-looking title "MIP* = RE", surprising and impacting not only complexity theory but also some areas of math and physics. Specifically, it resolved, in the negative, the "Connes' embedding conjecture" in the area of von-Neumann algebras, and the "Tsirelson problem" in quantum information theory. It further connects Turing's seminal 1936 paper which defined algorithms to Einstein's 1935 paper with Podolsky and Rosen which challenged quantum mechanics. You can find the paper here https://arxiv.org/abs/2001.04383. As it happens, both acronyms MIP* and RE represent proof systems, of a very different nature. To explain them, we'll take a meandering journey through the classical and modern definitions of proof. I hope to explain how the methodology of computational complexity theory, especially modeling and classification (of both problems and proofs) by algorithmic efficiency, naturally leads to the generation of new such notions and results (and more acronyms, like NP). A special focus will be on notions of proof which allow interaction, randomness, and errors, and their surprising power and magical properties. The talk does not require special mathematical background.
Matei
Hanu
FU Berlin
Sandro
Roch
Block coupling on k-heights
Convergence analysis of the nonoverlapping Robin-Robin method for nonlinear elliptic equations
Abstract.
The nonoverlapping Robin-Robin method is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. In this talk we present a convergence analysis for the Robin-Robin method applied to nonlinear elliptic equations with a p-structure, including degenerate diffusion equations governed by the p-Laplacian. The analysis relies on a new theory for nonlinear Steklov-Poincare operators based on the p-structure and the Lp-generalization of the Lions-Magenes spaces. This framework allows the reformulation of the Robin-Robin method into a Peaceman-Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. This is joint work with Emil Engström (Lund University)
Raul
Penaguiao
MPI Leipzig
Linear Tropical Geometry with Matroids
Abstract.
There are several ways of computing the degree of varieties, but the most basic one is to intersect your variety with a line. We will be doing the same in the tropical world. In this talk, we will introduce the Bergman fan of a matroid, our tropical variety of interest. We study its combinatorics and identify its degree. Interesting things arise when we apply the Cremona map to the Bergman fan, which flips x \to -x. This new fan recovers some interesting beta invariants with the number of so called nbc bases. With the techniques developed we will compute the likelihood degeneracy degree of a matroid and recover a result from Agostini et al. All these results are algorithmic, which is a new way of computing this degree.
Avi
Wigderson
IAS, Princeton
Approximate Sylvester-Gallai, and applications
Abstract.
Assume you have a finite set of points in real space, with the property that every line through any pair of these points meets a third point. This property is satisfied when all points are colinear, and the (so called) Sylvester-Gallai theorem states that this is the only way!In this talk we explore approximate versions of the assumption, in which the hypothesis holds "on average". I'll motivate this question from applications in coding theory and algebraic complexity. I'll then prove bounds on the dimension of the point set under such relaxed hypothesis. The proof in interesting in that in combines combinatorial, algebraic and analytic arguments, each of them simple to explain.Based on (old) joint works with Barak, Dvir, Saraf and Yehudayoff.The talk is self contained - no special background is assumed.
Marian
Aprodu
University of Bucharest
Resonance and vector bundles
Abstract.
Resonance varieties are algebro-geometric objects that emerged from the geometric group theory. They are naturally associated to vector subspaces in second exterior powers and carry natural scheme structures that can be non-reduced. In algebraic geometry, they made an unexpected appearance in connection with syzygies of canonical curves. In this talk, based on a joint work with G. Farkas, C. Raicu and A. Suciu, I report on some recent results concerning the geometry of resonance schemes in the vector bundle case.
Adam
Schweitzer
TU Berlin
The Chow ring of matroids and the Rota-Welsh conjecture - Part II
Abstract.
Recently methods in combinatorial Hodge theory have been used successfully to settle long standing conjectures about certain characteristic sequences of matroids.This talk will introduce some of the central concepts and theorems of this theory for the case of matroids, without delving into the underlying algebraic geometry, and use it to give parts of June Huh's proof of the Rota-Welsh conjecture.
Stefan
Sauter
Uni Zürich
The method of integral equations for acoustic transmission problems
with varying coefficients
Abstract.
In our talk we will derive an integral equation method which transforms a three-dimensional acoustic transmission problem with variable coefficients and mixed boundary conditions to a non-local equation on the two-dimensional boundary and skeleton of the domain. For this goal, we introduce and analyze abstract layer potentials as solutions of auxiliary coercive full space variational problems and derive jump conditions across domain interfaces. This allows us to formulate the non-local skeleton equation as a direct method for the unknown Cauchy data of the original partial differential equation. We develop a theory which inherits coercivity and continuity of the auxiliary full space variational problem to the resulting variational form of the skeleton equation without relying on an explicit knowledge of Green's function. Some concrete examples of full and half space transmission problems with piecewise constant coefficients are presented which illustrate the generality of our integral equation method and its theory. This talk comprises joint work with Francesco Florian, University of Zurich and Ralf Hiptmair, ETH Zurich.
April 2023
Marco
Flores
HU Berlin
What is an Eisenstein series?
Abstract.
Modular forms are a certain kind of holomorphic maps defined on the upper half-plane whose Fourier expansions, very much surprisingly, mantain an intimate relationship with number theory. One of the most striking and intricate manifestations of this relationship is the proof by Andrew Wiles and others of Fermat's Last Theorem, a result about an integral equation with 4 variables which resisted proof for over 350 years. Eisenstein series are a particular kind of modular forms which can be written down explicitly, thus being ideal for experimentation in the theory of modular forms. In this talk I will give some examples of Eisenstein series, and we will witness exactly how number theoretic information can be extracted from their structure.
Laurent
Clozel
U Paris-Sud
Topological realization of Eisenstein cohomology
Anna
Barkova
Geometrische Realisierbarkeit der Repräsentation von Graphen mit polygonalen Flächen
Manfred
Scheucher
TU Berlin
Using SAT Solvers in Combinatorics and Geometry
Abstract.
In this talk, we discuss how modern SAT solvers can be used to tackle mathematical problems. We discuss various problems to give the audience a better understanding, which might be tackled in this fashion, and which might not. Besides the naive SAT formulation further ideas are sometimes required to tackle problems. Additional constraints such as statements which hold 'without loss of generality' might need to be added so that solvers terminate in reasonable time.
Adam
Schweitzer
TU Berlin
The Chow ring of matroids and the Rota-Welsh conjecture - Part I
Abstract.
Recently methods in combinatorial Hodge theory have been used successfully to settle long standing conjectures about certain characteristic sequences of matroids.This talk will introduce some of the central concepts and theorems of this theory for the case of matroids, without delving into the underlying algebraic geometry, and use it to give parts of June Huh's proof of the Rota-Welsh conjecture.
Ram
Band
Technion, Haifa / U Potsdam
Spectral flows and the Robin count deficiency for metric graphs
Abstract
Samouil
Molcho
ETH Zürich
Extending Brill-Noether classes to the boundary of moduli space of curves
Abstract.
Inside the Jacobian of the universal curve of the moduli space of genus \(g\), \(n\)-pointed curves lie the Brill-Noether loci, parametrizing pairs of curves with line bundles that have more than expected sections. Pulling the (virtual fundamental classes of the) Brill-Noether loci to \(M_{g,n}\) by any section of the universal Jacobian produces interesting cycles in the tautological ring, which play a key role in its structure as a ring. Pagani, Ricolfi and van Zelm have proposed an extension of these classes to the Deligne-Mumford compactification \(\bar{M}_{g,n}\), and conjecture that they are also in the tautological ring. In this talk, I will explain a natural refinement of these classes from the perspective of logarithmic geometry, which allows us to prove the PRvZ conjecture.
Monika
Eisenmann
Lund University
A randomized operator splitting scheme inspired by stochastic optimization methods
Ignacio
Barros
University of Antwerp
Moduli spaces of hyperkähler varieties
Abstract.
I will discuss general aspects of the geometry of moduli spaces of hyperkähler varieties and talk about the state of the art regarding their birational classification. In the second half, I will present recent results on their Kodaira dimension generalizing to higher dimension results of Gritsenko-Hulek-Sankaran in the surface and fourfold K3-type cases. The talk is based on joint work with Pietro Beri, Emma Brakkee, and Laure Flapan.
Carla
Michini
The Price of Anarchy in Series-Parallel Network Congestion Games
Abstract.
We study the pure Price of Anarchy (PoA) of symmetric network congestion games defined over series-parallel networks, which measures the inefficiency of pure Nash equilibria. First, we consider affine edge delays. For arbitrary networks, Correa and others proved a tight upper bound of 5/2 on the PoA. On the other hand, Fotakis showed that restricting to the class of extension-parallel networks makes the worst-case PoA decreases to 4/3. We prove that, for the larger class of series-parallel networks, the PoA is at most 2, and that it is at least 27/19 in the worst case, improving both the best-known upper bound and the best-known lower bound. Next, we consider edge delays that are polynomial functions with highest degree p. We construct a family of symmetric congestion games over arbitrary networks which achieves the same worst-case PoA of asymmetric network congestion games given by Aland and others. We then establish that in games defined over series-parallel networks the PoA cannot exceed 2<sup>p+1</sup> - 1, which is considerably smaller than the worst-case PoA in arbitrary networks. We also prove that the worst-case PoA, which is sublinear in extension-parallel networks (as shown by Fotakis), dramatically degrades to exponential in series-parallel networks. Finally, we extend the above results to the case where the social cost of a strategy profile is computed as the maximum players’ cost. In this case, the worst-case PoA is in O(4<sup>p</sup>), which is considerably smaller than the worst-case PoA in arbitrary networks given by Christodoulou and Koutsoupias. Moreover, while in extension-parallel networks each pure Nash equilibrium is also a social optimum (as shown by Epstein and others), we construct instances of series-parallel network congestion games with exponential PoA.
Optimality conditions for problems with probabilistic state constraints
Abstract.
In this talk, we discuss optimization problems subject to probabilistic constraints. Our focus is on the setting in which the control variable belongs to a reflexive and separable Banach space, which is of interest, for instance, in physics-based models where the control acts on a system described by a partial differential equation (PDE). Incorporating uncertainty into such models has been of increasing interest, since in practice, one might only have access to empirical measurements or ranges of values for model parameters and inputs. We present different possibilities for incorporating uncertainty in state constraints and derive their optimality conditions. The conditions are applied to a simple example from PDE-constrained optimization under uncertainty. Perspectives for the numerical solution of these problems are discussed, as well as planned research directions.
Ruijie
Yang
MPI Bonn
The Riemann-Schoktty problem and Hodge theory
Abstract.
It is a classical problem, going back to Riemann, to decide which principally polarized abelian varieties come from the Jacobians of curves. In 2008, Casalaina-Martin proposed a question towards the Riemann-Schottky problem in terms of the codimension of multiplicity locus of the theta divisor, which includes Debarre’s conjecture as a special case. In this talk, I will talk about a partial affirmative answer to this question using a newly developed theory “Higher multiplier ideals”, which builds on Sabbah-Schnell’s theory of complex Hodge modules and Beilinson-Bernstein’s language of twisted D-modules. It is based on joint work with Christian Schnell.
Georgi
Mitsov
HU Berlin
Combining time-stepping θ-schemes with dPG-FEM for the solution of the heat equation
Omid
Nohadani
Nonlinear Decision Rules Made Scalable
Abstract.
Sequential decision-making often requires dynamic policies, which are computationally not tractable in general. Decision rules provide approximate solutions by restricting decisions to simple functions of uncertainties. In this talk, we consider a nonparametric lifting framework where the uncertainty space is lifted to higher dimensions to obtain nonlinear decision rules. Current lifting-based approaches require pre-determined functions and are parametric. We propose two nonparametric liftings, which derive the nonlinear functions by leveraging the uncertainty set structure and problem coefficients. Both methods integrate the benefits from lifting and nonparametric approaches, and hence, provide scalable decision rules with performance bounds. More specifically, the set-driven lifting is constructed by finding polyhedrons within uncertainty sets, inducing piecewise-linear decision rules with performance bounds. The dynamics-driven lifting, on the other hand, is constructed by extracting geometric information and accounting for problem coefficients. This is achieved by using linear decision rules of the original problem, also enabling to quantify lower bounds of objective improvements over linear decision rules. Using numerical comparisons with competing methods, we demonstrate superior computational scalability and comparable performance in objectives. These observations are magnified in multistage problems with extended time horizons, suggesting practical applicability of the proposed nonparametric liftings in large-scale dynamic robust optimization. This is a joint work with Eojin Han.
Jörg
Wolf
Chung-Ang University, Seoul
A new local condition for partial regularity on the vorticity of a local suitable weak solution to the Navier-Stokes equations
Robert
Gruhlke
FU Berlin
Langevin Dynamics: Bayesian inference, homotopy and generative modeling
Robert
Lauff
Three nice riddles
Christian
Stump
Ruhr-Universität Bochum
The Poincaré-extended ab-index
Abstract.
Motivated by a conjecture of Maglione-Voll concerning Igusa zeta functions, we introduce and study the Poincaré-extended ab-index. This polynomial generalizes both the ab-index and the Poincaré polynomial. For posets admitting R-labelings, we prove that the coefficients are nonnegative and give a combinatorial description of the coefficients. This proves Maglione-Voll’s conjecture as well as a conjecture of the Kühne-Maglione. We also recover, generalize, and unify results from Billera-Ehrenborg-Readdy, Ehrenborg, and Saliola-Thomas. This is joint work with Galen Dorpalen-Barry and Josh Maglione.
Leonid
Monin
EPFL
Combinatorics of toric bundles
Abstract.
Toric bundles are fibre bundles which have a toric variety as a fiber. One particularly studied class of toric bundles are horospherical varieties which are toric bundles over generalized flag varieties. Similar to toric varieties, toric bundles admit a combinatorial description via polyhedral geometry. In my talk, I will explain such a combinatorial description, and describe a couple of results which rely on it. In particular, I will present a generalization of the BKK theorem and the Fano criterion for toric bundles.
Benedikt
Gräßle
HU Berlin
The hierarchical Argyris AFEM with optimal convergence rates
Stefan
Felsner
Partial orders induced by rectangulations
Frank
Sottile
Texas A&M University
Galois groups in Enumerative Geometry and Applications
Abstract.
In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem. I will describe this background and discuss some work in a long-term project to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. A main focus is to understand Galois groups in the Schubert calculus, a well-understood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry.
Yao
Xie
Georgia Institute of Technology
Connecting dots, casually — using discrete events data over networks
Abstract.
Discrete events are sequential observations that record event time, location, and possibly “marks” with additional event information. Such event data is ubiquitous in modern applications, including social networks, neuronal spike trains, police reports, medical ICU data, power networks, seismic activities, and COVID-19 data. We are particularly interested in capturing the complex dependence of the discrete events data, such as the latent influence — triggering or inhibiting effects of the historical events on future events — a temporal causal relationship. I will present my recent research on this topic from estimation, uncertainty quantification, handling high-dimensional marks, and leveraging neural network representation power. The developed methods particularly consider computational efficiency and statistical guarantees, leveraging the recent advances in variational inequality for monotone operators that bypass the difficulty posed by the original non-convex model estimation problem. The performance of the proposed method is illustrated using real-world data: crime, power outage, hospital ICU, and COVID-19 data.
Jeffrey
Giansiracusa
Durham University
An E-infinity structure on the matroid Grassmannian
Abstract.
The matroid Grassmannian is the space of oriented matroids. 30 years ago MacPherson showed us that understanding the homotopy type of this space can have significant implications in manifold topology. In some easy cases, the matroid Grassmannian is homotopy equivalent to the ordinary real Grassmannian, but in most cases we have no idea whether or not they are equivalent. This is known as MacPherson's conjecture. I'll show that one of the important homotopical structures of the ordinary Grassmannians has an analogue on the matroid Grassmannian, namely the monoidal product tof direct sum that is commutative up to all higher homotopies.
March 2023
Johannes
Thürauf
Trier Universität
An Exact Method for Nonlinear Network Flow Interdiction Problems
Abstract.
We study network flow interdiction problems with nonlinear and nonconvex flow models. The resulting model is a max-min bilevel optimization problem in which the follower’s problem is nonlinear and nonconvex. In this game, the leader attacks a limited number of arcs with the goal to maximize the load shed and the follower aims at minimizing the load shed by solving a transport problem in the interdicted network. We develop an exact algorithm consisting of lower and upper bounding schemes that computes an optimal interdiction under the assumption that the interdicted network remains weakly connected. The main challenge consists of computing valid upper bounds for the maximal load shed, whereas lower bounds can directly be derived from the follower’s problem. To compute an upper bound, we propose solving a specific bilevel problem, which is derived from restricting the flexibility of the follower when adjusting the load flow. This bilevel problem still has a nonlinear and nonconvex follower’s problem, for which we then prove necessary and sufficient optimality conditions. Consequently, we obtain equivalent single-level reformulations of the specific bilevel model to compute upper bounds. Our numerical results show the applicability of this exact approach using the example of gas networks.
On the identification and optimization of nonsmooth superposition operators in semilinear elliptic PDEs
Abstract.
We study an infinite-dimensional optimization problem that aims to identify the Nemytskii operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation which minimizes the distance between the PDE-solution and a given desired state. In contrast to previous works, we consider this identification problem in a low-regularity regime in which the function inducing the Nemytskii operator is a-priori only known to be an element of H1loc. This makes the studied problem class a suitable point of departure for the rigorous analysis of training problems for learning-informed PDEs in which an unknown superposition operator is approximated by means of a neural network with nonsmooth activation functions (ReLU, leaky-ReLU, etc.). We establish that, despite the low regularity of the controls, it is possible to derive a classical stationarity system for local minimizers and to solve the considered problem by means of a gradient projection method. It is also shown that the established first-order necessary optimality conditions imply that locally optimal superposition operators share various characteristic properties with commonly used activation unctions: They are always sigmoidal, continuously differentiable away from the origin, and typically possess a distinct kink at zero.
Gregory
Sorkin
London School of Economics and Political Science
Extremal Cuts and Isoperimetry in Random Cubic Graphs
Abstract.
The minimum bisection width of random cubic (3-regular) graphs is of interest because it is one of the simplest questions imaginable in extremal combinatorics. An additional reason is that the minimum bisection of (general) cubic graphs plays a role in the construction of efficient exponential-time algorithms, and it seems likely that random cubic graphs are extremal.It is known that a random cubic graph has a minimum bisection of size at most 1/6 times its order (indeed this is known for all cubic graphs), and we reduce this upper bound to below 1/7 (to 0.13993) by analyzing an algorithm with a couple of surprising features. We increase the corresponding lower bound on minimum bisection using the Hamilton cycle model of a random cubic graph. (The Hamilton cycle approach had also decreased the upper bound on maximum cut, but this has since been improved upon by making rigorous a 2010 conjecture from statistical physics.)
Stefanie
Gerke
Royal Holloway, University of London
The k-th shortest s-t path in a weighted complete graph
Abstract.
Suppose we weight the edges of the complete graph Kn with independent exponential Exp(1) random weights, pick two distinct vertices s and t,and then successively construct and then remove the edges of minimal weight s-t paths. We describe asymptotically the distributions of the weights of the first k paths obtained in this process. In particular we show that the mean weight of the k-th path is (log n + γ + 2ζ(3) + 2ζ(5) + ... + 2ζ(2k − 1) + o(1))/n as n → when k is a constant, and where γ is the Euler--Mascheroni constant and ζ is the Riemann zeta function. This is joint work with P. Balister.
February 2023
Manfred
Scheucher
SAT encodings for planarity and other graph properties
Vadim
Semenov
NYU
The Gauss Image Problem
Abstract.
The Gauss Image problem is a generalization to the question originally posed by Aleksandrov who studied the existence of the convex body with the prescribed Aleksandrov's integral curvature. A simple discrete case of the Gauss Image Problem can be formulated as follows: given a finite set of directions in Euclidian space and the same number of unit vectors, does there exist a convex polytope in this space containing the origin in its interior with vertices at given directions such that each normal cone at the vertex contains exactly one of the given vectors. In this talk, we are going to discuss the discrete Gauss Image Problem, and its relation to other Minkowski-type problems. Two different proofs of the problem are going to be addressed: A smooth proof based on transportation polytopes and a discrete proof based on Helly’s theorem. Time permitting, we will also discuss the uniqueness statement for the problem.
Moritz
Grillo
TU Berlin
Topological Expressive Power of ReLU Neural Networks
Robin
Chemnitz
Markov chain methods for nowhere-zero flows
Thomas
Kahle
OvGU Magdeburg
No short polynomials in determinantal ideals
Abstract.
We discuss the problem of determining (algorithmically or otherwise) the shortest polynomial in an ideal of the polynomial ring. Here shortness is measured in the number of terms of polynomials. While the general problem of determining if an ideal contains a polynomial with at most t terms is unsolved, we show that the shortest polynomials in a determinantal ideal are determinants. This implies that the shortest polynomials vanishing on all rank-r matrices must have (r+1)! terms and are thus not particularly short.
The Hamilton-Jacobi Formulation of Optimal Path Planning for Autonomous Vehicles
Abstract.
We present a partial-differential-equation-based optimal path planning framework for simple self-driving cars. This formulation relies on optimal control theory, dynamic programming, and a Hamilton-Jacobi-Bellman equation, and thus provides an interpretable alternative to black-box machine learning algorithms. We design grid-based numerical methods used to resolve the solution to the Hamilton-Jacobi-Bellman equation and generate optimal trajectories. We then describe how efficient and scalable algorithms for solutions of high dimensional Hamilton-Jacobi equations can be used to solve similar problems in higher dimensions and in nearly real-time. We demonstrate all of our methods with several examples.
Jasmin
Lindner
Lohnt es sich, längste Ketten zu suchen? – Approximieren von Kettenzerlegungen mit Greedy und First-Fit
Fernando
Cortés Kühnast
On the Number of Arrangements of Pseudolines
Dustin
Ross
San Francisco State University
Putting the “volume” back in “volume polynomials”
Abstract.
Recent developments in tropical geometry and matroid theory have led to the study of “volume polynomials” associated to tropical varieties, the coefficients of which record all possible degrees of top powers of tropical divisors. In this talk, I’ll discuss a volume-theoretic interpretation of volume polynomials of tropical fans; namely, they measure volumes of polyhedral complexes obtained by truncating the tropical fan with normal hyperplanes. I’ll also discuss how this volume-theoretic interpretation inspires a general framework for studying an analogue of the Alexandrov-Fenchel inequalities for degrees of divisors on tropical fans. Parts of this work are joint with Anastasia Nathanson, Lauren Nowak, and Patrick O’Melveny.
Andrés
Rojas
HU Berlin
Cohomological rank functions on abelian surfaces via Bridgeland stability
Abstract.
In the context of abelian varieties, Z. Jiang and G. Pareschi have introduced interesting invariants called cohomological rank functions, associated to \(\mathbb{Q}\)-twisted (complexes of) coherent sheaves. We will show that, in the case of abelian surfaces, Bridgeland stability provides an alternative description of these functions. This helps to understand their general structure, and allows to compute geometrically meaningful examples. As a main application, we will give new results on the syzygies of abelian surfaces. This is a joint work with Martí Lahoz.
Philipp
Warode
TU Berlin
and
Guillaume
Sagnol
TU Berlin
Competitive Kill-and-Restart and Preemptive Strategies for Non-Clairvoyant Scheduling
Shpresim
Sadiku
TU Berlin, Zuse Institut Berlin
What is backpropagation?
Abstract.
Deep Neural Networks (DNNs) are a composition of several vector-valued functions. In order to train DNNs, it is necessary to calculate the gradient of the error function with respect to all parameters. As the error function of a DNN consists of several nonlinear functions, each with numerous parameters, this calculation is not trivial. We revisit the Backpropagation (BP) algorithm, widely used by practitioners to train DNNs. By leveraging the composite structure of the DNNs, we show that the BP algorithm is able to efficiently compute the gradient and that the number of layers in the network does not significantly impact the complexity of the calculation.
Stephen
Wright
U Wisconsin - Madison
Optimization in Data Science
Omid
Amini
École Polytechnique
Convexity in tropical geometry
Abstract.
What is a tropical polytope? Is it possible to associate a (usual) convex body to a tropical polytope? Which informations are preserved in passing? I will present results around the notion of convexity in tropical geometry, focusing on the local setting, that of tropical fans and their supports named tropicalfanfolds, with the aim of answering the above questions. Based on joint works with Matthieu Piquerez.
Iskander
Aliev
Cardiff University
Sparse integer points in rational polyhedra: bounds for the integer Carathéodory rank
Abstract.
We will give an overview of the recent results on sparse integer points (that is the integer points with relatively large number of zero coordinates) in the rational polyhedra of the form {x: Ax=b, x>=0}, where A is an integer matrix and b is an integer vector. In particular, we will discuss the bounds on the Integer Caratheodory rank in various settings and proximity/sparsity transference results.
Random Bifurcations in Chemical Reaction Networks
Silas
Rathke
FU Berlin
Pseudorandom Ramsey graphs
Abstract.
In recent years, the theory of pseudorandom graphs have received considerable attention in combinatorial research. One question raised was, how pseudorandom Ks-free graphs can be. A note by Mubayi and Verstraëte published in 2019 describes a link between pseudorandom graphs and Ramsey numbers: They show that if pseudorandom Ks-free graphs exist, then the Ramsey number r(s,t) can be bounded from below. In this talk, we will look at this note. We will try to understand the link in detail. Could pseudorandom graphs really be the "the central tool required to determine classical graph Ramsey numbers" as the authors suggest?
Tibor
Szabó
How many edges guarantee a small graph pattern?
Abstract.
A graph consist of a set of vertices and a set of edges, with each edge representing a pair of vertices. Graphs are a widely applicable model of symmetric binary relations which arise in various networks, let it be computer, transportation, or social.How many edges overall guarantee that a graph on n vertices surely contains some specific small graph pattern of edges, say forming a triangle? Or a cycle of length four? Or a complete graph on four vertices? How about other patterns?This is the theory of Turán numbers, a classical topic in Graph Theory, with many interesting theorems and even more tantalizing open problems. It provides a great variety of behaviours and a rich arsenal of proof methods. In the talk we aim an introduction into these. No familiarity with graphs will be assumed.
Gari Yamel
Peralta Alvarez
HU Berlin
An adelic formula of Chern-Weil type for the height of a toric variety
Abstract.
The philosophy behind height functions is to measure the complexity of objects. These functions have proven to be extremely useful when dealing with finitude statements of arithmetic objects. Gillet and Soulé developed an arithmetic version of intersection theory which produces height functions on arithmetic varieties. These heights depend on the choice of a smooth hermitian line bundle. While theoretically satisfactory, writing down smooth metrics can be extremely difficult. In the context of Shimura varieties, the natural metrics to consider are singular. This calls for an extension of the theory that admits these examples. Recently, Yuan and Zhang gave a general framework that is compatible with known extensions. In this talk, we work out the case of toric varieties. This case can be described explicitly, and we can compute heights for singular metrics with a formula that resembles Chern-Weil theory.
Sylvain
Spitz
TU Berlin
The Polyhedral Geometry of Truthful Auctions
Dongho
Chae
Chung-Ang University, Seoul
On the active vector model generalizing 3D Euler equations, Electron-MHD system and the surface quasi-geostrophic equations
Emmanuel
Silva
Consecutive pattern-avoidance in Catalan words according to the last symbol
Sofía
Garzón
FU Berlin
Weighted Ehrhart Theory
Abstract.
Since Ehrhart's original work in the late 1960's, Ehrhart theory has developed into a key topic at the intersection of polyhedral geometry, number theory, commutative algebra, algebraic geometry, enumerative combinatorics, and integer programming. Here we examine an approach where instead of counting lattice points in polytopes, we define a weight function and count the evaluations of this function at the lattice points instead. We will study which of the properties of the original h*-vector hold more generally in the weighted case for some nicely behaved functions and we will see some counterexamples as to why we need some hypothesis on the weights. This leads us to a generalization of Stanley’s celebrated theorem about the h*-polynomial of the Ehrhart Series having nonnegative coefficients for some specific families of weight functions.
Igor
Makhlin
Skoltech
Poset polytopes and toric degenerations
Abstract.
In the 1980s Stanley introduced the order polytope and chain polytope associated with a poset while Hibi considered the closely related notions of the Hibi ideal and Hibi ring (as they are now known) associated with a distributive lattice. During the subsequent decades it was realized that these concepts can be applied to the construction and study of toric degenerations of Grassmannians and flag varieties. Moreover, virtually all sufficiently explicit and sufficiently general constructions of such degenerations known today can be interpreted in terms of poset polytopes and Hibi ideals. I will give an overview of these notions and explain how they lead to toric degenerations, reviewing some foundational and some very recent results in this direction.
January 2023
Martín
Sombra
ICREA and Universitat de Barcelona
The zero set of the independence polynomial of a graph
Abstract.
In statistical mechanics, the independence polynomial of a graph \(G\) arises as the partition function of the hard-core lattice gas model on \(G\). The distribution of the zeros of these polynomials when \( G \to +\infty\) is relevant for the study of this model and, in particular, for the determination of its phase transitions. In this talk, I will review the known results on the location of these zeros, with emphasis on the case of rooted regular trees of fixed degree and varying depth \(k \ge 0\). In an on-going work with Juan Rivera-Letelier (Rochester, USA) we show that for these graphs, the zero sets of their independence polynomials converge as \(k \to \infty\) to the bifurcation measure of a certain family of dynamical systems on the Riemann sphere. In turn, this allows to show that the pressure function of this model has a unique phase transition, and that it is of infinite order.
Dingyu
Yang
HU Berlin
What is an infinity structure?
Abstract.
Producing a loop of unit length by joining two such ones is an operation which is almost associative. An infinity structure makes this precise. We explain how to think about such a higher structure, and illustrate using a couple of examples from linking of ancient Borromean rings to some glimpse of current research.
Paul
Breiding
Universität Osnabrück & MPI MiS Leipzig
Sampling from Algebraic Varieties
Abstract.
I will discuss recent progress on the problem of sampling from a nonlinear real smooth algebraic variety; that is a nonlinear smooth manifold defined as the zero set of polynomial equations.
Bruno
Vallette
U Sorbonne Paris Nord
Why higher structures?
Rimma
Hämäläinen
On Subdivision Numbers of Triangulations and Equiareality
Karin
Schaller
FU Berlin
Combinatorial identity relating LDP-polygons to the number 12
Abstract.
In [Batyrev-S. 2017], we give a combinatorial interpretation of the stringy Libgober-Wood identity in terms of generalized stringy Hodge numbers and intersection products of stringy Chern classes for arbitrary projective Q-Gorenstein toric varieties. Simultaneously we introduce stringy E-functions and stringy Chern classes in general for projective varieties with at worst log-terminal singularities. As an application we derive a novel combinatorial identity extending the well-known formula for reflexive polygons including the number 12 to LDP-polygons and toric log del Pezzo surfaces, respectively. In the mentioned work, our proof depends largely on the powerful connection between algebraic geometry and combinatorics of polytopes. Nevertheless, we will leave the connection behind and present a purely combinatorial proof for this identity that is equivalent to the stringy Libgober-Wood identity and relates LDP-polygons to the number 12. This talk is based on joint work in progress with Ulrike Bücking, Christian Hasse, and Jan-Hendrik de Wiljes.
Clemens
Sirotenko
Machine Learning for Quantitative MRI
Abstract.
The field of quantitative Magnetic Resonance Imaging aims at extracting physical tissue parameters from a sequence of highly under sampled MR images. One recently proposed approach attempts to solve this problem by estimating a set of unknown parameters in a system of ordinary differential equations. While classical approaches such as the Levenberg Marquardt algorithm or Landweber iteration yield good results under small noise levels, numerical experiments with low sampling rates and noise of large magnitude lead to unsatisfactory outcomes and unstable convergence behavior. Therefore, a spatial regularization approach based on coupled dictionary learning is proposed. From a mathematical viewpoint this ends up in a variety of non convex and non smooth optimization problems. Iterative schemes to solve these problems are discussed and convergence to equilibrium points is studied. Moreover numerical results and open questions are presented.
Adam
Afandi
University of Münster
An Ehrhart Theory For Tautological Intersection Numbers
Abstract.
The tautological intersection theory of the moduli space of stable pointed curves is an active field of research in modern algebraic geometry. In this talk, I'll explain how Ehrhart theory can give a novel perspective on tautological intersection numbers. In particular, one can arrange tautological intersection numbers into families of evaluations of Ehrhart polynomials of partial polytopal complexes. The proof of this result relies on a theorem of Breuer, which classifies the Ehrhart polynomials of partial polytopal complexes. Time permitting, I will also discuss my current attempts to generalize this Ehrhart phenomenon.
Angel
Rios Ortiz
MPI Leipzig
Asymptotic base loci on Hyperkähler varieties
Abstract.
Hyperkähler varieties can be thought as higher dimensional analogues of K3 surfaces. As such, it is generally expected that properties that hold for K3 surfaces can be generalised (appropriately) for this class of varieties as well. In this talk I will discuss ongoing joint work with Francesco Denisi (Bologna University) that characterizes the so-called asymptotic base loci of a big divisor in a Hyperkähler variety in terms of rational curves on it.
Yamaan
Attwa
FU Berlin
Towards the Erdős-Hajnal conjecture for P5
Abstract.
The Erdős-Hajnal Conjecture is one of the most well-known problems in extremal combinatorics. It states that in contrast to the general n-vertex graph, if one forbids any fixed graph H as an induced subgraph, instead of a tight guarantee of a logarithmic-size homogeneous set, one can guarantee a homogeneous set of polynomial size. While the conjecture is far from being fully resolved, it is known to be true for some forbidden subgraphs of small order. The smallest forbidden subgraph for which the conjecture remains open is the path on five vertices. In this talk I intend to review a recent paper by Blanco and Bucić guaranteeing a homogeneous set of size 2Ω(log(n)^2/3) in P5-free graphs. This result is an improvement to the 2Ω(log(n)^1/2) size of a homogeneous set guaranteed by a result of Erdős and Hajnal for any forbidden subgraph.
Daniel
Duan
On multidimensional first-order hyperbolic systems modeling structured populations
Andreas
Neuenkirch
University of Mannheim
Strong approximation of the CIR process: A never ending story?
Felix
Schröder
Introduction to Nowhere zero flows
Martina
Juhnke
Osnabrück
On cosmological polytopes
Abstract.
In this talk, I will consider cosmological polytopes, a class of lattice polytopes that can be constructed from a graph. Those polytopes were originally defined and studied by Arkani-Hamed, Benincasa, and Postnikov in their study of the wavefunction of the universe. The goal of this talk is to understand how geometric invariants, e.g., their volume and triangulations, of these polytopes are related to the combinatorics of the underlying graph. In particular, we will show that these polytopes have a regular unimodular triangulation by computing an explicit squarefree Gröbner basis for the corresponding toric ideals. We will describe the maximal cells in these triangulations by Feynman diagram like graphs. As a consequence, we are able to compute the normalized volume of the cosmological polytope for paths and cycles. This is joint work (in progress) with Liam Solus and Lorenzo Venturello.
Alberto
Espuny Díaz
TU Ilmenau
Threshold for (some) spanning trees in random geometric graphs
Abstract.
Consider the following model of random graphs: a total of n vertices are assigned to uniformly random positions on the unit square, independently of each other, and any two vertices are then joined by an edge if the distance between their positions is less than a given parameter r. This is called the random geometric graph G(n,r) and, similarly to the binomial random graph G(n,p), increasing properties exhibit thresholds (with respect to the parameter r) which we wish to understand. The behaviour of random geometric graphs, however, is very different from the behaviour of G(n,p). In this talk, I will highlight some of these differences and eventually focus on the case of balanced s-ary trees, for which we have established the threshold. This is based on joint work with Lyuben Lichev, Dieter Mitsche and Alexandra Wesolek.
Emre
Sertöz
Leibniz University of Hannover
Limit periods, arithmetic, and combinatorics
Abstract.
With Spencer Bloch and Robin de Jong, we recently proved that in a nodal degeneration of smooth curves, the periods of the resulting limit mixed Hodge structure (LMHS) contain arithmetic information. More precisely, if the nodal fiber is identified with a smooth curve C glued at two points p and q then the LMHS relates to the Neron--Tate height of p-q in the Jacobian of C. In making this relation precise, we observed that a "tropical correction term" is required that is based on the degenerate fiber. In this talk, I will explain this circle of ideas with the goal of arriving at the tropical correction term.
Joshua
Lam
HU Berlin
Infinitely many rank two motivic local systems
Abstract.
A natural question in the study of motivic local systems is whether there are infinitely many such with fixed rank and determinant on a fixed proper curve, or a punctured curve if we further fix the conjugacy classes at the punctures. I'll discuss joint work with Daniel Litt, in which we answer this in the positive in the punctured case. More precisely, we construct all the rank two motivic local systems on P^1-4 points, with unipotent monodromy at three points, and "1/2-unipotent"-monodromy at the remaining point. Time permitting, I'll show how this implies several conjectures of Sun, Yang and Zuo coming from the theory of Higgs-de Rham flow.
Jörg
Wolf
Chung-Ang University, Seoul
A note on the discretely self-similar solutions to the Euler equations
Andrea
Di Lorenzo
HU Berlin
What is a rational variety?
Abstract.
Starting from rational algebraic curves, I will discuss the notion of rational varieties, giving some concrete examples. Next, I will present a classical problem: how can we tell a rational variety from a non-rational (i.e. irrational) one? I will briefly sketch some strategies for tackling this problem and outline some open questions.
Sandro
Roch
Introduction to hyperplane arrangements
Robert
Lazarsfeld
Stony Brook U
How irrational is an irrational variety?
Martin
Winter
Warwick
On 4-flat graphs and the embeddability of 2-complexes in dimension four
Abstract.
I will discuss the following graph class introduced by van der Holst: for a graph G let C(G) denote the 2-dimensional (CW-)complex obtained by gluing 2-cells to all cycles of G. G is called 4-flat if C(G) embeds (piecewise linearly) into R^4. This graph class is a natural candidate for generalizing planar and linkless graphs and I briefly address the (partially conjectural) evidence for and against this claim. In our work we studied operations on 2-complexes that preserve embeddability into R^4. Based on this we were able to confirm two conjectures of van der Holst: 4-flat graphs are closed under doubling edges and under Delta-Y transforms. Whether 4-flat graphs are closed under Y-Delta transforms remains open. We furthermore showed that the following two graph classes are actually equivalent to 4-flat graphs, even though the requirement is seemingly much stronger (i) resp. weaker (ii): instead of attaching 2-cells to all cycles of G, we attach 2-cells to (i) all closed walk in G. (ii) all induced cycles in G. If time remains, I demonstrate how topological graph theory can be used to construct a 2-complex that embeds into R^4 piecewise linearly, but not smoothly. This talk is a result of joined work with Agelos Georgakopoulos (University of Warwick) and Tam Nguyen-Phan (Karlsruhe Institute of Technology).
Catherine
Babecki
University of Washington
Structure and Complexity of Graphical Designs for Weighted Graphs through Eigenpolytopes
Abstract.
We extend the theory of graphical designs, which are quadrature rules for graphs, to positively weighted graphs. Through Gale duality for polytopes, we show that there is a bijection between graphical designs and the faces of eigenpolytopes associated to the graph. This bijection proves the existence of graphical designs with positive quadrature weights, and upper bounds the size of a graphical design. We further show that any combinatorial polytope appears as the eigenpolytope of a positively weighted graph. Through this universality, we establish two complexity results for graphical designs: it is strongly NP-complete to determine if there is a graphical design smaller than the mentioned upper bound, and it is #P-complete to count the number of minimal graphical designs. Joint work with David Shiroma. https://arxiv.org/abs/2209.06349.
Robert
Lazarsfeld
Stony Brook
Measures of association for algebraic varieties
Abstract.
I will discuss joint work with Oliver Martin addressing the following (vague) question: given two projective varieties \(X\) and \(Y\) of the same dimension, how far are \(X\) and \(Y\) from being birationally isomorphic?
Aswin
Kannan
MATH+ Junior Research Group Leader
Multiobjective Learning in Solar Energy Prediction: Benefits and Algorithms
Abstract
Stephan
Schmidt
HU Berlin
Higher order methods for geometric inverse problems
Helmut
Harbrecht
University of Basel
Shape optimization for time-dependent domains
Tibor
Szabó
FU Berlin
Results, problems, and a couple of proofs from Oberwolfach
Peter
Friz
TU Berlin
Brownian motion on a singular surface
Kate
Smith-Miles
University of Melbourne; OPTIMA
Stress-testing algorithms via Instance Space Analysis
Abstract.
Instance Space Analysis (ISA) is a recently developed methodology to support objective testing of algorithms. Rather than reporting algorithm performance on average across a chosen set of test problems, as is standard practice, ISA offers a more nuanced understanding of the unique strengths and weaknesses of algorithms across different regions of the instance space that may otherwise be hidden on average. It also facilitates objective assessment of any bias in the chosen test instances, and provides guidance about the adequacy of benchmark test suites and the generation of more diverse and comprehensive test instances to span the instance space. This talk provides an overview of the ISA methodology, and the online software tools that are enabling its worldwide adoption in many disciplines. A case study comparing algorithms for university timetabling is presented to illustrate the methodology and tools, with several other applications to optimisation, machine learning, computer vision and quantum computing highlighted.
Robert
Lauff
Plus und Minus: Zur Kombinatorik von Signotopen
Wolfgang
König
Prof. Dr.
Wird ein Gigant erscheinen oder nicht?
Abstract.
Wir werfen zufällig Punkte in den Raum und verbinden zufällig je zwei Punkte mit einer Kante oder nicht. Hat dieser Graph eine unendlich große Komponente oder nicht? -- Wir werfen lauter zufällige Schlingen mit fester und sehr großer Gesamtlänge in eine große Box. Wird eine der Schlingen ganz besonders lang sein? -- Wir starten mit vielen Partikeln der Masse Eins und lassen sie paarweise immer wieder mit einander koagulieren. Gibt es später ein ganz besonders großes Partikel? -- Dies sind drei Typen von zufälligen Prozessen, in denen manchmal ein Mikro-Makro-Phasenübergang beobachtet werden kann: das Auftreten einer gigantisch großen Teilstruktur. Die drei Phasenübergänge in diesen Beispielen heißen Perkolation, Bose-- Einstein-Kondensation und Gelation. Um sie herum gibt es viele tolle mathematische Ergebnisse, aber noch mehr ist weithin offen. Daher gibt es viele Modellvarianten und viele hochinteressante Teilaufgaben, von denen etliche auch für Bachelorarbeiten geeignet sind.
Leonid
Monin
MPI Leipzig
Inversion of matrices, ML-degrees and the space of complete quadrics
Abstract.
What is the degree of the variety \(L^{-1}\) obtained as the closure of the set of inverses of matrices from a generic linear subspace \(L\) of symmetric matrices of size \(n\times n\)? Although this is an interesting geometric question in its own right, it is also motivated by algebraic statistics: the degree of \(L^{-1}\) is equal to the maximum likelihood degree (ML-degree) of a generic linear concentration model. In 2010, Sturmfels and Uhler computed the ML-degrees for \(\mathrm{dim}(L)\) less than 5 and conjectured that for the fixed dimension of \(L\) the ML-degree is a polynomial in \(n\). In my talk I will describe geometric methods to approach the computation of ML-degrees which in particular allow to prove the polynomiality conjecture. The talk is based on a joint works with Laurent Manivel, Mateusz Michalek, Tim Seynnaeve, Martin Vodicka, and Jaroslaw A. Wisniewski.
December 2022
Moritz
Kerz
Regensburg
The Picard-Lefschetz formula for normal crossings
Abstract.
In the study of semi-stable degeneration of Lefschetz pencils one is led to a generalization of the classical Picard-Lefschetz formula for certain perverse sheaves on normal crossing spaces. In the talk I will recall the formalism of nearby cycle and vanishing cycle functors and I will explain how Hodge theory allows one to obtain the normal crossing Picard-Lefschetz formula. Joint work with A. Beilinson and H. Esnault.
Mike
Theiss
Deriving a constrained Mean-Field Game
Abstract.
Mean-Field Games (MFGs) have a wide area of applications, i.e. crowd motion, flocking models, or behavior of investors. In most of these applications, it makes sense to assume constraints to the control or the state. We will start with some basic properties of a specific linear quadratic N-player game with mean field interaction. Afterward, we let the number of players N go to infinity, for deriving a "constrained MFG". Therefore we have to analyze the mean-field interaction, which describes the behavior of the whole group modeled as a flow of probability measures. Interesting is also the connection of the MFG to the original N-player problem. In the end, we will discuss some ideas on how to solve such constrained MFGs.
Silas
Rathke
FU Berlin
What is a k-tensor?
Abstract.
Don't Panic! $k$-Tensors are quite different to tensor products and I also won't write down any commutative diagram, I promise! Instead, $k$-tensors are used to answer questions in extremal combinatorics using linear algebra. We are going look at a beautiful proof for a surprising fact in extremal combinatorics and then try to generalize the main idea, which will lead us to $k$-tensors and its slice rank.
Lisa
Sauermann
MIT
On the cap-set problem and the slice rank polynomial method
Louis
Borchert
Hamiltonkreise in Mindestabstandsgraphen in der euklidischen Ebene
Responsibility without Causation – Probability Raising as an Independent Desert Base
Abstract
Marian
Margraf
Perfekt sichere Verschlüsselungsverfahren
Abstract.
Der Vortrag führt zunächst Verschlüsselungsverfahren ein, gibt einfache Beispiele und diskutiert verschiedene Sicherheitsbegriffe. Im Hauptteil des Vortrags wird auf den Begriff der perfekten Sicherheit eingegangen, ein Verfahren ist perfekt sicher, wenn selbst Angreifer mit unbeschränkten Ressourcen dieses Verfahren nicht brechen können. Wir definieren den Begriff formal, geben eine äquivalente Beschreibung an (ein Resultat von Shannon aus dem Jahr 1949) und zeigen, dass perfekt sichere Verschlüsselungsverfahren tatsächlich existieren. Allerdings haben sie den großen Nachteil, dass die für die Verschlüsselung eingesetzten Schlüssel mindestens so groß sein müssen wie der Informationsgehalt des zu verschlüsselnden Textes.
Christoph
Hertrich
LSE
Training Fully Connected Neural Networks is ∃R-Complete
Feliks
Nüske
Max Planck Institute
Koopman Analysis of Quantum Systems
Abstract.
Koopman operator theory has been successfully applied to problems from various research areas such as fluid dynamics, molecular dynamics, climate science, engineering, and biology. Most applications of Koopman theory have been concerned with classical dynamical systems driven by ordinary or stochastic differential equations. In this presentation, we will first compare the ground-state transformation and Nelson’s stochastic mechanics, thereby demonstrating that data-driven methods developed for the approximation of the Koopman operator can be used to analyze quantum physics problems. Moreover, we exploit the relationship between Schrödinger operators and stochastic control problems to show that modern data-driven methods for stochastic control can be used to solve the stationary or imaginary-time Schrödinger equation. Our findings open up a new avenue towards solving Schrödinger’s equation using recently developed tools from data science.
Leticia
Mattos
FU Berlin
Bounds For Essential Covers Of The Cube
Abstract.
An essential cover of the vertices of the n-cube {-1,1}n by hyperplanes is a minimal covering where no hyperplane is redundant and every variable appears in the equation of at least one hyperplane. Linial and Radhakrishnan gave a construction of an essential cover with n/2+1 hyperplanes and showed that √n hyperplanes are required. Recently, Yehuda and Yehudayoff improved the lower bound by showing that any essential cover of the n-cube contains at least n0.52 hyperplanes. In this talk, we show that n5/9 hyperplanes are needed. This is based on a joint work with Araujo and Balogh.
Jieao
Song
HU Berlin
Geometry and moduli of Debarre-Voisin hyperkähler fourfolds
Abstract.
Debarre-Voisin varieties are one of the few known locally complete families of projective hyperkähler fourfolds, constructed inside the Grassmannian \(\mathrm{Gr}(6,10)\). Motivated by the case of varieties of lines on cubic fourfolds, we study their geometry as well as that of two associated Fano varieties, focusing on some special divisors in the moduli. This is a joint work with Vladimiro Benedetti.
Roberto
Gualdi
Regensburg
How to guess the height of the solutions of a system of
polynomial equations
Abstract.
A beautiful result due to Bernstein and Kushnirenko allows to predict the number of solutions of a system of Laurent polynomial equations from the combinatorial properties of the defining Laurent polynomials. In a joint work with Martín Sombra (ICREA and Universitat de Barcelona), we give intuitions for an arithmetic version of such a theorem. In particular, in the easy case of the planar curve x + y + 1 = 0, we show how to guess the height of its intersection with a twist of itself by a torsion point. The talk will involve the Arakelov geometry of toric varieties, special values of the Riemann zeta function and, unexpectedly, the most famous detective of the world literature.
Annette
Lutz
TU Darmstadt
Incremental Optimization of Potential Based Flows
Peter
Friz
Weierstrass Institute/TU Berlin
Why rough stuff matters for UQ
Paul
Brommer-Wierig
FU Berlin
What is tame geometry?
Abstract.
In the 80's Grothendieck claimed that general topology was unfit for the study of the shape of geometric objects since it is possible to realise arbitrary wild pathologies within it. He proposed a tame geometry which should not exhibit such phenomena. Around the same time, model theorists interested in a seemingly unrelated question developed the now widely accepted framework of o-minimal structures. In this talk, I will explain the notion of an o-minimal structure and discuss basic examples of it. I will then continue explaining why they lead to the correct notion of tame geometry. I will finish the talk by sketching applications of o-minimal structures in algebraic geometry.
Bruno
Klingler
HU Berlin
Hodge theory, between algebraicity and transcendence
Alex
Black
UC Davis
Realizable Standard Young Tableaux
Abstract.
Given two vectors u and v, their tropical outer product (or outer sum) is given by the matrix A with entries A_{jk} = u_{j} + v_{k}. If the entries of u and v are increasing and sufficiently generic, the total ordering of the entries of the matrix is a standard Young tableau of rectangular shape. Similarly, given a generic configuration of n points in two-dimensions, the total ordering of all the slopes between pairs of points gives rise to a standard Young tableau of staircase shape. We call standard Young tableaux that arise in either way realizable. Realizable tableaux appear independently in many contexts including sorting algorithms, quantum computing, random sorting networks, reflection arrangements, fiber polytopes, and Goodman and Pollack's theory of allowable sequences. However, little is known about their combinatorial structure. In particular, a basic question is: How many are there? Equivalently, what is the probability that a random standard Young tableaux is realizable? In this talk, I will discuss my recent joint work with Igor Araujo, Amanda Burcroff, Yibo Gao, Robert Krueger, and Alex McDonough in which we use algebraic and polyhedral geometry to address these questions asymptotically.
Felix
Schröder
Completing Partial Generalized Signotopes is NP-hard
November 2022
Zafeirakis
Zafeirakopoulos
Gebze Technical University
Polyhedral Omega (+ applications)
Abstract.
Polyhedral Omega is an algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon’s iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decomposition and Barvinok’s short rational function representations. This synthesis of ideas makes Polyhedral Omega by far the simplest algorithm for solving linear Diophantine systems available to date. After presenting the algorithm, we will see some applications and generalizations.
Laurentiu
Maxim
University of Wisconsin
On variants of the Singer-Hopf conjecture in complex geometry
Abstract.
The conjectures of Singer and Hopf predict the sign of the topological Euler characteristic of a closed aspherical manifold. In this talk I will discuss various generalizations (e.g., for singular spaces or Hodge enhancements) and partial results concerning the conjectures of Singer and Hopf in the context of Kähler geometry.
Laurentiu
Maxim
Wisconsin
On the topology of aspherical complex projective manifolds and related questions
Abstract.
I will report on recent progress on various open problems involving aspherical complex projective manifolds, including the Singer-Hopf conjecture and the Bobadilla-Kollar conjecture. The first part will be very informal, stating the main problems, historical developments, and some recent results. In the second part, I will introduce the main technical tools and sketch proofs of some of the recent results. Note: This is the first of a series of two talks. The second talk will be in the algebraic geometry seminar on Wednesday, see here.
Michael
Feischl
TU Wien
Local equivalence of residuals in random PDEs
Remo
Kretschmann
University of Würzburg
Optimal and Bayesian hypothesis testing in statistical inverse problems
Alexandru
Suciu
Polyhedral products, duality properties, and cohomology jump loci
Abstract.
The polyhedral product is a functorial construction that assigns to each simplicial complex $K$ on $n$ vertices, and to each pair of topological spaces, $(X,A)$, a certain subspace, $\mathcal{Z}_K(X,A)$, of the Cartesian product of $n$ copies of $X$. I will discuss some of the relationships between the duality properties of these spaces and the Cohen-Macaulay property of the original simplicial complex, and the interplay between the cohomology jump loci of right-angled Artin groups and Bestvina--Brady groups.
Georg
Loho
University of Twente
Generalized permutahedra and positive flag Dressians
Abstract.
Generalized permutahedra appear in various areas from submodular function optimization to the study of Grassmannians and Flag varieties. I will give an introduction to some of their geometric and combinatorial properties leading to the study of their subdivisions. Those arise naturally in Tropical Geometry and Discrete Convex Analysis. I present a new characterization of the corresponding height functions and show how to refine this for the study of positivity. The talk is based on joint work with Michael Joswig, Dante Luber and Jorge Alberto Olarte (https://arxiv.org/abs/2111.13676).
Aldo
Kiem
ZIB
Flag Algebras for Edge-Colored Graphs
Abstract.
Flag Algebras is a highly successive formalism to study problems in extremal combinatorics. It has achieved the best known bounds for Turán’s (3,4)-problem as well as tight bounds for the three color Ramsey multiplicity of the triangle, Erdös’s Pentagon conjecture and a conjecture of Erdös and Sós on the asymptotic frequency of Rainbow triangles. What is special about Flag Algebras is that a significant amount of work in finding a Flag Algebra proof certificate can be automated by relaxing the given problem to a semidefinite program (SDP). In this talk, we will formally define Flag Algebras with many examples from the theory of c-edge-colored complete graphs. We will then see how to set up a Flag Algebra SDP and will discuss methods, old and new, to reduce the size of the SDP as well as splitting it up into smaller blocks. Finally, we will present our newest calculation for the four color Ramsey multiplicity of the triangle. This is joint work with Prof. Pokutta (TU Berlin and ZIB) and Dr. Christoph Spiegel (ZIB).
Katherina
von Dichter
TU München / Cottbus
Mean inequalities for symmetrizations of convex sets
Abstract.
The arithmetic-harmonic mean inequality can be generalized for convex sets, considering the intersection, the harmonic and the arithmetic mean, as well as the convex hull of two convex sets. We study those relations of symmetrization of convex sets, i.e., dealing with the means of some convex set C and -C. We determine the dilatation factors, depending on the asymmetry of C, to reverse the containments between any of those symmetrizations, and tighten the relations proven by Firey and show a stability result concerning those factors near the simplex.
Aswin
Kannan
HU Berlin
Multiobjective Learning in Solar Energy Prediction: Benefits and Algorithms
Alex
Suciu
Northeastern University
Topological invariants of groups and tropical geometry
Abstract.
There are several topological invariants that one may associate to a finitely generated group \(G\) -- the characteristic varieties, the resonance varieties, and the Bieri–Neumann–Strebel invariants -- that keep track of various finiteness properties of certain subgroups of \(G\). These invariants are interconnected in ways that makes them both more computable and more informative. I will describe in this talk one such connection, made possible by tropical geometry, and I will provide examples and applications pertaining to complex geometry and low-dimensional topology.
Siddarth
Mathur
Orsay
Searching for the impossible Azumaya algebra
Abstract.
In two 1968 seminars, Grothendieck used the framework of étale cohomology to extend the definition of the Brauer group to all schemes. Over a field, the objects admit a well-known algebro-geometric description: They are represented by \( \mathbb{P}^n\)-bundles (equivalently: Azumaya Algebras). Despite the utility and success of Grothendieck's definition, an important foundational aspect remains open: Is every cohomological Brauer class over a scheme represented by a \( \mathbb{P}^n\)-bundle? It is not even known if smooth proper threefolds over the complex numbers have enough Azumaya algebras! In this talk, I will outline a strategy to construct a Brauer class that cannot be represented by an Azumaya algebra. Although the candidate is algebraic, the method will leave the category of schemes and use formal-analytic line bundles to create Brauer classes. I will then explain a strange criterion for the existence of a corresponding Azumaya Algebra. At the end, I will reveal the unexpected conclusion of the experiment.
Svenja
Griesbach
TU Berlin
Improved Approximation Algorithms for the Expanding Search Problem
Thu Hien
Nguyen
Leipzig University and V.N. Karazin Kharkiv National University
Some conditions for entire functions to belong to the Laguerre-Polya class in terms of their Taylor coefficients
Manfred
Scheucher
Isomorphism of point sets
Omid
Amini
\'Ecole Polytechnique
Polyhedral geometry in higher rank and non-linear settings
Abstract.
New types of polyhedral geometry and combinatorial structures emerge in the study of asymptotic geometry of complex manifolds. This includes a multi-scale version of polyhedral geometry, with links to a theory of higher rank combinatorial structures, which combine parts living in different scales at infinity, and a theory of convexity in a piecewise linear setting. The talk presents some features of these recent developments, and discusses applications in complex geometry. Based on joint works with Hernan Iriarte, with Noema Nicolussi, and with Matthieu Piquerez.
Tensor Correspondence Analysis (CA) – A multidisciplinary investigation of linguistic changes in Ancient Egypt
Abstract
Olaf
Parczyk
FU Berlin
A general approach to transversal versions of Dirac-type theorems
Abstract.
Given a collection of m hypergraphs on the same vertex set, an m-edge graph F is a transversal if there is each edge comes from a different graph of the collection. How large does the minimum degree of each graph in the collection need to be so that it necessarily contains a copy of F that is a transversal? Each graph in the collection could be the same hypergraph, hence the minimum degree of each of them needs to be large enough to ensure that it individually contains F. In this talk, we discuss a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac-type results for (powers of) Hamilton cycles. This is joint work with Pranshu Gupta, Fabian Hamann, Alp Müyesser, and Amedeo Sgueglia.
Annette
Huber-Klawitter
Freiburg
Period Numbers
Abstract.
Period numbers are complex numbers like \( \pi, \log(2) \) or \( \zeta(5)\). They can be described as values of integrals. As apparent from the examples, such numbers appear in many places in mathematics and are of great interest in transcendence theory. More recently, a variant involving also the exponential function has come into focus. I am going to explain the definition and its conceptual interpretation.
Martin
Skutella
TU Berlin
A Note on the Quickest Minimum Cost Transshipment Problem
Marc
Sedjro
Monge-Ampere equations on a free boundary domain
Han
Lie
University of Potsdam
Frechet derivatives of path functionals of stochastic differential equations
Stefan
Felsner
The flip-graph of arrangements of pairwise intersecting pseudocircles
Andreas
Kretschmer
OvGU Magdeburg
Thin polytopes: lattice polytopes with vanishing local h*-polynomial
Abstract.
I will introduce the local h*-polynomial of a lattice polytope and give its main properties, like palindromicity and non-negativity, and how it interacts with the usual h*-polynomial and combinatorial invariants of the polytope. We call a lattice polytope thin if its local h*-polynomial vanishes, and I am going to explain the classification of thin polytopes in dimension 3 and a characterization in the Gorenstein case of arbitrary dimension. I will also mention an open question relating thinness to the width of a lattice polytope.
Analysis of stochastic gradient descent in continuous time
Abstract.
Optimisation problems with discrete and continuous data appear in statistical estimation, machine learning, functional data science, robust optimal control, and variational inference. The 'full' target function in such an optimisation problem is given by the integral over a family of parameterised target functions with respect to a discrete or continuous probability measure. Such problems can often be solved by stochastic optimisation methods: performing optimisation steps with respect to the parameterised target function with randomly switched parameter values. In this talk, we discuss a continuous-time variant of the stochastic gradient descent algorithm. This so-called stochastic gradient process couples a gradient flow minimising a parameterised target function and a continuous-time 'index' process which determines the parameter. We first briefly introduce the stochastic gradient processes for finite, discrete data which uses pure jump index processes. Then, we move on to continuous data. Here, we allow for very general index processes: reflected diffusions, pure jump processes, as well as other Lévy processes on compact spaces. Thus, we study multiple sampling patterns for the continuous data space. We show that the stochastic gradient process can approximate the gradient flow minimising the full target function at any accuracy. Moreover, we give convexity assumptions under which the stochastic gradient process with constant learning rate is geometrically ergodic. In the same setting, we also obtain ergodicity and convergence to the minimiser of the full target function when the learning rate decreases over time sufficiently slowly.
Omid
Amini
Ecole Polytechnique Paris
The tropical Hodge conjecture
Abstract.
The aim of the talk is to present the formulation of the Hodge conjecture for tropical varieties, and to explain a proof in the case the tropical variety is triangulable. This provides a partial answer to a question of Kontsevich. The proof uses Hodge theoretic properties of tropical varieties established in our companion works, which will be reviewed in the talk. These results generalize to the global setting the work of Adiprasito-Huh-Katz on combinatorial Hodge theory, by going in the local setting beyond the case of matroids, and provide answers to conjectures of Mikhalkin and Zharkov. Based on joint works with Matthieu Piquerez.
Giulia
Codenotti
FU Berlin
Flatness constants and lattice-reduced convex bodies
Abstract.
The lattice width of a convex body measures how "thin" the body is in lattice directions. The so-called flatness theorem states that in each fixed dimension there is an upper bound for the lattice width of a special class of convex bodies, those which are hollow. In this talk we introduce these definitions and certain generalizations thereof and look at the few known extremal hollow convex bodies achieving maximum lattice width. This leads us to a conjecture about extremal convex bodies, namely that they are lattice-reduced. This class of convex bodies has not been previously studied, and we present (many) questions and (some) answers about such lattice-reduced convex bodies.
Riccardo
Pengo
MPI Bonn
On the Northcott property for special values of L-functions
Abstract.
According to Northcott's theorem, each set of algebraic numbers whose height and degree are bounded is finite. Analogous finiteness properties are also satisfied by many other heights, as for instance the Faltings height. Given the many (expected and proven) links between heights and special values of L-functions (with the BSD conjecture as the most remarkable example), it is natural to ask whether the special values of an L-functions satisfy a Northcott property. In this talk, based on a joint work with Fabien Pazuki, and on another joint work in progress with Jerson Caro and Fabien Pazuki, we will show how this Northcott property is often satisfied at the left of the critical strip, and not satisfied on the right. We will also overview the links between these Northcott properties and those of the motivic heights defined by Kato, and also some effective aspects of our work, which aim at giving some explicit bounds for the cardinality of the finite sets that we come across.
Abi
Srikumar
UNSW Sydney
Approximating distribution functions in uncertainty quantification using quasi-Monte Carlo methods
Gari Yamel
Peralta Alvarez
HU Berlin
What is a Grothendieck topology?
Abstract.
A Grothendieck topology is a structure on a category that mimics the properties of open covers of topological spaces. These properties are particularly useful to define geometric structures in terms of local information, for example, an atlas on a manifold or a sheaf on an algebraic variety. The goal of this talk is to explain in simple terms how Grothendieck topologies and sheaves generalize familiar geometric objects.
Peter
Scholze
MPI Bonn
Condensed Mathematics
Helena
Bergold
Holes in Convex Drawings of $K_n$
Martin
Henk
TU Berlin
Linear and affine subspace concentration conditions (for centered polytopes)
Abstract.
The linear subspace concentration conditions play a crucial role in the log-Minkowski problem which, in the discrete setting, asks for a characterization of the cone-volumes of polytopes, i.e., the volumes of the cones given by the convex hull of a facet and the origin, which is supposed to be an interior point of the polytope. This problem is only solved in the symmetric case, but it is known that also centered polytopes, i.e, the centroid is at the origin, satisfy certain linear subspace concentration conditions. Based on these linear conditions, K.-Y. Wu introduced recently affine subspace concentration conditions and proved them for centered, reflexive, smooth lattice polytopes. We extend this result to arbitrary centered polytopes. This is joint work with Ansgar Freyer (TU Vienna) and Christian Kipp (TU Berlin).
Raman
Sanyal
Goethe-Universität Frankfurt
Monotone paths on matroids, pivot rules, and flag polymatroids
Abstract.
The well-known greedy algorithm on matroids was adapted by Edmonds to polymatroid polytopes or, essentially equivalent, generalized permutahedra. For a given objective function, the greedy algorithm traces a monotone path in the graph of the polymatroid polytope. The starting point of this work is the observation that this path is precisely the path taken by the shadow-vertex simplex algorithm. It turns out that all monotone paths are greedy paths and all are coherent in the sense of Billera-Sturmfels. Thus, monotone paths are parametrized by the associated monotone path polytope. We show that these monotone path polytope are again a generalized permutahedron and in the case of matroids are the underlying flag matroids. In this talk I will explain the geometry of these “flag polymatroids” and the connection to certain nestohedra associated to lattices of flats via so-called pivot rule polytopes. If time permits, I will also discuss relations to sorting networks and the enumeration of Young tableaux. This is joint work with Alex Black.
Irene
Spelta
University of Barcelona
Prym maps and generic Torelli theorems: the case of plane quintics.
Abstract.
The talk deals with Prym varieties and Prym maps. Prym varieties are polarized abelian varieties associated with finite morphisms between smooth curves. Prym maps are accordingly defined as maps from the moduli space of coverings to the moduli spaces of polarized abelian varieties. Once recalled the classical generic Torelli theorem for the Prym map of étale double coverings, we will move to the more recent results on the ramified Prym map \(P_{g,r}\) associated with ramified double coverings. For most of the values of \((g,r)\) a generic Torelli theorem holds and, furthermore, a global Torelli theorem holds when \(r\) is greater (or equal to) 6. At the same time, it is known that \(P_{g,2}\) and \(P_{g,4}\) have positive dimensional fibres when restricted to the locus of coverings of hyperelliptic curves. But this is not a characterization: the study of the differential \(d P_{g,r}\) shows that there are also other configurations to be considered. We will focus on the case of degree 2 coverings of plane quintics ramified in 2 points. We will show that the restriction of \(P_{g,r}\) here is generically injective. This is joint work with J.C. Naranjo.
David
Fabian
FU Berlin
Graph bootstrap percolation of random graphs
Abstract.
Given graphs H and G the H-bootstrap process on G is the process that starts with G and in which at each step every edge that is the only missing edge in a copy of H is added. Any such process stabilises after a finite number of steps. The maximum running time MH(n) is the largest number of steps of an H-bootstrap process when H is fixed and G ranges over all graphs on n vertices. In this talk we investigate MH(n) when H is distributed as the random graph G(k,p). We will show that for p = ω(log(k)/k) the maximum running time is bounded from below by ckn² for some ck>0. This is joint work with Patrick Morris and Tibor Szabó.
Constantin
Podelski
HU Berlin
The degree of the Gauss map for certain Prym varieties
and the Schottky problem
Abstract.
One approach to the Schottky problem is to study the stratification of the moduli space \( \mathcal{A}_g \) of principally polarized abelian varieties by the dimension of the singular locus of the theta divisor: Andreotti and Mayer have shown that the locus of Jacobians is an irreducible component of the stratum \( \mathcal{N}_{g-4} \subset \mathcal{A}_g\). In a similar spirit one can stratify \(\mathcal{A}_g \) by the degree of the Gauss map: Recently, Codogni and Krämer have shown that the locus of Jacobians is also an irreducible component of the corresponding Gauss stratum. Motivated by this, we study the Gauss map on various other irreducible components of \(\mathcal{N}_{g-4}\) that parametrize certain Prym varieties, and we show that for each \( g \ge 4\) one of these components has the same Gauss degree as Jacobians.
Moritz
Grillo
TU Berlin
Dynamic Programming and Semi-Coalgebras
October 2022
Klaus
Altmann
Die Infizierung des Z2
Abstract.
Wir betrachten spezielle Konfigurationen endlich vieler Punkte in der Ebene, deren Lage streng reglementiert ist. Wir studieren, wie sich diese Konfigurationen nach bestimmten Regeln ausbreiten können, um schließlich die gesamte Ebene zu „infizieren“. Wir starten mit dem kleinsten Spezialfall von vier Punkten – hier kann man schnell und spielerisch einen ersten Eindruck von der Problemstellung bekommen. Während sich das zweidimensionale Problem umfassend und klar lösen lässt, zeigen wir auch, wie die natürliche Verallgemeinerung auf drei Dimensionen offene Fragen aufwirft. Der kleinste Spezialfall handelt dabei von gewissen räumlichen Anordnungen von acht Punkten. Der Hintergrund dieses kombinatorischen Themas kommt aus der algebraischen Geometrie. Die Punkte in der Ebene repräsentieren dann Geradenbündel auf torischen Varietäten vom Picardrang 2.
Lassi
Roininen
LUT University
Cauchy Markov random field priors for Bayesian inversion
Nils
Hinrichs
Charité – Universitätsmedizin Berlin
Short-term vital parameter forecasting in the intensive care unit
Manfred
Scheucher
Plane Hamiltonian Cycles in Convex Drawings of $K_n$
Felix
Goldmann
HU Berlin
Discontinuous Galerkin method for the parabolic obstacle problem with a functional a posteriori error estimator
Andrea
Di Lorenzo
HU Berlin
Degenerations of twisted maps to algebraic stacks
Abstract.
Line bundles over curves, cyclic covers, elliptic surfaces: what these objects have in common is that they can all be regarded as maps from a family of curves to some moduli stack. Therefore, to have a controlled way to degenerate maps to algebraic stacks means having a controlled way to degenerate all the objects above, and more. How can this be obtained and made precise will be the main focus of this talk, which is based on a joint work with Giovanni Inchiostro.
Stephane
Gaubert
INRIA / CMAP École Polytechnique
Ambitropical convexity, injective hulls of metric spaces, and mean-payoff games
Abstract.
Hyperconvexity was introduced by Aronszajn and Panichpadki in the 50', to study nonexpansive mappings between metric spaces. We study a new kind of convexity, defined in terms of lattice properties. We call it ``ambitropical'' as it includes both tropical convexity and its dual, and we relate it with hyperconvexity and mean-payoff games.Ambitropical convex sets coincide with hyperconvex sets with an additional requirement: stability by an additive group action of the real numbers. They also coincide with the fixed-point sets of Shapley operators, i.e., of dynamic programming operators of (undiscounted) zero-sum games. In this way, ambitropical convex sets provide geometric representations of (calibrated) optimal stationary strategies of mean-payoff games. Moreover, there is a notion of ``ambitropical hull'', unique up to isomorphism, which we construct as the range of a tropical analogue of the Petrov-Galerkin projector, providing an alternative to the ``tight-span'' construction of Isbell and Dress for the injective hull, in the special case of an ambitropical convex set. We finally consider ambitropical polyhedra, defined as ambitropical convex sets that have a cell decomposition in terms of alcoved polyhedra. We relate them with deterministic games and order preserving retracts of the Boolean hypercube.This talk is based on a joint work with Marianne Akian and Sara Vannucci. A preliminary account of the results appeared in arXiv:2108.07748.
Marco
D'Addezio
Jussieu
Hecke orbits on Shimura varieties
Abstract.
I will talk about the proof of the Hecke orbit conjecture, proposed by Chai and Oort. I will mainly focus on two of the tools that are used. The first main ingredient is a new local result on the monodromy groups of F-isocrystals, which enhances Crew's parabolicity conjecture. Another one is the Cartier-Witt stack constructed by Bhatt-Lurie. This is a joint work with Pol van Hoften.
Javier
Cembrano
TU Berlin
Optimal Impartial Correspondences
Jens A.
Griepentrog
Well-posedness of the Maxwell equations with nonlinear Ohm law
Kyle
Huang
FU Berlin
What is a Semialgebraic Set?
Abstract.
Semialgebraic sets are a generalization of polyhedra, where instead of linear constraints we take polynomial inequalities. As the fundamental object of real algebraic geometry and capable of modeling diverse real-world phenomena, they are of interest to both pure and applied mathematicians. We'll begin with an introduction to the motivations and definitions surrounding real algebraic geometry, after which we will discuss the Tarski-Seidenberg Principle and see an illustrative application to robotics. With computer algebra!
Harri
Hakula
Aalto University
Eigenlocking – Parameter-dependent loss of convergence rate
Rimma
Hämäläinen
A Solution to a Problem About Bichromatic Configurations in the Plane
Carlos Enrique
Améndola Cerón
TU Berlin
Algebraic Statistics: Past, Present and Future
Giulia
Codenotti
An overview of classifications of lattice polytopes
Abstract.
In this survey talk we will touch on classifications of different classes of lattice polytopes, such as those of low volume, hollow polytopes, or empty simplices. We will explore the special role played by lattice width in these classifications and discuss some tools coming from Geometry of Numbers used in these contexts. A particular focus will be on empty simplices and open questions regarding their h^* vectors and related invariants.
Philipp
Bringmann
HU Berlin
A (hybrid) discontinuous least-squares finite element method with built-in a posteriori error estimation
Leonid
Monin
MPI MiS Leipzig
Polyhedral models for K-theory
Abstract.
One can associate a commutative, graded algebra which satisfies Poincare duality to a homogeneous polynomial f on a vector space V. One particularly interesting example of this construction is when f is the volume polynomial on a suitable space of (virtual) polytopes. In this case the algebra A_f recovers cohomology rings of toric or flag varieties. In my talk I will explain these results and present their recent generalizations. In particular, I will explain how to associate an algebra with Gorenstein duality to any function g on a lattice L. In the case when g is the Ehrhardt function on a lattice of integer (virtual) polytopes, this construction recovers K-theory of toric and full flag varieties.
Michaela
Borzechowski
FU Berlin
A Universal Construction for Unique Sink Orientations
Abstract.
A Unique Sink Orientation (USO) is an orientation of the hypercube graph, such that the induced subgraph of every non-empty subcube contains exactly one sink. USOs can be used to capture the combinatorial structure of many essential algebraic and geometric problems. Unfortunately, there is no known algorithm that verifies if a given orientation is actually USO and which runs in polynomial time in the dimension of the cube. Out of the many possible orientations for a given cube only few of them are USO. But still, for a cube of a fixed dimension, the number of USOs is exponential in the dimension. To generate bounds on the total number of USOs, construction techniques are needed. They are also useful to find counterexamples to suspected properties of USOs. Furthermore, families of ''bad'' USOs provide examples to show lower bounds for the worst-case runtime of algorithms. While there are some construction methods for USOs, until now they have not been able to generate all possible USOs. In this talk we present a new construction framework which can be applied to all USOs and with which we can generate every n-dimensional USO using only USOs of dimension n-1 or lower. Our universal construction was inspired by techniques from cube tilings of space. This is joint work with Joseph Doolittle (TU Graz) and Simon Weber (ETH Zürich)
Martin
Knaack
TU Berlin
Maximizing a Submodular Function with Bounded Curvature under an Unknown Knapsack Constraint
Mattes
Mollenhauer
FU Berlin
Nonparametric approximation of conditional expectation operators
Ilja
Klebanov
FU Berlin
QMC and sparse grids beyond uniform distributions on cubes: transport maps to mixture distributions
Sandro
Roch
Introduction to Cluster Algebras
Yulia
Alexandr
UC Berkeley
Logarithmic Voronoi cells
Abstract.
Logarithmic Voronoi cells are convex sets, which arise as fibers of the maximum likelihood estimation map. For discrete models, they live inside their log-normal polytopes, and for certain families of statistical models, the two sets coincide. In particular, logarithmic Voronoi cells for such families are polytopes. I will introduce these notions and investigate when this property holds. I will also talk about how this theory generalizes to Gaussian models, with lots of examples. This talk is based on joint works with Alex Heaton and Serkan Hosten.
Fabrizio
Catanese
Bayreuth
Manifolds with vanishing Chern classes and some questions by Severi/Baldassari
Abstract.
We give a negative answer to a question posed by Severi in 1951, whether the Abelian Varieties are the only manifolds with vanishing Chern classes. We exhibit Hyperelliptic Manifolds which are not Abelian varieties (nor complex tori) and whose Chern classes are zero not only in integral homology, but also in the Chow ring. We prove moreover the surprising result that Bagnera de Franchis manifolds ( quotients \(T/G\) where \(T\) is a torus and \(G\) is cyclic) have topologically trivial tangent bundle. Motivated by a more general question addressed by Mario Baldassarri in 1956, we discuss the Hyperelliptic Manifolds, the Pseudo- Abelian Varieties introduced by Roth, and we introduce a new notion, of Manifolds Isogenous to a \(k\)-Torus Product: the latter have the last \(k\) Chern classes trivial in rational homology and vanishing Chern numbers. We show that the latter class is the correct substitute for some incorrect assertions by Enriques, Dantoni, Roth and Baldassarri: in dimension 2 these are the surfaces with \(K_X\) nef and \(c_2(X) = 0\). A similar picture does not hold in higher dimension, unless we consider manifolds (isogenous to manifolds) whose tangent (resp. cotangent bundle) has a trivial summand. We survey old and new results on Kähler manifolds whose tangent (resp. cotangent bundle) has a trivial summand, and pose some open problems.
Mathias
Staudigl
Maastricht University
On the iteration complexity of first and second-order Hessian barrier algorithms for non-convex non-smooth conic optimization
Abstract.
A key problem in mathematical imaging, signal processing and computational statistics is the minimization of non-convex objective functions over conic domains, which are continuous but potentially non-smooth at the boundary of the feasible set. For such problems, we propose a new family of first and second-order interior-point methods for non-convex and non-smooth conic constrained optimization problems, combining the Hessian barrier method with quadratic and cubic regularization techniques. Our approach is based on a potential-reduction mechanism and attains a suitably defined class of approximate first- or second-order <abbr title="Karush–Kuhn–Tucker conditions">KKT</abbr> points with worst-case iteration complexity O(ϵ−2) and O(ϵ−3/2), respectively. Based on these findings, we develop a new double loop path-following scheme attaining the same complexity, modulo adjusting constants. These complexity bounds are known to be optimal in the unconstrained case, and our work shows that they are upper bounds in the case with complicated constraints as well. A key feature of our methodology is the use of self-concordant barriers to construct strictly feasible iterates via a disciplined decomposition approach and without sacrificing on the iteration complexity of the method. To the best of our knowledge, this work is the first which achieves these worst-case complexity bounds under such weak conditions for general conic constrained optimization problems. This is joint work with Pavel Dvurechensky (WIAS Berlin) and based on the paper: [arXiv:2111.00100 [math.OC]](https://arxiv.org/abs/2111.00100).
Matthias
Schymura
BTU Cottbus-Senftenberg
On the maximal number of columns of a Delta-modular integer matrix
Abstract.
We are interested in a combinatorial question related to the recent and ongoing interest in understanding the complexity of integer linear programming with bounded subdeterminants. Given a number Delta and a full-rank integer matrix A with m rows such that the absolute value of every m-by-m minor of A is bounded by Delta, at most how many pairwise distinct columns can A have? The case Delta = 1 is the classical result of Heller (1957) saying that the maximal number of pairwise distinct columns of a totally unimodular integer matrix with m rows equals m^2 + m + 1. If m >= 3 and Delta >= 3, the precise answer to this combinatorial question is not known, but bounds of order O(Delta^2 m^2) have been proven by Lee et al. (2021). In the talk, I will discuss our approach to the problem which rests on counting columns of A by residue classes of a suitably defined integer lattice. As a result, we obtain the first bound that is linear in Delta and polynomial (of low degree) in the dimension m. Moreover, I explain how one can approach the corresponding classification problem and how this results in the construction of counterexamples to the previously conjectured value of the maximum above. The talk is based on a joint work with Gennadiy Averkov, see https://arxiv.org/abs/2111.06294.
Paul
Jungeblut
The Air-Pressure Method for Area-Universality
September 2022
Jesús A.
De Loera
UC Davis
Open problems arising from trying to understand the Simplex Method
Abstract.
The simplex method is one of the most famous and popular algorithms in optimization, it was even named on the top 10 algorithms of the 20th century by SIAM and IEEE. But despite its great success and fame we do not fully understand it. There are several natural open questions arising from the analysis of its performance This talk highlights several fascinating geometric and combinatorial problems we wish to answer. Many open questions will be available for the eager young researcher. I promise!In the first part I re-introduce natural geometric-topological structure one can associate to the set of all possible monotone paths of a linear program, I will then use this structure, first introduced by Billera and Sturmfels, to study How long can the longest monotone paths on a linear program become? How many different monotone paths can there be on a linear program? We report on two papers, the first joint work with M. Blanchard and Q. Louveaux, and another with C. Athanasiadis and Z. Zhang. Next, the engine of any version of the simplex method is a *pivot rule* that selects the outgoing arc for a current vertex. Pivot rules come in many forms, definitions, and types, but after 80 years we are still ignorant whether there is one that can make the simplex method run in polynomial time. Can we classify all pivot rules? How many possible different pivot rules can there be on a linear Program? Do these questions even make sense? In the second half of my talk will present two recent positive results: For 0/1 polytopes there are explicit pivot rules for which the number of non-degenerate pivots is polynomial and even linear (joint work with A. Black, S. Kafer, L. Sanita). I also present a parametric analysis for all pívot rules. We construct a polytope, the pivot rule polytope, that parametrizes all memoryless pívot rules of a given LP. Its construction is a generalization of the Fiber polytope construction of Billera Sturmfels. This is an attempt to get a complete picture of the structure “space of all pivot rules of an LP” (joint work with A. Black, N.Lütjeharms, and R. Sanyal).
Felix
Schröder
Improper colorings of planar graphs
Stefan
Felsner
Orthogonal convexity
Melanie
Reihl
Zur Geradeüberdeckungszahl von Graphen
Daniel
Blankenburg
University of Bonn
Resource Sharing Revisited
Abstract.
We revisit the (block-angular) min-max resource sharing problem, which is a well-known generalization of fractional packing and the maximum concurrent flow problem. It consists of finding an ℓ<sub>∞</sub>-minimal element in a Minkowski sum X=∑<sub>c∈C</sub>X<sub>c</sub> of non-empty closed convex sets X<sub>c</sub>⊆ℝ<sup>R</sup><sub>≥0</sub>, where C and R are finite sets. We assume that an oracle for approximate linear minimization over X<sub>c</sub> is given. We improve on the currently fastest known FPTAS in various ways. A major novelty of our analysis is the concept of local weak duality, which illustrates that the algorithm optimizes (close to) independent parts of the instance separately. Interestingly, this implies that the computed solution is not only approximately ℓ<sub>∞</sub>-minimal, but among such solutions, also its second-highest entry is approximately minimal. Based on a result by Klein and Young, we provide a lower bound of 𝛺(((|C|+|R|) log |R|)/𝛿²) required oracle calls for a natural class of algorithms. Our FPTAS is optimal within this class — its running time matches the lower bound precisely, and thus improves on the previously best-known running time for the primal as well as the dual problem.
Alex
Nedev
Partially ordered sets: structure and compact encoding
Hiroshi
Kera
Chiba University
Approximate computation of vanishing ideals
Abstract.
The vanishing ideal of points is the set of all polynomials that vanish over the points. Any vanishing ideal can be generated by a finite set of vanishing polynomials or generators, and the computation of approximate generators has been developed at the intersection of computer algebra and machine learning in the last decade under the name of approximate computation of vanishing ideals. In computer algebra, the developed algorithms are supported by theories more deeply, whereas, in machine learning, the algorithms have been developed toward applications at a cost of some theoretical properties. In this talk, I will present a review on the development of approximate computation of vanishing ideals in two fields, particularly from the perspective of the spurious vanishing problem and normalization, which are recently suggested as a new direction of development.
August 2022
Vladimir
Kolmogorov
IST Austria
On discrete optimization problems and Gomory-Hu tree construction algorithms
Abstract.
I will consider the problem of minimizing functions of discrete variables represented as a sum of "tractable" subproblems. First, I will briefly review recent theoretical results characterizing complexity classification of discrete optimization in the framework of "Valued Constraint Satisfaction Problems" (VCSPs). Then I will talk about algorithms for solving Lagrangian relaxations of such problems. I will describe an approach based on the Frank-Wolfe algorithm that achieves the best-known convergence rate. I will also talk about practical implementation, and in particular about in-face Frank-Wolfe directions for certain combinatorial subproblems. Implementing such directions for perfect matching subproblems boils down to computing a Gomory-Hu (GH) tree of a given graph. Time permitting, I will describe a new approach for computing GH tree that appears to lead to a state-of-the-art implementation.
Benjamin
Schröter
KTH
Valuative Invariants for Matroids
Abstract.
Valuations on polytopes are maps that combine the geometry of polytopes with relations in a group. It turns out that many important invariants of matroids are valuative on the collection of matroid base polytopes, e.g., the Tutte polynomial and its specializations or the Hilbert–Poincaré series of the Chow ring of a matroid. In this talk I will present a framework that allows us to compute such invariants on large classes of matroids, e.g., sparse paving and elementary split matroids, explicitly. The concept of split matroids introduced by Joswig and myself is relatively new. However, this class appears naturally in this context. Moreover, (sparse) paving matroids are split. I will demonstrate the framework by looking at Ehrhart polynomials and Hilbert–Poincaré series of elementary split matroids. This talk is based on the preprint `Valuative invariants for large classes of matroids' which is joint work with Luis Ferroni.
Karen
Aardal
TU Delft
Integer optimization: Branching based on lattice information
Abstract.
There has been enormous progress in the branch-and-bound methods in the past couple of decades. In particular, much effort has been put into the so-called variable selection problem, i.e. the problem of choosing which variable to branch on in the current search node. Recently, many researchers have investigated the potential of using machine learning to find good solutions to this problem by for instance trying to mimic what good, but computationally costly, heuristics do. The main part of this research has been focused on branching on so-called elementary disjunctions, that is, branching on a single variable. Theory, such as the results by H.W. Lenstra, Jr. and by Lovász & Scarf, tells us that we in general need to consider branching on general disjunctions, but due in part to the computational challenges to implement such methods, much less work in this direction has been done. Some heuristic results in this direction have been presented. In this talk we discuss both theoretical and heuristic results when it comes to branching on general disjunctions with an emphasis on lattice based methods. A modest computational study is also presented. In the last part of the talk we also give a short description of results from applying machine learning to the variable selection problem. The talk is based on joint work with Laurence Wolsey.
Vijay
Vazirani
University of California, Irvine
Online Bipartite Matching and Adwords
Abstract.
Over the last three decades, the online bipartite matching (OBM) problem has emerged as a central problem in the area of Online Algorithms. Perhaps even more important is its role in the area of Matching-Based Market Design. The resurgence of this area, with the revolutions of the Internet and mobile computing, has opened up novel, path-breaking applications, and OBM has emerged as its paradigmatic algorithmic problem. In a 1990 joint paper with Richard Karp and Umesh Vazirani, we gave an optimal algorithm, called RANKING, for OBM, achieving a competitive ratio of (1 – 1/e); however, its analysis was difficult to comprehend. Over the years, several researchers simplified the analysis. We will start by presenting a “textbook quality” proof of RANKING. Its simplicity raises the possibility of extending RANKING all the way to a generalization of OBM called the adwords problem. This problem is both notoriously difficult and very significant, the latter because of its role in the AdWords marketplace of Google. We will show how far this endeavor has gone and what remains. We will also provide a broad overview of the area of Matching-Based Market Design and pinpoint the role of OBM.
Sébastien
Designolle
University of Geneva
Abstract.
In quantum mechanics, performing a measurement is an invasive process which generally disturbs the system. Due to this phenomenon, there exist incompatible quantum measurements, i.e., measurements that cannot be simultaneously performed on a single copy of the system. In this talk we will explain the robustness-based approach generally used to quantify this incompatibility and how it can be cast, for finite-dimensional systems, as a semidefinite programming problem. With this formulation at hand we analytically investigate the incompatibility properties of some high-dimensional measurements and we tackle, for an arbitrary fixed dimension, the question of the most incompatible pairs of quantum measurements, showing in particular optimality of Fourier-conjugated bases.
July 2022
Raphael
Steiner
Odd Hadwiger for line graphs
Shagnik
Das
NTW
Tight bounds for divisible subdivisions
Abstract.
Alon and Krivelevich proved that for every n-vertex subcubic graph H and every integer q ≥ 2 there exists a (smallest) integer f = f(H,q) such that every Kf-minor contains a subdivision of H in which the length of every subdivision-path is divisible by q. Improving their superexponential bound, we show that f(H, q) ≤ 10.5qn + 8n + 14q, which is optimal up to a constant multiplicative factor. This is joint work with Nemanja Draganić and Raphael Steiner.
Markus
Kirchweger
A SAT Attack on Rota's Basis Conjecture
Anton
Dochtermann
Homomorphism complexes and reconfiguration for digraphs
Anton
Dochtermann
Texas State University
Anton
Dochtermann
Texas State University
Root polytopes, tropical types, and toric edge ideals
Abstract.
We consider arrangements of tropical hyperplanes, where the apices of the hyperplanes can be `taken to infinity' in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is defined by its `type' data, analogous to the covectors of an oriented matroid. By work of Develin-Sturmfels and Fink-Rincon it is known that these `tropical complexes' are dual to (regular) subdivisions of root polytopes, which in turn are in bijection with mixed subdivisions of certain generalized permutohedra. Extending previous work with Joswig-Sanyal, we show how a natural monomial labeling of these complexes describes polynomial relations (syzygies) among `type ideals' which arise naturally from the combinatorial data of the arrangement. In particular, we show that the cotype ideal is Alexander dual to a squarefree initial ideal of the lattice ideal of a root polytope corresponding to the Stanley-Reisner ideal of a regular triangulation of the root polytope. This in turn leads to novel ways of studying algebraic properties of various monomial and lattice ideals. For instance, our methods of studying the dimension of a tropical complex provide new bounds on homological invariants of toric edge ideals of bipartite graphs, which have been extensively studied in the commutative algebra community. This is joint work with Ayah Almousa and Ben Smith.
Lorenz T. (Larry)
Biegler
Carnegie Mellon University
Nonlinear Optimization with Data-Driven and Surrogate Models
Abstract.
Optimization models for engineering design and operation, are frequently described by complex models of black-box simulations. The integration, solution, and optimization of this ensemble of large-scale models is often difficult and computationally expensive. As a result, model reduction in the form of simplified or data-driven surrogate models is widely applied in optimization studies. While the application to machine learning and AI approaches has lead to widespread optimization studies with surrogate models, less attention has been paid to validating these results on the optimality of high-fidelity, i.e., ‘truth’ models. This talk describes a surrogate-based optimization approach based on a trust-region filter (TRF) strategy. The TRF method substitutes surrogates for high-fidelity models, thus leading to simpler optimization subproblems with sampling information from truth models. Adaptation of the subproblems is guided by a trust region method, which is globally convergent to the local optimum of the original high-fidelity problem. The approach is suitable for broad choices of surrogate models, ranging from neural networks to physics-based shortcut models. The TRF approach has been implemented on numerous optimization examples in process and energy systems, with complex high fidelity models. Three case studies will be presented for Real-Time Optimization (RTO) for oil refineries, chemical processes and dynamic adsorption models for CO<sub>2</sub> capture, which demonstrate the effectiveness of this approach.
Haoyang
Guo
Max Planck Institute for Mathematics
Prismatic approach to p-adic local systems
Abstract.
Let X be a smooth proper scheme over a p-adic field that admits a good reduction. Inspired by the de Rham comparison theorem in complex geometry, Grothendieck asked if there is a "mysterious functor", relating étale cohomology of the generic fiber and crystalline cohomology of the special fiber. The question was subsequently answered by Fontaine, Faltings and many others' work, which was one of the foundational results in the p-adic Hodge theory. In particular, this motivates the definition of a p-adic local system being crystalline, generalizing the representational property of the etale cohomology of X as above. In this talk, we will give an overview for crystalline representations and crystalline local systems. Building on the recent advance of Bhatt-Scholze, we then introduce the prismatic approach to crystalline local systems. This is a joint work with Emanuel Reinecke.
Thomas
Lesgourgues
UNSW
Minimum degree of minimal Ramsey graphs for cliques
Abstract.
A graph G is r-Ramsey for a graph H, if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property. In 1976, Burr, Erdős and Lovász introduced the study of the smallest minimum degree sr(k) of a graph G such that G is r-Ramsey-minimal for a complete graph of size k. They were able to show the rather surprising exact result, that with two colours, then s2(k) = (k-1)2. The behaviour of this function is still not so well understood for more than 2 colours. In 2016, Fox, Grinshpun, Liebenau, Person, and Szabó showed that for r > 2, sr(k) is at most 8(k-1)6r3. The speaker, together with John Bamberg and Anurag Bishnoi, have recently used a group theoretic model of generalised quadrangles introduced by Kantor in 1980, coupled with a probabilistic approach, to improve this bound.
Ya
Deng
U Lorraine
Jörg
Wolf
Chung-Ang University, Seoul
The role of the parabolic Lipschitz truncation for existence of solutions systems of PDEs with non standard growth
Andrea
Jiménez
Universidad de Varparaíso, Chile
Groundstates of the Ising Model on antiferromagnetic triangulations
Abstract.
We discuss a dual version of a problem about perfect matchings in cubic graphs posed by Lovasz and Plummer. The dual version is formulated as follows "Every triangulation of an orientable surface has exponentially many groundstates'', where groundstates are the states at the lowest energy in the antiferromagnetic Ising Model.According to physicists, this dual formulation holds. In this talk, I show a counterexample to the dual formulation, a method to count groundstates which gives a better bound (for the original problem) on the class of Klee-graphs, the complexity of the related problems and, if time allows, some open problems. This is joint work with Marcos Kiwi and Martin Loebl.
Herman
Haverkort
Universität Bonn
Space-filling curves: properties, applications and challenges
Abstract.
A space-filling curve is a continuous, surjective map from [0,1] to a d-dimensional unit volume (for example, a cube or a simplex). Space-filling curves are usually constructed following a recursive tessellation of the unit volume that gives the curve useful structural properties. The most prominent of these properties is that the curve tends to preserve locality: points that are close to each other along the curve are (usually) close to each other in d-dimensional space and (usually) vice versa. This can be exploited to speed up algorithms, in practice and sometimes even in theory, by processing or storing data points in order along the curve. In this lecture I will show how space-filling curves can be described, how they get their useful properties, and I will show examples of their applications. This brings us to the question what would be the optimal space-filling curves for these applications. We will encounter a number of open questions on tessellations in 2D and 3D and on how to measure the quality of a space-filling curve.
Ekin
Ergen
TU Berlin
What is the generalized Poincaré conjecture?
Abstract.
We are going to explore a higher-dimension version of the Poincaré conjecture. In dimension three, this corresponds to the only Millennium Prize problem that is solved as of today. Roughly, the conjecture tells us that, if a manifold is homotopy equivalent to a sphere of its dimension, it is a sphere. We are going to discuss the history of this conjecture, and sketch a proof of the higher-dimension version via the $h$-cobordism theorem, due to Smale (1960). We are also going to introduce handle decompositions and the so-called Whitney trick, due to Whitney, which helps us tidy up handles. Prepare to see a lot of great achievements and of course, a lot of pictures!
Andreas
Rupp
LUT University
Introduction to DG methods
Arunima
Ray
MPI Bonn
Embedding surfaces in 4-manifolds
Büsra
Sert
The Half-Plane Property of Matroids, Related Concepts, and an Algorithm
Abstract.
Studies on homogeneous polynomials with the half-plane property were initially motivated by the physical theory of electrical networks. They later became of interest in convex optimization. In particular, a homogeneous polynomial with the half-plane property is hyperbolic with respect to every point in the positive orthant, having an associated hyperbolicity cone. Feasible sets of semi-definite programming, i.e., spectrahedral cones, are hyperbolicity cones for some polynomials. For the converse, the generalized Lax conjecture states that every hyperbolicity cone is spectrahedral. Considering that the basis generating polynomials of matroids are homogeneous and multiaffine, a natural approach is investigating the mentioned properties of matroids. For instance, the combinatorial structure of matroids was used by Brändén to give a counter example for a stronger version of the conjecture. In this talk, we present our results on the operations on matroids that preserve related properties. Moreover, we give an algorithm for testing the half-plane property of matroids that uses Macaulay2 packages SumsOfSquares, Matroids and the Julia package Homotopy Continuation.jl. We illustrate the outcome of the classification of matroids on at most 8 elements with respect to the half-plane property and provide our test results on matroids on 9 elements. This is joint work with Mario Kummer.
Sarah
Morell
TU Berlin
Single Source Unsplittable Flows and their Application in Machine Scheduling
Dimas
Arsaputra
On Minimizing the Weighted Number of Late Jobs
Warut
Suksompong
National University of Singapore
The Price of Connectivity in Fair Division
Abstract.
We study the allocation of indivisible goods that form an undirected graph and quantify the loss of fairness when we impose a constraint that each agent must receive a connected subgraph. Our focus is on well-studied fairness notions including envy-freeness and maximin share fairness. We introduce the price of connectivity to capture the largest gap between the graph-specific and the unconstrained maximin share, and derive bounds on this quantity which are tight for large classes of graphs in the case of two agents and for paths and stars in the general case. For instance, with two agents we show that for biconnected graphs it is possible to obtain at least 3/4 of the maximin share with connected allocations, while for the remaining graphs the guarantee is at most 1/2. In addition, we determine the optimal relaxation of envy-freeness that can be obtained with each graph for two agents, and characterize the set of trees and complete bipartite graphs that always admit an allocation satisfying envy-freeness up to one good (EF1) for three agents. Our work demonstrates several applications of graph-theoretic tools and concepts to fair division problems. Joint work with Xiaohui Bei, Ayumi Igarashi, and Xinhang Lu.
Atharva
Kelkar
FU Berlin
Studying the Effect of Chemical Patterning on the Hydrophobicity of Chemically Heterogeneous Surfaces via Active Learning
Haoxiang
Yang
CUHK-Shenzhen
Robust Optimization with Continuous Decision-Dependent Uncertainty
Abstract.
In this talk, we consider a robust optimization problem with continuous decision-dependent uncertainty (RO-CDDU), which has two new features: an uncertainty set linearly dependent on continuous decision variables and a convex piecewise-linear objective function. We prove that RO-CDDU is NP-hard in general and reformulate it into an equivalent mixed-integer nonlinear program (MINLP) with a decomposable structure to address the computational challenges. Such an MINLP model can be further transformed into a mixed-integer linear program (MILP) given the uncertainty set's extreme points. We propose an alternating direction algorithm and a column generation algorithm for RO-CDDU. We model a robust demand response (DR) management problem in electricity markets as RO-CDDU, where electricity demand reduction from users is uncertain and depends on the DR planning decision. Extensive computational results demonstrate the promising performance of the proposed algorithms in both speed and solution quality. The results also shed light on how different magnitudes of decision-dependent uncertainty affect the demand response decision.
Amal
Alphonse
Some aspects of elliptic quasi-variational inequalities
Abstract.
Quasi-variational inequalities (QVIs) can be thought of as generalisations of variational inequalities where the constraint set in which the solution is sought depends on the unknown solution itself. In this talk, I'll discuss various aspects of elliptic quasi-variational inequalities of obstacle type including existence results, sensitivity analysis of the source-to-solution map as well as optimal control problems with QVI constraints and associated stationarity systems.
Bill
Zwicker
Union College, USA
Higher-order Condorcet cycles
Abstract.
In an ordinary Condorcet cycle one can identify, for each candidate, a second candidate preferred, by a majority of voters, to the first. In a Condorcet cycle of order 2 one can identify, for each pair of candidates, a third candidate preferred, by a majority of voters, to both. We construct two Condorcet cycles of order 2. The first, with 11 alternatives and 11 voters, improves the example of 15 alternatives and 15 voters given in [1]. The second, with 7 alternatives and 21 voters, shows that the lower bound on alternatives established in [4] and [3] (and independently in [1]) is sharp. Both our constructions use the method of horizontal rotation, introduced here, which generalizes the more typical form of rotation used to construct standard Condorcet cycles. The second example also makes use of a beautifully symmetric tournament constructed in [3]. William S. Zwickera (joint work with Davide P. Cervoneb) keywords: Condorcet cycle of order 2, Condorcet winning set, tournament [1] Elkind, E., Lang, J., and Saffidine, A., Condorcet Winning Sets, Soc Choice Welf 44, 493-517 (2015) [2] Erdös, P., On a problem of graph theory, Math Gaz 47, 220-223 (1963) [3] Graham, R.L. and Spencer, J.H., A constructive solution to a tournament problem, Can Math Bul 14, 45-48 (1971) [4] Szekeres, E. and Szekeres,G., On a problem of Schütte and Erdös, Math Gaz 49, 290-293 (1965) aWilliam D Williams Professor of Mathematics Emeritus, Union College, New York; and Murat Sertel Center for Advanced Economic Studies, Istanbul Bilgi University bMathematics Department, Union College, New York
Upanshu
Sharma
FU Berlin
Aizhan
Issagali
FU Berlin
On tensor-based training of neural networks
Stefan
Felsner
Convex tilings with given slopes
Anna
Hartkopf
FU Berlin
and
Erin
Henning
FU Berlin
Mathematical Science Communication
Abstract.
Quality science journalism is a crucial tool in science communication to build a relationship between scientific mathematics and society that is based on mutual trust. The MIP.labor is a research project for science journalism in mathematics, computer science and physics at the Freie Universität Berlin. In our talk, we will first outline the theoretical foundations of science communication and briefly introduce a classification of its practice. In the second part of the talk, we will present the operating principle of the research being done at the MIP.labor, which includes research on the development of new formats of science journalism as well as scientifically evaluating them. In collaboration with the National Institute for Science Communication (NaWik), instruments are being tested to analyze the reception of these formats. With this cooperation, MIP.labor aims to critically reflect on the developed formats in order to advance the connection between the theory and practice of science communication.
Xiaolei
Zhao
UCSB
Gushel-Mukai varieties, stability conditions, and moduli of stable objects
Abstract.
Gushel-Mukai varieties are smooth Fano varieties of dimension between 3 to 6, Picard rank 1, degree 10 and co-index 3. Depend on the parity of the dimension, their derived categories contain a so called Kuznetsov component, behaving similarly to an Enriques surface or a K3 surface. I will survey the construction of Bridgeland stability conditions on the Kuznetsov component, and some properties about moduli of stable objects on it. Applications to the geometry of Fano varieties will also be explained. This is based on joint work with Alex Perry and Laura Pertusi.
Bjarne
Schülke
Caltech
A local version of Katona's intersection theorem
Abstract.
Katona's intersection theorem states that every intersecting family ℱ ⊆ [n](k) satisfies |∂ℱ| ≥ |ℱ|, where ∂ℱ = {F\x : x∊F∊ℱ} is the shadow of ℱ. Frankl conjectured that for n>2k and every intersecting family ℱ⊆ [n](k), there is some i∊[n] such that |∂ℱ(i)| ≥ |ℱ(i)|. Here, we prove this conjecture in a very strong form for n > (k+1)k/2. In particular, our result implies that for any j∊[k], there is a sequence a1,...,aj∊[n] such that |∂ℱ(a1,...,aj)| ≥ |ℱ(a1,...,aj)|. Further, Frankl conjectured that for n > k+ℓ and cross-intersecting families G⊆ [n](ℓ) and ℋ⊆ [n](k), there is some i∊[n] such that |∂G(i)| ≥ |G(i)| or |∂ℋ(i)| ≥ |ℋ(i)|. We prove this conjecture again in a very strong form for n > kℓ. This is joint work with Marcelo Sales.
Gari Yamel
Peralta Alvarez
HU Berlin
Arithmetic intersection theory on toric varieties with singular metrics
Abstract.
In the context of arithmetic intersection theory, the height of a variety with respect to a hermitian line bundle is an arithmetic analogue of the usual degree. Gillet and Soulé have developed a general construction for line bundles equipped with smooth metrics. In many interesting examples, the natural metrics to consider are not smooth. This called for extensions of this theory that admit such metrics. In this talk we focus on the case of toric varieties, for which Burgos, Philippon and Sombra gave an explicit description that admits continuous semipositive metrics. We sketch a generalization of these results to a broader type of singularities. This is based on the theory of adelic line bundles of Yuan and Zhang.
Svenja
Griesbach
TU Berlin
Public Signals in Network Congestion Games
Marcel
Celaya
ETH, Zürich
Improving the Cook et al. Proximity Bound Given Integral Valued Constraints
Abstract.
Given an optimal solution to a linear program, how far away can a nearest optimal integral solution be? In 1986 Cook, Gerards, Schrijver, and Tardos gave a bound for this distance, known as proximity, which depends only on the dimension and the largest possible magnitude of any subdeterminant of the corresponding constraint matrix. In this talk I will briefly survey this problem, describe some long standing related conjectures, and highlight some recent developments including a recent improvement to the Cook et al. bound when the dimension is at least 2. This is joint work with Joseph Paat, Stefan Kuhlmann, and Robert Weismantel.
Myfanwy
Evans
Universität Potsdam
Enumerating triply-periodic tangles
Abstract.
Using periodic surfaces as a scaffold is a convenient route to making periodic entanglements. I will present a systematic way of enumerating new tangled periodic structures, using low-dimensional topology and combinatorics, posing the question of how to best characterise these new patterns. I will also give an insight into applications of these structures.
Mahmut Levent
Doğan
TU Berlin
What is a probabilistically checkable proof?
Abstract.
The complexity class NP consists of search problems where the sought object may be hard to find, but once the object is provided by an all-knowing oracle, it is possible to verify it efficiently and deterministically. However, if we allow ourselves interaction and small probability of error in our verification processes, we can reach higher complexity classes than NP. These “probabilistic proof systems” consist of interactive proofs, zero-knowledge proofs and the main object of the talk, the probabilistically checkable proofs. In this talk, we will provide a humble introduction to probabilistic proof systems and talk about one of the most important theorems of complexity theory, the PCP theorem. If time allows, we will also show how the PCP theorem implies inapproximability of certain combinatoric optimizatiton problems.
Simon
Weissmann
University of Heidelberg
Rimma
Hämäläinen
Graph Drawings with Few Slopes
Irit
Dinur
Weizmann Institute of Science
P, NP, and Probabilistically Checkable Proofs as part of the Millenium Festival Event and BMS Certificate CeremonyLink to the live-stream
June 2022
Matthias
Beck
San Francisco State University
f^*- and h^*-vectors
Abstract.
If P is a lattice polytope (i.e., P is the convex hull of finitely many integer points in R^d), Ehrhart's theorem asserts that the integer-point counting function L_P(m) = #(mP \\cap Z^d) is a polynomial in the integer variable m. Our goal is to study structural properties of Ehrhart polynomials - essentially asking variants of the (way too hard) question which polynomials are Ehrhart polynomials? Similar to the situations with other combinatorial polynomials, it is useful to express L_P(m) in different bases. E.g., a theorem of Stanley (1976) says that L_P(m), expressed in the polynomial basis \\binom(m,d), \\binom(m+1,d), …, \\binom(m+d,d), has nonnegative coefficients; these coefficients form the h^*-vector of P. More recent work of Breuer (2012) suggests that one ought to also study L_P(m) as expressed in the polynomial basis \\binom(m-1,0), \\binom(m-1,1), \\binom(m-1,2), …; the coefficients in this basis form the f^*-vector of P. We will survey some old and new results (the latter joint work with Danai Deligeorgaki, Max Hlavaczek, and Jéronimo Valencia) about f^*- and h^*-vectors, including analogues and dissimilarities with f- and h-vectors of polytopes and polyhedral complexes.
Junliang
Shen
Yale
A tale of three moduli spaces of sheaves
Abstract.
I will discuss cohomological structures for three moduli spaces of sheaves: the moduli of vector bundles on a curve, the moduli of Higgs bundles on a curve, and the moduli of 1-dimensional torsion sheaves on \(\mathbb{P}^2\). These moduli spaces have been studied intensively from various perspectives. In recent years, enumerative geometry and string theory sheds new lights on the cohomological structure of these classical moduli spaces. In the talk I will discuss some results and conjectures in this direction; this concerns the \(\chi\)-independence phenomenon, tautological generators, and the \(P=W\) conjecture. Based on joint works with Davesh Maulik, Weite Pi, and an on-going project with Yakov Kononov and Weite Pi.
Markus
Pflaum
U Colorado
Persistent homology and the Morse-Smale complex as tools in topological data analysis and an application in chemistry
Abstract
Fabrizio
Barroero
U Roma Tre
On the Zilber-Pink conjecture for complex abelian varieties and distinguished categories
Abstract.
The Zilber-Pink conjecture is a very general statement that implies many well-known results in diophantine geometry, e.g., Manin-Mumford, Mordell-Lang and André-Oort. I will report on recent joint work with Gabriel Dill in which we proved that the Zilber-Pink conjecture for a complex abelian variety A can be deduced from the same statement for its trace, i.e., the largest abelian subvariety of A that can be defined over the algebraic numbers. This gives some unconditional results, e.g., the conjecture for curves in complex abelian varieties (over the algebraic numbers this is due to Habegger and Pila) and the conjecture for arbitrary subvarieties of powers of elliptic curves that have transcendental j-invariant. While working on this project we realised that many definitions, statements and proofs were formal in nature and we came up with a categorical setting that contains most known examples and in which (weakly) special subvarieties can be defined and a Zilber-Pink statement can be formulated. We obtained some conditional as well as some unconditional results.
Manuel
Radons
TU Berlin
Generalized Perron Roots and Solvability of the Absolute Value Equation
Michaela
Borzechowski
Freie Universität Berlin
Unique Sink Orientations of Grids is in Unique End of Potential Line
Abstract.
The complexity classes Unique End of Potential Line (UEOPL) and its promise version PUEOPL were introduced in 2018 by Fearnly et al. PUEOPL captures search problems where the instances are promised to have a unique solution. UEOPL captures total search versions of these promise problems. The promise problems can be made total by defining violations that are returned as a short certificate of an unfulfilled promise. GridUSO is the problem of finding the sink in a grid with a unique sink orientation. It was introduced by Gärtner et al. in 2008. We describe a promise preserving reduction from GridUSO to UniqueForwardEOPL, a UEOPL-complete problem. Thus, we show that GridUSO is in UEOPL and its promise version is in PUEOPL.
Wolfgang
Mulzer
Freie Universität Berlin
Long Alternating Paths Exist
Abstract.
Let P be a set of 2n points in convex position, such that n points are colored red and n points are colored blue. A non-crossing alternating path on P of length l is a sequence p_1, ..., p_l of l points from P so that (i) all points are pairwise distinct; (ii) any two consecutive points p_i, p_{i+1} have different colors; and (iii) any two segments p_i p_{i+1} and p_j p_{j+1} have disjoint relative interiors, for i /= j. We show that there is an absolute constant eps > 0, independent of n and of the coloring, such that P always admits a non-crossing alternating path of length at least (1 + eps)n. The result is obtained through a slightly stronger statement: there always exists a non-crossing bichromatic separated matching on at least (1 + eps)n points of P. This is a properly colored matching whose segments are pairwise disjoint and intersected by a common line. For both versions, this is the first improvement of the easily obtained lower bound of n by an additive term linear in n. The best known published upper bounds are asymptotically of order 4n/3 + o(n). Based on joint work with Pavel Valtr.
Alexey
Garber
University of Texas Rio Grande Valley
Concrete polytopes may not tile the space
Abstract.
Pick’s formula gives a simple way to connect area of lattice polygon $P$ with lattice points inside $P$. Particularly, Pick’s theorem claims that the discrete volume of $P$ coincides with the Euclidean volume of $P$. Unfortunately, there is not generalization of this property for lattice polytopes in higher dimensions. In an attempt to extend Pick’s theorem to higher dimensions, Brandolini et al. conjectured that equality of discrete and Euclidean volumes implies a tileability property for lattice polytopes. In the talk I will present a construction of family of counterexamples to this conjecture. The construction is based on McMullen’s theory of valuations of lattice polytopes and Dehn’s invariant. The talk is based on joint result with Igor Pak (UCLA).
Torsten
Mütze
The Hamilton compression of highly symmetric graphs
Frédéric
Meunier
École Nationale des Ponts et Chaussées
Coloring complements of line graphs
Abstract.
Our purpose is to show that complements of line graphs enjoy nice coloring properties. For instance, all graphs in this class have their local chromatic number equal to their (usual) chromatic number. We also provide a sufficient condition for the chromatic number to be equal to a natural upper bound. A consequence of this latter condition is a complete characterization of all induced subgraphs of the Kneser graph KG(n, 2) that have a chromatic number equal to its chromatic number, namely n − 2. In addition to the upper bound, a lower bound is provided by Dol’nikov’s theorem, a classical result of the topological method in graph theory. We prove the NP-hardness of deciding the equality between the chromatic number and any of these bounds. The topological method is especially suitable for the study of coloring properties of complements of line graphs of hypergraphs. Nevertheless, all proofs are elementary and we provide a short discussion on the ability for the topological methods to cover some of our results. Joint work with Hamid Reza Danehspajouh and Guilhem Mizrahi.
Matthias
Beck
SFSU
\(f^*\)- and \(h^*\)-vectors
Abstract.
If \(P\) is a lattice polytope (i.e., \(P\) is the convex hull of finitely many integer points in \({\bf R}^d\)), Ehrhart's theorem asserts that the integer-point counting function \(L_P(m) = \#(mP \cap {\bf Z}^d)\) is a polynomial in the integer variable $m$. Our goal is to study structural properties of Ehrhart polynomials---essentiallly asking variants of the (way too hard) question which polynomials are Ehrhart polynomials? Similar to the situations with other combinatorial polynomials, it is useful to express \(L_P(m)\) in different bases. E.g., a theorem of Stanley (1976) says that \(L_P(m)\), expressed in the polynomial basis \(\binom m d, \binom{m+1} d, \dots, \binom{m+d} d\), has nonnegative coefficients; these coefficiencts form the \(h^*\)-vector of \(P\). More recent work of Breuer (2012) suggests that one ought to also study \(L_P(m)\) as expressed in the polynomial basis \(\binom {m-1} 0, \binom {m-1} 1, \binom {m-1} 2, \dots\); the coefficiencts in this basis form the \(f^*\)-vector of \(P\). We will survey some old and new results (the latter joint work with Danai Deligeorgaki, Max Hlavaczek, and Jerónimo Valencia) about \(f^*\)- and \(h^*\)-vectors, including analogues and dissimiliarieties with \(f\)- and \(h\)-vectors of polytopes and polyhedral complexes.
Denis
Korolev
Model order reduction techniques for electrical machines
Abstract.
In this talk, I will discuss model order reduction methods for parameterized elliptic and parabolic partial differential equations and their application to the modelling of magnetic fields in electrical machines. If time permits, modern deep learning methods of model order reduction will be discussed.
Carl
Lian
HU Berlin
Degenerations of complete collineations and geometric Tevelev degrees of \(\mathbb{P}^r\)
Abstract.
This is a report on work in progress. We will discuss a complete answer, in terms of Schubert calculus, to the problem of enumerating maps of degree \(d\) from a fixed general curve of genus \(g\) to \(\mathbb{P}^r\) satisfying incidence conditions at the appropriate number of marked points, that is, we compute the geometric Tevelev degrees of \(\mathbb{P}^r\). Previously, after the work of many people, the answers were known only when \(r = 1\), or when d is large compared to \(r\), \(g\); in the latter case, the answers agree with virtual counts in Gromov-Witten theory, but when \(d\) is small, the situation is considerably more subtle. The method proceeds by reduction to genus 0 via limit linear series, and then by an analysis of certain Schubert-type cycles on moduli spaces of complete collineations upon further degeneration.
Jan-Hendrik
de Wiljes
FU Berlin
Peg Solitaire on Graphs
Abstract.
The single-player board game (Peg) Solitaire has been around for centuries and eventually attracted researchers from mathematics and computer science. Recently, but certainly very naturally, peg solitaire was extended to graphs. The game rules are as follows: Place pegs on all vertices but one of a (connected) finite simple graph G. If u,v and w are vertices of G such that uv and vw are edges of G and there are pegs on u and v but not on w, then it is possible to jump with the peg from u onto w removing the peg from v in the process. Usually one tries to eliminate all but one peg from the graphs vertices by using such jumps, calling a graph solvable if that is possible. This talk serves as an introduction to the game and provides an overview of major developments and open problems. Moreover, typical methods used in the proofs of known results will be presented. Several variations of the game, for example a misère version, were also considered recently, some of which will be discussed in this talk as well.
José Ignacio
Burgos Gil
ICMAT
Chern-Weil theory and Hilbert-Samuel theorem for semi-positive
singular toroidal metrics on line bundles
Abstract.
In this talk I will report on joint work with A. Botero, D. Holmes and R. de Jong. Using the theory of b-divisors and non-pluripolar products we show that Chern-Weil theory and a Hilbert Samuel theorem can be extended to a wide class of singular semi-positive metrics. We apply the techniques relating semipositive metrics on line bundles to b-divisors to study the line bundle of Siegel-Jacobi forms with the Peterson metric. On the one hand we prove that the ring of Siegel-Jacobi forms of constant positive relative index is never finitely generated, and we recover a formula of Tai giving the asymptotic growth of the dimension of the spaces of Siegel-Jacobi modular forms.
Christian
Kipp
Technische Universität Berlin
Affine Subspace Concentration Conditions for Polytopes
Abstract.
Given an n-dimensional polytope P and one of its facets F, the cone volume corresponding to F is the volume of conv(0,F). P is said to satisfy the subspace concentration condition w.r.t. a d-dimensional linear subspace L if the total cone volume of the facets with normal vectors in L is at most d/n*vol(P). The subspace concentration condition plays an important role in the context of the (discrete) logarithmic Minkowski problem, i.e., the question: What conditions ensure that a given list of normal vectors and cone volumes can be realized by a polytope? Recently, an affine version of the subspace concentration condition was introduced by Wu for certain lattice polytopes. In this talk, I will focus on the affine case and discuss possible generalizations. This is joint work with Ansgar Freyer and Martin Henk.
Vu
Nguyen
Amazon Research Australia
Bayesian Optimization with Categorical and Continuous Variables
Abstract.
Bayesian optimization (BO) has demonstrated impressive success in optimizing black-box functions. However, there are still challenges in dealing with black-boxes that include both continuous and categorical inputs. I am going to present our recent works in optimizing the mixed space of categorical and continuous variables using Bayesian optimization \[B. Ru, A. Alvi, V. Nguyen, M. Osborne, and S. Roberts. “Bayesian optimisation over multiple continuous and categorical inputs.” ICML 2020] and how to scale it up to higher dimensions \[X. Wan, V. Nguyen, H. Ha, B. Ru, C. Lu, and M. Osborne. “Think Global and Act Local: Bayesian Optimisation over High-Dimensional Categorical and Mixed Search Spaces.” ICML 2021] and population-based AutoRL setting \[J. Parker-Holder, V. Nguyen, S. Desai, and S. Roberts. “Tuning Mixed Input Hyperparameters on the Fly for Efficient Population Based AutoRL”. NeurIPS 2021].
Karoly
Böröczky
Rényi Institute, Budapest
Facets of the Brascamp-Lieb Inequality and its Reverse form
Abstract.
The Brascamp-Lieb inequality, a generalization of Holder's inequality, is introduced, together with its reverse form generalizing the Prekopa-Leindler due to Barthe. Under certain conditions, the optimal factor in either of inequalities can be obtained using Gaussian test functions. These conditions give rise to the so-called Brascamp Lieb polytope. Algorithmic aspects of approximating the optimal factor are also discussed.
Jonathan
Eckstein
Rutgers Business School
Solving Stochastic Programming Problems by Operator Splitting
Abstract.
This talk describes the solution of convex optimization problems that include uncertainty modeled by a finite but potentially very large multi-stage scenario tree. In 1991, Rockafellar and Wets proposed the progressive hedging (PH) algorithm to solve such problems. This method has some advantages over other standard methods such as Benders decomposition, especially for problems with large numbers of decision stages. The talk will open by showing that PH is an application of the Alternating Direction Method of Multipliers (ADMM). The equivalence of PH to the ADMM has long been known but not explicitly published. The ADMM is an example of an "operator splitting" method, and in particular of a principle called "Douglas–Rachford splitting". I will briefly explain what is meant by an "operator splitting method". Next, the talk will apply a different, more recent operator splitting method called "projective splitting" to the same problem. The resulting method is called "asynchronous projective hedging" (APH). Unlike most decomposition methods, it does not need to solve every subproblem at every iteration; instead, each iteration may solve just a single subproblem or a small subset of the available subproblems. Finally, the talk will describe work integrating the APH algorithm into `mpi-sppy`, a Python package for modeling and distributed parallel solution of stochastic programming problems. `mpi-sppy` uses the Pyomo Python-based optimization modeling sytem. Our experience includes using up to 2,400 processor cores to solve 2-stage and 4-stage test problem instances with as many as 1,000,000 scenarios. Portions of the work described in this talk are joint with Patrick Combettes (North Carolina State University), Jean-Paul Watson (Lawrence Livermore National Laboratory, USA), and David Woodruff (University of California, Davis).
Martha
Nansubuga
HU Berlin
What is a McKean-Vlasov process?
Abstract.
A McKean-Vlasov process is a stochastic process described by a stochastic differential equation where the coefficients depend on the distribution of the process itself. The story of these processes started with a stochastic toy model for the Vlasov equation of plasma proposed by Mark Kac in his paper "Foundations of kinetic theory (1956)". They have been applied in several areas like physics, finance, social interactions and extra. In order to understand McKean-Vlasov processes, I will give a brief introduction to stochastic differential equations, too.
Christoph
Reisinger
U Oxford
Simulation and control of stochastic mean-field models: from starlings over neurons and traders to supercooling
Hany
Ibrahim
Edge Contraction and Forbidden Induced Subgraph
Julian
Pfeifle
UPC
Coming soon to polymake: (sometimes) disproving convex realizability of
spheres
Abstract.
Every so often, we find a family of simplicial spheres with interesting combinatorial properties, and would like to know if they are realizable as boundaries of convex polytopes. For instance, this happened recently to Zheng; to Novik and Zheng; and to Criado and Santos.We focus on algorithmically *disproving* the convex realizability of a given simplicial sphere. All previous successful approaches made essential use of integer or linear programming, which severely limits the practical applicability due to memory and runtime constraints. In our new implementation, we instead perform a combinatorial search using optimized data structures, which lets us decide in seconds examples that are completely out of reach of the best alternative implementation by Gouveia, Macchia, and Wiebe.Our code has now undergone the third complete rewrite, and is almost ready to be merged into the polymake tree. If time permits, we will discuss some of the lessons learned during the implementation, and mention further conceptual improvements that bring us tantalizingly close to deciding the long-standing open problem of convex realizability of multitriangulations.
Adam
Schweitzer
Berlin Mathematical School
Clique number of Xor products of Kneser graphs
Abstract.
Given two graphs, G and H, let us denote by V(G), V(H) and E(G), E(H) their vertex and edge sets respectively. Define the Xor product, s the graph with the vertex set V=V(G) × V(H), and two vertices (g,h) and (g',h') are connected if and only if among the statements gg' ∊ E(G) and hh' ∊ E(H) exactly one occurs. In this talk we examine the clique number of the Xor product of two isomorphic KG(N,k) Kneser graphs, denote this number with f(k,N). We discuss the background of the problem and we give lower and upper bounds on f(k,N). Furthermore we determine f(k,N) up to a constant deviation depending only on k, and find the exact value for f(2,N) if N is large enough. We also compute that f(k,k2) is asymptotically equivalent to k2.
Christopher
Deninger
U Münster
Dynamical systems for arithmetic schemes
Abstract.
For any arithmetic scheme \(X\) we construct a continuous time dynamical system whose periodic orbits come in compact packets that are in bijection with the closed points of \( X \). All periodic orbits in a given packet have the same length equal to the logarithm of the order of the residue field of the corresponding closed point. For \( X = \mathrm{Spec}\, \mathbb{Z}\) we get a dynamical system whose periodic orbits are related to the prime numbers. The construction uses new ringed spaces which are constructed from rational Witt vector rings. In the zero-dimensional case we recover a construction of Kucharczyk and Scholze who realized certain Galois groups as étale fundamental groups of ordinary topological spaces. A p-adic version of our construction turns out to be closely related to the Fargues-Fontaine curve of p-adic Hodge theory.
Javier
Cembrano
TU Berlin
Impartial Selection with Additive Guarantees via Iterated Deletion
Andrei
Comăneci
Technische Universität Berlin
Tropical Medians by Transportation
Abstract.
The Fermat-Weber problem seeks a point that minimizes the average distance from a given sample. The problem was studied by Lin and Yoshida (2018) using the standard tropical metric with the purpose of analyzing phylogenetic data. In this talk, we argue that using a related asymmetric distance we have better geometric and algorithmic properties. The new formulation is strongly related to tropical convexity and is equivalent to a transportation problem. This gives a geometric perspective to the transportation problem, which was exploited by Tokuyama and Nakano (1995) to obtain efficient algorithms. At the end, we will see an application to computational biology: a new method for computing consensus trees. The talk is based on joint work with Michael Joswig.
Matthias
Beck
San Francisco State University
Boundary h*-polynomials of rational polytopes
Abstract.
If P is a lattice polytope (i.e., P is the convex hull of finitely many integer points in R^d), Ehrhart's famous theorem asserts that the integer-point counting function |mP∩Z^d| is a polynomial in the integer variable m. Equivalently, the generating function \sum_{m \ge 0} |mP∩Z^d| t^m is a rational function of the form h*(t)/(1-t)^{d+1}; we call h*(t) the Ehrhart h*-polynomial of P. We know several necessary conditions for h*-polynomials, including results by Hibi, Stanley, and Stapledon, who used an interplay of arithmetic (integer-point structure) and topological (local h-vectors of triangulations) data of a given polytope. We introduce an alternative ansatz to understand Ehrhart theory through the h*-polynomial of the boundary of a polytope, recovering all of the above results and their extensions for rational polytopes in a unifying manner. This is joint work with Esme Bajo (UC Berkeley).
Gabriel
Ribeiro
École Polytechnique Paris
Cartier duality, character sheaves, and generic vanishing
Abstract.
Since Green and Lazarsfeld's seminal work, generic vanishing theorems have influenced many fields ranging from birational geometry to analytic number theory. In this talk, I will present a new generic vanishing theorem for holonomic D-modules, that may shed new light on both of these worlds. Following ideas of Laumon, I'll explain how "character sheaves" play the role of topologically trivial line bundles and construct their moduli space based on a stacky version of Cartier duality.
Martin
Seyferth
On Discretizations in a Weighted Total Variation Model for Image Processing
David
Steurer
ETH Zürich
Beyond Parallel Pancakes: Quasi-Polynomial Time Guarantees for Non-Spherical Gaussian Mixtures
Abstract.
We consider mixtures of k≥2 Gaussian components with unknown means and unknown covariance (identical for all components) that are well-separated, i.e., distinct components have statistical overlap at most k<sup>-C</sup> for a large enough constant C≥1. Previous statistical-query lower bounds \[Ilias Diakonikolas, Daniel M. Kane, and Alistair Stewart, Statistical query lower bounds for robust estimation of high-dimensional Gaussians and Gaussian mixtures (extended abstract), 58th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2017, pp. 73–84] give formal evidence that, even for the special case of colinear means, distinguishing such mixtures from (pure) Gaussians may be exponentially hard (in k). We show that, surprisingly, this kind of hardness can only appear if mixing weights are allowed to be exponentially small. For polynomially lower bounded mixing weights, we show how to achieve non-trivial statistical guarantees in quasi-polynomial time. Concretely, we develop an algorithm based on the sum-of-squares method with running time quasi-polynomial in the minimum mixing weight. The algorithm can reliably distinguish between a mixture of k≥2 well-separated Gaussian components and a (pure) Gaussian distribution. As a certificate, the algorithm computes a bipartition of the input sample that separates some pairs of mixture components, i.e., both sides of the bipartition contain most of the sample points of at least one component. For the special case of colinear means, our algorithm outputs a k-clustering of the input sample that is approximately consistent with all components of the underlying mixture. We obtain similar clustering guarantees also for the case that the overlap between any two mixture components is lower bounded quasi-polynomially in k (in addition to being upper bounded polynomially in k). A significant challenge for our results is that they appear to be inherently sensitive to small fractions of adversarial outliers unlike most previous algorithmic results for Gaussian mixtures. The reason is that such outliers can simulate exponentially small mixing weights even for mixtures with polynomially lower bounded mixing weights. A key technical ingredient of our algorithms is a characterization of separating directions for well-separated Gaussian components in terms of ratios of polynomials that correspond to moments of two carefully chosen orders logarithmic in the minimum mixing weight.
Vicky
Holfeld
ITWM
Steven
Simon
Bard College
Nicola
Tarasca
Virginia Commonwealth University
Incident varieties of algebraic curves and canonical divisors
Abstract.
The theory of canonical divisors on curves has witnessed an explosion of interest in recent years, motivated by recent developments in the study of limits of canonical divisors on nodal curves. Imposing conditions on canonical divisors allows one to construct natural geometric subvarieties of moduli spaces of pointed curves, called strata of canonical divisors. These strata are the projection on moduli spaces of curves of incidence varieties in the projectivized Hodge bundle. I will present a formula for the class of such incident varieties over the locus of pointed curves with rational tails. The formula is expressed as a linear combination of tautological classes indexed by decorated stable graphs, with coefficients enumerating appropriate weightings. I will conclude discussing applications to the study of relations in the tautological ring. Joint work with Iulia Gheorghita.
Clemens
Sirotenko
Dictionary learning for quantitative MRI
Abstract.
A nonlinear inverse problem related to quantitative Magnetic Resonance Imaging (qMRI) is under consideration. In general, qMRI summarizes techniques that aim at extracting physical tissue parameters from a sequence of highly under sampled MR images. Recently, a mathematical setup was introduced that addresses this problem by estimating a set of unknown parameters in a system of ODEs called Bloch equations. While classical approaches such as the Levenberg Marquardt algorithm or Landweber iteration yield good results under small noise levels, numerical experiments show that low sampling rates and noise of large magnitude lead to unsatisfactory outcomes and unstable convergence behavior. Therefore, a spatial regularization approach based on (coupled) dictionary learning is proposed, which has already shown excellent results in the linear inverse problem of classical MRI. From a mathematical viewpoint this ends up in a variety of non convex and non smooth optimization problems. Iterative schemes to solve these problems are discussed and convergence to equilibrium points is studied. Moreover numerical results are presented and open questions such as regularization properties, parameter choice and acceleration strategies are discussed.
Walner
Mendonça
ISTA
Tiling edge-coloured graphs with few monochromatic bounded-degree graphs
Abstract.
Gerencsér and Gyárfás proved in 1967 that for any 2-colouring of the edges of the complete graph with n vertices, Kn, it is possible to partition V(Kn) into two monocromatic paths. This result, which has a straightforward proof, motivated many other challenging problems that have been extensively studied in the last years. For instance, an open conjecture of Erdős, Gyárfás and Pyber from 1991 states that for any r-colouring of the edges of Kn there are r monochromatic paths partitioning V(Kn). We can also find in the literature other versions of the problem where instead of partitioning into paths, we are interested in partitioning into trees, cycles, or even power of cycles. Grinshpun and Sárközy studied a more general version of the problem where they were interested in partitioning V(Kn) into few monochromatic subgraphs which are copies of a given family of bounded degree graphs. They proved that for any family of graphs S = {F1, F2, F3, ...} such that Fi has exactly i vertices and maximum degree at most D, the following holds: for any 2-colouring of edges of Kn, there is a partition of V(Kn) into at most exp(O(D log D)) monochromatic subgraphs that are copies of graphs from S. They conjectured that for any r-colouring of the edges of Kn, it is possible to partition V(Kn) into exp(DC) monochromatic subgraphs that are copies of graphs from S, where C=C(r) is a constant that only depends on r. In this talk, we present the first progress towards Grinshpun-Sárközy conjecture by establishing a super-exponential bound.
Morten
Lüders
U Hannover
Milnor K-theory of p-adic rings and motivic cohomology
Abstract.
We explain a joint work with Matthew Morrow on \(p\)-adic Milnor K-theory. Our main theorem is a comparison of mod \(p^r\) Milnor K-groups of \(p\)-henselian local rings with the Milnor range of a newly defined syntomic cohomology theory by Bhatt, Morrow and Scholze. We begin by putting our result into context. Then we sketch the proof which builds on an analysis of a filtration on Milnor K-groups and a new technique called the left Kan extension from smooth algebras.
Margarete
Wohlleber
Gomory-Hu Trees on Special Classes of Parametric Graphs
Angela
Ortega
HU Berlin
What is a linear system of plane curves?
Abstract.
The interpolation problem can be roughly stated as follows. Given a set of points in the plane find a polynomial $f(x, y)$ with these points as roots. More generally, one can also ask for the vector space of all interpolating polynomials with bounded degree and with specified vanishing order at each one of the points. In this talk we will look at the problem of estimating the dimension of such space and will explain why does it become hard to give a general answer to this question. We will spice the exposition with a couple of examples.
Khai Van
Tran
TU Berlin
What is submodularity?
Abstract.
Submodularity is a property of functions that assign values to subsets of a ground set. It can be characterized by the inequality $f(X) + f(Y) \geq f(X \cap Y) + f(X \cup Y)$. In this talk, we will explain why submodularity is regarded as a discrete equivalent to both convexity and concavity. Furthermore, we will demonstrate some common techniques making use of submodular functions.
Vesa
Kaarnioja
FU Berlin
QMC and kernel interpolation
Johannes
Obenaus
Edge Partitions of Complete Geometric Graphs
Thomas
McCormick
UBC
Why Should You Care About Submodularity?
Ron
Aharoni
Technion
Enis
Kaya
MPI MiS Leipzig
Determining reduction types of Picard curves via tropical invariants
Abstract.
For a separable binary form of degree n over a complete non-archimedean field, there is a canonical metric tree on n leaves associated to it. Furthermore, for every degree n, there are only finitely many tree types, which gives rise to a partition of the space of all non-archimedean binary forms of degree n.In this talk, we will focus on the particular case where n = 5. Then, there are exactly 3 unmarked and 5 marked tree types. We will give a set of tropical invariants, valuations of certain elements coming from invariant theory for binary quintics, and show that these invariants allow us to distinguish the tree types algorithmically. As an application, we will express the reduction types of Picard curves in terms of tropical invariants of the associated binary quintics.This is joint work with Yassine El Maazouz and Paul Alexander Helminck.
Ziquan
Zhuang
MIT
Finite generation and Kähler-Ricci soliton degenerations of Fano varieties
Abstract.
By the Hamilton-Tian conjecture on the limit behavior of Kähler-Ricci flows, every complex Fano manifold degenerates to a Fano variety that has a Kähler-Ricci soliton. In this talk, I'll discuss the algebro-geometric analogue of this statement and explain its connection to certain finite generation results in birational geometry. Based on joint work with Harold Blum, Yuchen Liu and Chenyang Xu.
Fabian
Telschow
MATH+ Junior Research Group Leader
EF3-10: True Excursion Discovery Inference for Imaging Data
Abstract
Nina
Kamčev
University of Zagreb
The Turán density of 3-uniform tight cycles
Abstract.
Turán-type problems for hypergraphs are famously interesting and difficult. For example, for 3-uniform hypergraphs, the Turán density of F is known for very few hypergraphs F. The topic of this talk are Turán-type problems for 3-uniform tight cycles Ck, where the number of vertices k is not divisible by 3. The Turán density of a family of 3-graphs F, denoted π(F) , is the limit of the maximum density of an n-vertex 3-graph not containing a member of F. There is an iterated construction of a C5-free 3-graph Hn with edge density asymptotic to 2√3−3 ≈ 0.464 due to Mubayi and Rödl. Razborov showed that π(C5) ≤ 0.468, indicating that Hn might be optimal for C5. We show that Hn is asymptotically optimal for an enlarged family of graphs - namely, π(FK) = 2√3−3, where FK is the family of tight cycles whose length k is not divisible by 3 and is bounded by an absolute constant K. One of our main tools, which may be of independent interest, is a 3-uniform analogue of the statement `a graph is bipartite if and only if it does not contain an odd cycle'. Joint work with Shoham Letzter and Alexey Pokrovskiy.
May 2022
Sarah
Essadi
From N-player games to mean-field games
Abstract.
We consider deterministic differential games with a large, but finite, population of symmetric interacting players. The interaction term is of mean-field type and exhibits heterogeneity both via the linear dynamics of the players and in their non-smooth cost functionals. We proceed on a first-step with only constraints on the control and with no additional state constraints. We characterise optimal solutions by deriving first-order optimality conditions. However, due to the non-smoothness of the objectives, set-valued mappings appear in the adjoint equation. To overcome this issue, we make use of a Huber-type regularisation. Furthermore, we aim at analysing the asymptotic behaviour of this system, for infinitely many players. This limiting analysis renders possible the construction of approximate Nash equilibria for the N-player games based on a solution of the corresponding mean-field game.
Max
Klimm
TU Berlin
Equilibria in Multiclass and Multidimensional Atomic Congestion Games
Imre
Bárány
Alfréd Rényi Mathematical Institute of the Hungarian Academy of Science, Budapest
Cells in the box and a hyperplane
Abstract.
It is well known that a line can intersect at most 2n−1 cells of the n×n chessboard. What happens in higher dimensions: how many cells of the d-dimensional [0,n]^d box can a hyperplane intersect? We answer this question asymptotically. We also prove the integer analogue of the following fact. If K,L are convex bodies in R^d and K ⊂ L, then the surface area K is smaller than that of L. This is joint work with Péter Frankl.
Pier Luigi
Dragotti
Imperial College London
Computational Imaging and Sensing: Theory and Applications
Abstract.
The revolution in sensing, with the emergence of many new imaging techniques, offers the possibility of gaining unprecedented access to the physical world, but this revolution can only bear fruit through the skilful interplay between the physical and computational realms. This is the domain of computational imaging which advocates that, to develop effective imaging systems, it will be necessary to go beyond the traditional decoupled imaging pipeline where device physics, image processing and the end-user application are considered separately. Instead, we need to rethink imaging as an integrated sensing and inference model. In the first part of the talk we highlight the centrality of sampling theory in computational imaging and investigate new sampling modalities which are inspired by the emergence of new sensing mechanisms. We discuss time-based sampling which is connected to event-based cameras where pixels behave like neurons and fire when an event happens. We derive sufficient conditions and propose novel algorithms for the perfect reconstruction of classes of non-bandlimited functions from time-based samples. We then develop the interplay between learning and computational imaging and present a model-based neural network for the reconstruction of video sequences from events. The architecture of the network is model-based and is designed using the unfolding technique, some element of the acquisition device are part of the network and are learned with the reconstruction algorithm. In the second part of the talk, we focus on the heritage sector which is experiencing a digital revolution driven in part by the increasing use of non-invasive, non-destructive imaging techniques. These new imaging methods provide a way to capture information about an entire painting and can give us information about features at or below the surface of the painting. We focus on Macro X-Ray Fluorescence (XRF) scanning which is a technique for the mapping of chemical elements in paintings and introduce a method that can process XRF scanning data from paintings. The results presented show the ability of our method to detect and separate weak signals related to hidden chemical elements in the paintings. We analyse the results on Leonardo's 'The Virgin of the Rocks' and show that our algorithm is able to reveal, more clearly than ever before, the hidden drawings of a previous composition that Leonardo then abandoned for the painting that we can now see. This is joint work with R. Alexandru, R. Wang, Siying Liu, J. Huang and Y.Su from Imperial College London; C. Higgitt and N. Daly from The National Gallery in London and Thierry Blu from the Chinese University of Hong Kong. Bio:Pier Luigi Dragotti is Professor of Signal Processing in the Electrical and Electronic Engineering Department at Imperial College London and Fellow of the IEEE. He received the Laurea Degree (summa cum laude) in Electronic Engineering from the University Federico II, Naples, Italy, in 1997; the Master degree in Communications Systems from the Swiss Federal Institute of Technology of Lausanne (EPFL), Switzerland in 1998; and PhD degree from EPFL, Switzerland, in 2002. He has held several visiting positions. In particular, he was a visiting student at Stanford University, Stanford, CA in 1996, a summer researcher in the Mathematics of Communications Department at Bell Labs, Lucent Technologies, Murray Hill, NJ in 2000, a visiting scientist at Massachusetts Institute of Technology (MIT) in 2011 and a visiting scholar at Trinity College Cambridge in 2020. Dragotti was Editor-in-Chief of the IEEE Transactions on Signal Processing (2018-2020), Technical Co-Chair for the European Signal Processing Conference in 2012, Associate Editor of the IEEE Transactions on Image Processing from 2006 to 2009. He was also Elected Member of the IEEE Computational Imaging Technical Committee and the recipient of an ERC starting investigator award for the project RecoSamp. Currently, he is IEEE SPS Distinguished Lecturer. His research interests include sampling theory, wavelet theory and its applications, computational imaging and sparsity-driven signal processing.
János
Pach
Alfréd Rényi Mathematical Institute of the Hungarian Academy of Science, Budapest
Facets of Simplicity
Abstract.
We discuss some notoriously hard combinatorial problems for large classes of graphs and hypergraphs arising in geometric, algebraic, and practical applications. These structures are of bounded complexity: they can be embedded in a bounded-dimensional space, or have small VC-dimension, or a short algebraic description. What are the advantages of low complexity? I will suggest a few possible answers to this question, and illustrate them with classical examples.
Doris
Schneider
FAU
Data-based modeling of the cellular response to oxidative stress -- A Bayesian approach for model selection and parameter identification in (bio)chemical networks
Zoe
Wellner
Carnegie Mellon University
Colorful Borsuk-Ulam Results, Their Generalizations and Applications
Abstract.
In combinatorics and discrete geometry there are many colorful and rainbow results. Additionally topological tools have proven to be useful in generating interesting combinatorial results. In this line of approach we prove a colorful generalization of the Borsuk–Ulam theorem. We give a short proof of Ky Fan’s Lemma and generalize it from involutions to larger symmetry groups Z/p for prime p. We also present colorful generalizations of these nonexistence results for equivariant maps. As consequences we derive a colorful generalization of the Ham–Sandwich theorem and the necklace splitting theorem among other applications of Borsuk-Ulam. This is joint work with Florian Frick.
Philippe
Eyssidieux
Grenoble
\(L_2\) Cohomology for Hodge Modules
Abstract.
The talk will be based on arxiv:2203.06950[math.AG]. For an infinite Galois covering space of a compact Kähler manifold, we define \(L_p\) cohomology for \(1\le p<+\infty\) for various types of coefficients (perverse sheaves, coherent D-modules, Mixed Hodge Modules) and explain how to control it when \(p=2\). The formalism encompasses both my 2000 work on \(L_2\)-coherent cohomology (see also the independant and simultaneous work of Campana-Demailly) and Dingoyan's 2013 work on \(L_2\)-De Rham cohomology of an open subset. We describe a conjectural MHS on the reduced \(L_2\)-cohomology of a MHM and explain what can be proved with today's technology.
Reinier
Kramer
University of Alberta
The spin Gromov-Witten/Hurwitz correspondence
Abstract.
There are two important ways to calculate the number of maps between Riemann surfaces with given conditions: Hurwitz theory is over a hundred years old and uses the monodromy representation to transport the problem to symmetric groups, representation theory, and the Kadomstev-Petviashvili (KP) integrable hierarchy. Gromov-Witten theory for curves recasts the problem as intersection theory on the moduli space of such (stable) maps. By work of Okounkov-Pandharipande, there is a strict correspondence between the two. Both of these sides have an analogue with spin: this takes into account a bundle on the source which squares to the canonical bundle. On the Hurwitz side, this has relations to spin-symmetric or Sergeev groups, and the BKP hierarchy, while on the Gromov-Witten side, these invariants arise from localising the invariants of surfaces with smooth canonical divisor. I will explain that, at least for P^1, there is a correspondence between these two sides as well, which hinges on a spin analogue of the Ekedahl-Lando-Shapiro-Vainshtein formula and cosection localisation. This is joint work (partially in progress) with Alessandro Giacchetto, Danilo Lewański, and Adrien Sauvaget.
Peter
Allen
LSE
Packing degenerate graphs
Abstract.
A simple random greedy algorithm can almost perfectly pack graphs with bounded degeneracy and almost linear maximum degree into the complete graph. I will explain how the algorithm works and sketch the analysis. In some situations, one can add further ideas to get from almost perfect packings to perfect packings. I will describe two ways, which together prove the Gýarfás Tree Packing Conjecture (which says that T1,...,Tn pack into Kn, where each Ti has i vertices) with a restriction cn/log n on the maximum degree of the trees. This is joint work with Julia Böttcher, Dennis Clemens, Jan Hladký, Diana Piguet and Anusch Taraz.
Sarah
Zerbes
ETH Zürich
Euler systems and the Birch—Swinnerton-Dyer conjecture for abelian surfaces
Abstract.
Euler systems are one of the most powerful tools for proving cases of the Bloch-Kato conjecture, and other related problems such as the Birch and Swinnerton-Dyer conjecture. I will recall a series of recent works (variously joint with Loeffler, Pilloni, Skinner) giving rise to an Euler system in the cohomology of Shimura varieties for \( \mathrm{GSp}(4)\), and an explicit reciprocity law relating the Euler system to values of \( L\)-functions. I will then explain recent work with Loeffler, where we use this Euler system to prove new cases of the BSD conjecture for modular abelian surfaces over \( \mathbb{Q} \), and for modular elliptic curves over imaginary quadratic fields.
Katharina
Jochemko
KTH Stockholm
The Eulerian transformation
Abstract.
Many polynomials arising in combinatorics are known or conjectured to have only real roots. One approach to these questions is to study transformations that preserve the real-rootedness property. This talk is centered around the Eulerian transformation which is the linear transformation that sends the i-th standard monomial to the i-th Eulerian polynomial. Eulerian polynomials appear in various guises in enumerative and geometric combinatorics and have many favorable properties, in particular, they are real-rooted and symmetric. We discuss how these properties carry over to the Eulerian transformation. In particular, we disprove a conjecture by Brenti (1989) concerning the preservation of real roots, extend recent results on binomial Eulerian polynomials and provide enumerative and geometric interpretations. This is joint work with Petter Brändén.
Kateryna
Buryachenko
Vasyl' Stus Donetsk National University
On the qualitative properties of solutions to fourth order hyperbolic equations in plane domains
Nicholas
Schmidt
HU Berlin
What is a zeta function?
Abstract.
Riemann, Dedekind, Hecke, Artin, Selberg, Ruelle,... the list of names associated to "zeta functions" is long, nearly as long the list of objects to which zeta functions have been attached, ranging from number fields over riemannian manifolds to partially ordered sets, dynamical systems and — finally — groups. This talk will try to give a conceptual answer to the question "what is a zeta function" by "categorifying" Dirichlet series, and to look at some recurring themes connecting various kinds of zeta functions.
Claudia
Totzeck
University of Wuppertal
Helena
Bergold
Unavoidable patterns in complete simple topological graphs
Shira
Zerbib
Iowa State University
Line transversals in families of connected sets in the plane
Abstract.
We prove that if a family of compact connected sets in the plane has the property that every three members of it are intersected by a line, then there are three lines intersecting all the sets in the family. This answers a question of Eckhoff from 1993, who proved that under the same condition there are four lines intersecting all the sets. We also prove a colorful version of this result under weakened conditions on the sets, improving results of Holmsen from 2013. Our proofs use the topological KKM theorem. Joint with Daniel McGinnis.
Niels
Lindner
ZIB
On the tropical and zonotopal geometry of periodic timetabling
Abstract.
The timetable is the core of every public transportation system. The standard mathematical tool for optimizing periodic timetabling problems is the Periodic Event Scheduling Problem (PESP). A solution to PESP consists of three parts: a periodic timetable, a periodic tension, and integer periodic offset values. While the space of periodic tension has received much attention in the past, we explore geometric properties of the other two components, establishing novel connections between periodic timetabling and discrete geometry. Firstly, we study the space of feasible periodic timetables, and decompose it into polytropes, i.e., polytopes that are convex both classically and in the sense of tropical geometry. We then study this decomposition and use it to outline a new heuristic for PESP, based on the tropical neighbourhood of the polytropes. Secondly, we recognize that the space of fractional cycle offsets is in fact a zonotope. We relate its zonotopal tilings back to the hyperrectangle of fractional periodic tensions and to the tropical neighbourhood of the periodic timetable space. Finally, we also use this new understanding to give tight lower bounds on the minimum width of an integral cycle basis in terms of the number of spanning trees.
Arturo
Merino
TU Berlin
From Combinatorial Optimization to Gray codes
Abstract.
Let X⊆{0,1}<sup>n</sup> be a set of binary vectors of length n. Given a weight function w∈R<sup>n</sup>, a prescription-optimization problem for X is an optimization problem over X where some of the coordinates of the vectors have been prescribed to be either 0 or 1. More formally, we are given two subsets of coordinates I, J⊆[n] and the prescription-optimization problem for X is to solve: <math display="block"><mtable columnalign="right left"> <mtr><mtd><mo>min</mo></mtd> <mtd><mi>wᵗ</mi><mi>x</mi></mtd></mtr> <mtr><mtd><mtext>s.t.</mtext></mtd> <mtd><mi>x</mi><mo>∈</mo><mi>X</mi></mtd></mtr> <mtr><mtd></mtd><mtd><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mn>1</mn> <mspace width="1em"/><mtext>for all</mtext><mspace width=".5em"/> <mi>i</mi><mo>∈</mo><mi>I</mi></mtd></mtr> <mtr><mtd></mtd><mtd><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mn>0</mn> <mspace width="1em"/><mtext>for all</mtext><mspace width=".5em"/> <mi>i</mi><mo>∈</mo><mi>J</mi></mtd></mtr> </mtable> </math> In this talk, we present the following surprising relationship between prescription-optimization problems and exhaustive generation: Theorem. If the prescription optimization problem for X can be solved in time T, then there is an exhaustive generation Algorithm for X which takes O(n+T⋅log n) time between generated objects in X. Furthermore, this Algorithm outputs X in the order of a Hamilton path in conv(X). Then we will see some applications of this Theorem to the 0/1 vertex enumeration problem, matroid bases Gray codes, and graph matching Gray codes. In many cases, we obtain new generation algorithms with the best-known worst-case delay. Furthermore, the algorithm guaranteed by the Theorem at each step acts greedily on the prefix and generalizes a recent greedy-generation algorithm for matroid bases by Merino, Mütze, and Williams. This is work in progress.
Sandro
Roch
Technische Universität Berlin
Arrangements of Pseudocircles: On Digons and Triangles
Abstract.
A pseudocircle is a simple closed curve in the plane. An intersecting arrangement of pseudocircles is a finite collection of pseudocircles so that any two intersect in exactly two points where they cross. Grünbaum conjectured in the 1970's that in the case of simple arrangements there are at most 2n - 2 digon cells, i.e. cells which have exactly two crossings on its boundary. I will present a result by Agarwal et al. (2004) which proves this conjecture for the special case of cylindrical arrangements. Based on that we show that the conjecture also holds whenever the arrangement contains three pseudocircles which pairwise form a digon cell. Moreover, I will present a result concerning the number of triangles in digon free arrangements, which disproves another conjecture by Grünbaum. (joint with S.Felsner and M.Scheucher)
Torsten
Ueckerdt
Karlsruher Institut für Technologie, KIT
Stack and Queue Layouts of Planar Graphs
Abstract.
A colored linear layout of a graph is a total ordering of its vertices together with a partition of its edges into color classes. In a stack layout each color class is crossing-free, in a queue layout each color class is nesting-free, while in both cases our goal is to minimize the number of colors. In this talk we discuss on a higher level approaches to find good stack or queue layouts for planar graphs, including some recent breakthroughs and open problems.
Ji Hoon
Chun
TU Berlin
What is a sphere packing?
Abstract.
The problem of finding the densest way to pack equally sized spheres in Euclidean space has been studied for hundreds of years. The densest packing is only known in dimensions 1, 2, 3, 8, and 24, with the last two being recently solved by Viazovska (2016) and Cohn et al. (2016). This talk will give an overview of several lower and upper bounds for the maximum packing density. We will also show examples of dense sphere packings in small and high dimensions and explain their properties and connections to other areas of mathematics.
Toni
Karvonen
University of Helsinki
Gaussian processes for uncertainty quantification and error estimation
Maryna
Viazovska
EPFL
Sphere packings, universal optimality, and Fourier interpolation
Rosna
Paul
Perfect matchings with crossings
Amzi
Jeffs
Carnegie Mellon University
Enumerating interval graphs and d-representable complexes
Abstract.
Given a collection of n convex sets in R^d, one can record how they intersect using a simplicial complex on the vertex set [n] = {1, 2, ..., n} called the nerve. Simplicial complexes that arise this way are called d-representable, and enjoy a variety of interesting combinatorial properties, such as d-collapsibility. However, the number of these complexes has not been previously studied, except in the case d=1, which amounts to counting interval graphs with n vertices. We show that the number of d-representable complexes on vertex set [n] is exp(Theta(n^d log n)), and provide bounds on the constants involved. As a consequence, we show that d-representable complexes comprise a vanishingly small fraction of d-collapsible complexes. In the case d=1 our results are more precise, and improve the previous best asymptotics for the number of interval graphs. These results are joint work with Boris Bukh.
Erling
Andersen
Mosek
The value of conic optimization for analytics practitioners
Abstract.
Linear optimization, also known as linear programming, is a modelling framework widely used by analytics practitioners. The reason is that many optimization problems can easily be described in this framework. Moreover, huge linear optimization problems can be solved using readily available software and computers. However, a linear model is not always a good way to describe an optimization problem since the problem may contain nonlinearities. Nevertheless such nonlinearities are often ignored or linearized because a nonlinear model is considered cumbersome. Also there are issues with local versus global optima and in general it is just much harder to work with nonlinear functions than linear functions. Over the last 15 years a new paradigm for formulating certain nonlinear optimization problems called conic optimization has appeared. The advantage of conic optimization is that it allows the formulation of a wide variety of nonlinearities while almost keeping the simplicity and efficiency of linear optimization. Therefore, in this presentation we will discuss what conic optimization is and why it is relevant to analytics practitioners. In particular we will discuss what can be formulated using conic optimization, illustrated by examples. We will also provide some computational results documenting that large conic optimization problems can be solved efficiently in practice. To summarize, this presentation should be interesting for everyone interested in an important recent development in nonlinear optimization.
Andrei
Bud
HU Berlin
The Prym-Brill-Noether divisor
Abstract.
Understanding the birational geometry of the moduli space \(\mathcal{R}_g\) parametrizing Prym curves has been the subject of several papers, with great insight into this problem coming from the work of Farkas and Verra. Of particular importance for this study is finding divisors of small slope on the space \(\overline{\mathcal{R}}_g\). Drawing parallels with the situation on \(\overline{\mathcal{M}}_g\), we consider the Prym-Brill-Noether divisor and compute (some relevant coefficients of) its class. We will highlight the role of strongly Brill-Noether loci in understanding Prym-Brill-Noether loci. A consequence of our study is that the space \(\mathcal{R}_{14,2}\) parametrizing \(2\)-branched Prym curves of genus \(14\) is of general type.
Henri Elad
Altman
Dirichlet Postdoc
Long-time behaviour of additive functionals of rough stochastic processes through local times
Abstract
Marco
Flores
HU Berlin
A cohomological approach to formal Fourier-Jacobi series
Abstract.
Siegel modular forms admit various expansions, one of the most important being the Fourier-Jacobi expansion. Algebraically, these expansions take the form of a series whose coefficients are Jacobi forms satisfying a certain symmetry condition. One then poses the following modularity question: does every formal series of that shape come from a Siegel modular form? Bruinier and Raum answered the question affirmatively, over the complex numbers, in 2014. In this talk I will consider this question over the ring of integers, and reformulate it as a matter of cohomological vanishing. I will present a weaker version of the desired cohomological vanishing, and a result highlighting how special the case of genus g=2 potentially is.
Ekin
Ergen
TU Berlin
Lower Bounds for Approximation Algorithms for the Steiner Tree Problem
Abstract.
The Steiner tree problem inquires for a minimum weight subgraph that connects a given subset of vertices of a graph. It is known to be NP-complete, and many constant-factor approximation algorithms are known. In this talk, we present lower bounds of two of these approximation algorithms. The first algorithm is the k-Loss Contraction Algorithm of Robins and Zelikovsky. It was known to have an approximation ratio at least 1.233 and at most 1.55. We improve the lower bound to 1.25, and present some further upper and lower bounds for special cases. The second algorithm is the new local search based algorithm of Traub and Zenklusen. It is known to have an approximation ratio less than 1.39 in general instances. We show a lower bound of 1.25 for the approximation ratio. We also provide an example that comes arbitrarily close to the upper bound 73/60 of quasi-bipartite instances, showing that the analysis is tight in that case.
Marcel
Celaya
ETH, Zürich
Improving the Cook et al. Proximity Bound Given Integral Valued Constraints
Abstract.
Given an optimal solution to a linear program, how far away can a nearest optimal integral solution be? In 1986 Cook, Gerards, Schrijver, and Tardos gave a bound for this distance, known as proximity, which depends only on the dimension and the largest possible magnitude of any subdeterminant of the corresponding constraint matrix. In this talk I will briefly survey this problem, describe some long standing related conjectures, and highlight some recent developments including a recent improvement to the Cook et al. bound when the dimension is at least 2. This is joint work with Joseph Paat, Stefan Kuhlmann, and Robert Weismantel.
Andrew
Newman
Carnegie Mellon University, Pittsburgh
Complexes of nearly maximum diameter
Abstract.
The combinatorial diameter of a simplicial complex is the diameter of its dual graph. Using a probabilistic approach we determine the right first-order asymptotics for the maximum possible diameter among all d-complexes on n vertices as well as among all d-pseudomanifolds on n vertices. This is joint work with Tom Bohman.
Philipp
Wacker
FU Berlin
MAP estimators in l^p
Markus
Kirchweger
A Beam Search for the Shortest Common Supersequence Problem Guided by an Approximate Expected Length Calculation
Samantha
Fairchild
University of Osnabrück and MPI Leipzig
Counting pairs of saddle connections
Abstract.
A translation surface is a collection of polygons in the plane with parallel sides identified by translation to form a Riemann surface with a singular Euclidean structure. A saddle connection is a special type of closed geodesic, and the set of saddle connections can be associated to a discrete subset of the complex plane. I will discuss recent work showing that for almost every translation surface the number of pairs of saddle connections with bounded virtual area has quadratic asymptotic growth. No previous knowledge of translation surfaces or counting problems will be assumed. This is based on joint work with Jayadev Athreya and Howard Masur.
Anita
Liebenau
UNSW
Uncommon and Sidorenko systems of equations
Abstract.
A system of linear equations L over the finite field Fq is common if the number of solutions to L in any two-colouring of Fqn is asymptotically (as n→∞) at least the expected number of monochromatic solutions in a random colouring of Fqn. For example, a Schur triple x+y=z was shown to be common by Cameron, Cilleruelo and Serra in 2007. Another heavily studied specific example is that of an arithmetic progression of length four (4-AP), which can be described by two equations of the form x - 2y + z = 0 and y - 2z + w = 0. Wolf showed that these are uncommon (over ZN and over F5). Motivated by these existing results on specific systems, as well as extensive research on common and Sidorenko graphs, the systematic study of common systems of linear equations was recently initiated by Saad and Wolf. Building on earlier work of Cameron, Cilleruelo and Serra, common linear equations have been fully characterised by Fox, Pham and Zhao. In this talk, we discuss recent progress towards characterising common systems of two or more equations. In particular, we prove that any system containing a 4-AP is uncommon, confirming a conjecture of Saad and Wolf. We also discuss the related concept of Sidorenko systems of equations. Joint work with Nina Kamčev and Natasha Morrison.
Alex
Elzenaar
MPI MiS Leipzig
Aspects of computation in hyperbolic geometry
Abstract.
Low-dimensional topology has the advantage that it is relatively easy to visualise-the key word being relatively. Some people have amazing powers of visualisation (famously, William Thurston); but most of us need to rely on visual aids to try to understand what is going on. There is a long history of computer experimentation in the area of hyperbolic geometry and Kleinian groups (groups which arise dually to hyperbolic 3-manifolds), from the Thurston days (Robert Riley's work on knot groups in the 1970s and Jeff Weeks' study of triangulations of 3-manifolds in the 1980s) to the modern era (including but not limited to work by David Mumford, Caroline Series, and David Wright; Masaaki Wada; and Sabetta Matsumoto and Henry Segerman). This talk will discuss some of the highlights of computation in hyperbolic geometry with an emphasis on visualisation, focusing on groups and manifolds related to two-bridge knots (parameterised by the so-called Riley slice).
Ruijie
Yang
HU Berlin
Singularities of theta divisors of hyperelliptic curves
Abstract.
It is a classical result that the theta divisor on the Jacobian variety associated to a smooth projective hyperelliptic curve has maximal dimension of singularities among indecomposable principally polarized abelian varieties. A version of the Schottky problem asks if this condition on the dimension of singularities characterizes hyperelliptic Jacobians. Motivated by this problem, it is natural to study the singularities of hyperelliptic theta divisors in more details to understand why they are special. In this talk, I will explain a natural and explicit embedded resolution of hyperelliptic theta divisors inside Jacobians by successively blowing up (proper transforms of) Brill-Noether subvarieties. A key observation is that we can use the geometry of Abel-Jacobi maps and secant varieties of rational normal curves to avoid the analysis of singularities in the blow-up process. From the point of view of singularities of pairs, we obtain all the essential information. If time permits, I will discuss how this resolution can be used to understand the mixed Hodge module structure on the vanishing cycle of a hyperelliptic theta divisor, using a global version of Esnault-Viehweg’s cyclic covering construction, limiting mixed Hodge structures on normal crossing divisors, and twisted D-modules. This is joint work (partially in progress) with Christian Schnell.
Patrick
Morris
UPC
Schur properties of randomly perturbed sets
Abstract.
A set A of integers is said to be Schur if any two-colouring of A results in monochromatic x,y and z with x+y=z. We discuss the following problem: how many random integers from [n] need to be added to some A ⊆ [n] to ensure with high probability that the resulting set is Schur? Hu showed in 1980 that when |A| > ⌈4n/5⌉, no random integers are needed, as A is already guaranteed to be Schur. Recently, Aigner-Horev and Person showed that for any dense set of integers A ⊆ [n], adding ω(n1/3) random integers suffices, noting that this is optimal for sets A with |A| ≤ ⌈n/2⌉. We close the gap between these two results by showing that if A ⊆ [n] with |A| = ⌈n/2⌉+t <⌈4n/5⌉, then adding ω(min{n1/3,nt-1}) random integers will with high probability result in a set that is Schur. Our result is optimal for all t, and we further initiate the study of perturbing sparse sets of integers A by providing nontrivial upper and lower bounds for the number of random integers that need to be added in this case. Joint work with Shagnik Das and Charlotte Knierim.
Dougal
Davis
U Edinburgh
Mixed Hodge modules on flag varieties and representations of real reductive groups
Abstract.
In this talk, I will give a gentle introduction to applications of mixed Hodge modules in the representation theory of real reductive groups. Since the work of Beilinson-Bernstein, Kazhdan-Lusztig and Lusztig-Vogan, it has been understood how to realise representations of real groups in terms of twisted D-modules on flag varieties, and how weight filtrations coming from the corresponding mixed sheaves over a finite field can be used to gain strong control over their structure. Much more recently, it was suggested by Schmid and Vilonen that passing instead to mixed Hodge modules reveals more information about representations than is accessible over a finite field, through Hodge filtrations and polarisations. I will sketch this broad story, including the main conjecture of Schmid-Vilonen relating Hodge structures to unitarity, and, as time permits, explain some concrete results appearing in recent joint work of myself and Vilonen (arXiv:2202.08797) on the computation of the Hodge structures in Kazhdan-Lusztig-Vogan theory and their connection with the unitarity algorithm of Adams-van Leeuwen-Trapa-Vogan.
Jana
Novotna
University of Warsaw
Taming Creatures
Abstract.
A graph class is tame if it admits a polynomial bound on the number of minimal separators, and feral if it contains infinitely many graphs with exponential number of minimal separators. The former entails the existence of polynomial-time algorithms for Maximum Weight Independent Set, Feedback Vertex Set, and many other problems, when restricted to an input graph from a tame class, by a result of Fomin, Todinca, and Villanger [SIAM J. Comput. 2015]. In the talk, we show a full dichotomy of hereditary graph classes defined by a finite set of forbidden induced subgraphs into tame and feral. To obtain the full dichotomy, we confirm a conjecture of Gartland and Lokshtanov [arXiv:2007.08761]: if for a hereditary graph class C there exists a constant k such that no member of C contains a k-creature or a k-skinny-ladder as an induced minor, then there exists a polynomial p such that every graph G from C contains at most p(|V(G)|) minimal separators. Joint work with Jakub Gajarský, Lars Jafke, Paloma T. Lima, Marcin Pilipczuk, Paweł Rzążewski, and Uéverton S. Souza.
Tomáš
Masarik
Charles University Prague
Max Weight Independent Set in graphs with no long claws: An analog of the Gyárfás' path argument
Abstract.
We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw St,t,t as an induced subgraph [Chudnovsky, Pilipczuk, Pilipczuk, Thomassé, SODA 2020] and provide a subexponential-time algorithm with improved running time 2O(n√logn) and a quasipolynomial-time approximation scheme with improved running time 2O(ε−1log5n). The Gyárfás' path argument, a powerful tool that is the main building block for many algorithms in Pt-free graphs, ensures that given an n-vertex Pt-free graph, in polynomial time we can find a set P of at most t−1 vertices, such that every connected component of G−N[P] has at most n/2 vertices. Our main technical contribution is an analog of this result for St,t,t-free graphs: given an n-vertex St,t,t-free graph, in polynomial time we can find a set P of O(tlogn) vertices and an extended strip decomposition (an appropriate analog of the decomposition into connected components) of G−N[P] such that every particle (an appropriate analog of a connected component to recurse on) of the said extended strip decomposition has at most n/2 vertices. In the talk, we first show how Gyárfás' path argument works on Pt-free graphs. Then we will sketch-prove our main result with as few technical details as possible. Joint work with: Konrad Majewski, Jana Novotná, Karolina Okrasa, Marcin Pilipczuk, Paweł Rzążewski, Marek Sokołowski
Marco
Maculan
IMJ Paris
Counting rational points on varieties with large fundamental group
Abstract.
A nonsingular projective curve of genus at least 2 on a number field admits only finitely many rational points. Elliptic curves might have infinitely many rational points (as the projective line does), but “way less” than the projective line. In a joint work with Y. Brunebarbe, inspired by a recent result of Ellenberg-Lawrence-Venkatesh, we prove an analogous statement in higher dimension: projective varieties with large fundamental group in the sense of Kollár-Campana have “way less” rational points than Fano varieties.
Wolfgang
Lück
U Bonn
A Panorama of L2-Invariants
April 2022
Johannes
Zonker
FU Berlin
What is epidemic modeling?
Abstract.
While the spreading of infectious diseases has already been a popular research area for a long time, since the beginning of the COVID19 pandemic almost everyone (including non-mathematicians) has at least heard about the term epidemic modeling. The goal of this talk is to provide an introduction to the basic concepts and core assumptions of epidimic models as well as an overview of common model types. Among the different model types we will focus on the widely used compartmental ODE models and related network and metapopulation approaches. Slides are available here.
Eneas
Hensel
University of Mannheim
Variational inference for Bayesian neural networks
Max
von Kleist
RKI
Data science for COVID research and in policy design
Manfred
Scheucher
SAT, ASP, and Symmetry Breaking
Florian
Frick
FU Berlin and Carnegie Mellon University
Embeddings, chirality, triangulations
Abstract.
An object is chiral if it has a left-handed and a right-handed version, where one cannot be deformed into the other. A space that is chiral in (d+1)-space cannot embed into d-space. I will explain how to extend classical non-embeddability results to chirality results using triangulations of spaces. In fact, one can derive bounds for the topology of the space of embeddings this way. This is joint work with M. Harrison (IAS).
Thomas
Blomme
University of Geneva
Enumeration of tropical curves in abelian surfaces
Abstract.
Tropical geometry is a powerful tool that allows one to compute enumerative algebraic invariants through the use of some correspondence theorem, transforming an algebraic problem into a combinatorial problem. Moreover, the tropical approach also allows one to twist definitions to introduce mysterious refined invariants, obtained by counting tropical curves with polynomial multiplicities. So far, this correspondence has mainly been implemented in toric varieties. In this talk we will study enumeration of curves in abelian surfaces and use the tropical geometry approach to prove a multiple cover formula that enables an simple and elegant computation of enumerative invariants of abelian surfaces.
Thomas
McCormick
Parametric Min Cut Complexity
Abstract.
Consider the Max Flow / Min Cut problem where the arc capacities depend on one or more parameters. Then the max flow value as a function of the parameter(s) is a piecewise linear concave function. The complexity of a class of problems is the worst-case number of pieces in this function. It has been know since Carstensen (1983) that in general this function has exponential complexity. But Gallo, Grigoriadis, and Tarjan (1989) showed that when we restrict to Source-Sink Monotone (SSM) networks and a single parameter, the complexity is only linear, and this has been generalized by various authors. SSM networks are a special case of a general theory of parametric optimization by Topkis, which applies equally to more than one parameter. Our main result is that even with just two parameters, the complexity of SSM Min Cut is exponential.
Sandro
Roch
Constructions in combinatorics via neuronal networks
Eleonore
Bach
FU Berlin
Cographic hyperplane arrangements, flow zonotopes and their Ehrhart polynomials
Abstract.
Geometrically carrying a trove of information about the underlying simple graph, the graphic hyperplane arrangement $H_G$ yields an interesting mathematical object to study a simple graph $G$. For example, one proves that the regions of $H_G$ are in one-to-one correspondence to the acyclic orientations of $G$ and that the normal vectors of $H_G$ are linearly independent if and only if they induce forests on $G$. The latter is related to the independent sets of the graphic matroid which is the matroid associated to the graphic hyperplane arrangement. Furthermore, we associate with graphic hyperplane arrangements graphic zonotopes whose Ehrhart polynomials are related to the independent sets of the graphic matroid. In this talk we explain how the cographic hyperplane arrangement arises as a dual of its graphic counterpart. This is accomplished through matroid duality, graph duality and dualizing the chain complex associated to the faces of a special polytope. Completing this dual picture, a final result shown in this talk describes the coefficients of the Ehrhart polynomial of the flow zonotope, which is the zonotope associated to the cographic hyperplane arrangement.
Guilherme Oliveira
Mota
USP
Constrained colourings of random graphs
Abstract.
Given graphs G, H1 and H2, let G --> (H1,H2) denote the property that in every edge-colouring of G there is a monochromatic copy of H1 or a rainbow copy of H2. The constrained Ramsey number, defined as the minimum n such that Kn --> (H1,H2), exists if and only if H1 is a star or H2 is a forest. We determine the threshold for the property G(n,p) --> (H1,H2) when H2 is a forest. This is a joint work with Maurício Collares, Yoshiharu Kohayakawa and Carlos Gustavo Moreira.
Michael
Joswig
TU Berlin
Lattice polygons and real roots
Abstract.
It is known from theorems of Bernstein, Kushnirenko and Khovanskii from the 1970s that the number of complex solutions of a system of multivariate polynomial equations can be expressed in terms of subdivisions of the Newton polytopes of the polynomials. For very special systems of polynomials Soprunova and Sottile (2006) found an analogue for the number of real solutions. In joint work with Ziegler we could give a simple combinatorial formula and an elementary proof for the signature of foldable triangulation of a lattice polygon. Via the Soprunova-Sottile result this translates into lower bounds for the number of real roots of certain bivariate polynomial systems.
Tanh Long
Phan
On PDE Constraint Regularizations in Image Processing
Malte
Renken
TU Berlin
Connectivity thresholds in random temporal graphs
Abstract.
A graph whose edges only appear at certain points in time is called a temporal graph. In a temporal graph, two vertices are said to be connected if there is a temporal path between them, that is a path which traverses edges in chronological order. We consider a simple model of a random temporal graph, obtained by assigning to every edge of an Erdős–Rényi random graph G<sub>n,p</sub> a uniformly random presence time in the real interval [0,1]. We study several connectivity properties of this random temporal graph model and uncover a surprisingly regular sequence of sharp thresholds for these properties. Finally, we discuss how our results can be transferred to other random temporal graph models.
Steven-Marian
Stengl
Combined Regularization and Discretization of Equilibrium Problems and Primal-Dual Gap Estimators
Abstract.
In this talk, we adress the treatment of finite element discretizations of a class of equilibrium problems involving moving constraints. Therefore, a Moreau-Yosida based regularization technique, controlled by a parameter, is discussed. A generalized Γ-convergence concept is utilized to obtain a priori results. The same technique is applied to the discretization and the combination of both. In addition, a primal-dual gap technique is used for the derivation of error estimators and a strategy for balancing between a refinement of the mesh and an update of the regularization parameter is established. The theoretical findings are illustrated for the obstacle problem as well as numerical experiments are performed for two quasi-variational inequalities with application to thermoforming and biomedicine, respectively.
Dávid
Matolcsi
Eötvös Loránd University
Avoiding right angles and certain Hamming distances
Abstract.
In this paper we show that the largest possible size of a subset of Fqn avoiding right angles, that is, distinct vectors x,y,z such that x−z and y−z are perpendicular to each other is at most O(nq−2). This improves on the previously best known bound due to Naslund and refutes a conjecture of Ge and Shangguan. A lower bound of nq/3 is also presented. It is also shown that a subset of Fqn avoiding triangles with all right angles can have size at most O(n2q−2). Furthermore, asymptotically tight bounds are given for the largest possible size of a subset A ⊂ Fqn for which x−y is not self-orthogonal for any distinct x,y ∊ A. The exact answer is determined for q=3 and n ≡ 2 (mod 3).Our methods can also be used to bound the maximum possible size of a binary code where no two codewords have Hamming distance divisible by a fixed prime q. Our lower- and upper bounds are asymptotically tight and both are sharp in infinitely many cases.
András
Imolay
Eötvös Loránd University
Multicolor Turán Numbers
Abstract.
We consider a natural generalisation of Turán's forbidden subgraph problem and the Ruzsa-Szemerédi problem by studying the maximum number exF(n,G) of edge-disjoint copies of a fixed graph F can be placed on an n-vertex ground set without forming a subgraph G whose edges are from different F-copies. We determine the pairs {F,G} for which the order of magnitude of exF(n,G) is quadratic and prove several asymptotic results using various tools from the regularity lemma and supersaturation to graph packing results.
Krishnendu
Bhowmick
Methods of Hypergraph Containers
Philippe
Eyssidieux
Grenoble
Reshetikhin-Turaev representations as Kähler local systems
Abstract.
Joint work, partially in progress, with Louis Funar. In "Orbifold Kähler Groups related to Mapping Class groups", arXiv:2112.06726, we constructed certain orbifold compactifications of the moduli stack of stable pointed curves labelled by an integer \(p\) such that the corresponding Reshetikhin-Turaev representation of the mapping class group descend to a representation of the orbifold fundamental group. I will explain the construction of that orbifold and why it is uniformizable. I will then report on a work in progress on the uniformization of these orbifolds. I will sketch a proof of the steiness of its universal covering \(p\) odd large enough. An interesting new quantum topological consequence is that the image of the fundamental group of the smooth base of a non isotrivial complex algebraic family of smooth complete curves of genus greater than 2 by the Reshetikhin-Turaev representation is infinite (generalizing the Funar-Masbaum and the Koberda-Santharoubane-Funar-Lochak infiniteness theorems). If time allows, I will explain why the corresponding complex projective fundamental group satisfies the Toledo conjecture if \(p\) is divisible enough.
Felix
Schröder
Matroid depth parameters for integer programming
March 2022
Michele
Pernice
Scuola Normale Superiore, Pisa
Results about the Chow ring of moduli of stable curves of genus three
Abstract.
In this talk, we will discuss some results concerning the Chow ring of \(\overline{\mathcal{M}}_3\), the moduli stack of stable curves of genus three. In particular, we will describe the main new idea, which consists of enlarging the moduli stack of stable curves by adding curves with worse singularities, like cusps and tacnodes. This will make the geometry of the stack uglier but in turn its Chow ring will be easier to compute. The Chow ring of \(\overline{\mathcal{M}}_3\) can then be recovered by excising the locus of non-stable curves.
Meghana M.
Reddy
Lions and Contamination
Marcel
Boesl
Rechts-Links Pfade und ein Kriterium für Planarität
Marie
Brandenburg
MPI MiS Leipzig
Competitive Equilibrium and Lattice Polytopes
Abstract.
The question of existence of a competitive equilibrium is a game theoretic question in economics. It can be posed as follows: In a given auction, can we make an offer to all bidders, such that no bidder has an incentive to decline our offer? We consider a mathematical model of this question, in which an auction is modelled as weights on a simple graph. The existence of an equilibrium can then be translated to a condition on certain lattice points in a lattice polytope. In this talk, we discuss this translation to the polyhedral language. Using polyhedral methods, we show that in the case of the complete graph a competitive equilibrium is indeed guaranteed to exist. This is based on joint work with Christian Haase and Ngoc Mai Tran.
Zev
Woodstock
TU/ZIB
Proximity operators and nonsmooth optimization
Abstract.
Proximity operators are tools which use first-order information to solve optimization problems. However, unlike gradient-based methods, algorithms involving proximity operators are guaranteed to work in nonsmooth settings. This expository talk will discuss the mathematical and numerical properties of proximity operators, how to compute them, algorithms involving them, and advice on implementation.
Olivier
Huber
Topics in gas transport: Nash equilibrium and constrained exact boundary controllability
Abstract.
We present two results related to the transport of gas: the existence of a solution to a Generalized Nash Equilibrium Problem (GNEP) arising from the modeling of the gas market as an oligopoly, that is only the producers are players, and the consumers just react to the quantity of gas available. In a second part, the constrained exact boundary controllability of a semilinear hyperbolic PDE is investigated. The existence of an absolutely continuous solution and boundary control will be shown, under appropriate assumptions.
Lorenzo
Venturello
KTH
On the gamma-vector of symmetric edge polytopes
Abstract.
Symmetric edge polytopes (SEPs) are special lattice polytopes constructed from a simple graph. They play a role in physics, in the study of metric spaces and optimal transport. We study a numerical invariant of symmetric edge polytopes, called the gamma-vector, whose entries are linear combinations of the \(h^*\) numbers. In particular, we target a conjecture of Ohsugi and Tsuchiya which predicts nonnegativity of the gamma-vector of every SEP. In a joint work with Alessio D'Alì, Martina Juhnke-Kubitzke and Daniel Köhne, we prove nonnegativity for one of the entries of the gamma-vector, and descriube the graphs whose SEP attains equality. Moreover, we consider random graphs in the Erdős–Rényi model, and we obtain asymptotic nonnegativity of the whole gamma-vector in certain regimes.
Chiara
Fusar Bassini
A time-expanded Knapsack Problem with quadratic constraints
Abstract.
This work analyses a time-expanded version of the binary packing problem with convex quadratic constraints (TKP-QC), where items can be distributed over multiple time steps. In the first chapter, a formal definition of the model and some application examples are provided. We prove the NP-Hardness of the problem by reduction to the Maximum-Weight Independent Set Problem (MWIS) and show that if the number of time steps is part of the input size, the problem is APX-Hard. In subsequent chapters, different approximation techniques are discussed. First, we present two different multi-step strategies and their implications on the problem structure. Worse-case bounds for a related, uniformly distributing strategy across allowed time steps are derived. Then, we present theoretical results for two different algorithms with time-dependent constant-factor approximation bounds: a pipage rounding technique combined with a two-step relaxation procedure and a greedy strategy. Finally, some empirical experiments are discussed. We apply two of the introduced techniques to randomly-generated instances and to a real-life gas flow network, comparing them to the solution of an exact solver and a stochastic algorithm.
Melanie
Koser
A discrete-to-continuous Gamma-Limit of a two-dimensional frustrated Spin System
February 2022
Axel
Kröner
Optimal control of a semilinear heat equation subject to state and control constraints
Abstract.
In this talk we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional and the cost functional is of tracking type and contains a linear term in the control variable. We derive second-order necessary and sufficient conditions relying on the concept of alternative costates, quasi-radial critical directions, and the Goh transformation.
Maximilian
Gorsky
Matching Theory, Hamiltonicity, and Barnette's Conjecture
Rahul
Pandharipande
ETH Zürich
Log Abel-Jacobi theory
Abstract.
The cohomological study of the Abel-Jacobi map can be viewed at three level: the standard double ramification cycle, the universal double ramification cycle, and the logarithmic double ramification cycle. I will discuss the log DR cycle and its recent calculation (joint work with Holmes, Molcho, Pixton, Schmitt). The method involves the geometry of the universal Jacobian over the moduli space of curves, which I will also discuss.
Anastasija
Pesic
and
Melanie
Koser
and
Lukas
Abel
Probevorträge für BMS students' conference
Svenja
Griesbach
Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages
Simona
Boyadzhiyska
FU Berlin
Fixed-point cycles and envy-free division
Abstract.
Given an edge-labeling of the complete bidirected graph K⃡n with functions from [d] to itself, we call a cycle in K⃡n a fixed-point cycle if composing the labels of its edges results in a map that has a fixed point; the labeling is fixed-point-free if no fixed-point cycle exists. In this talk we will consider the following question: for a given d, what is the largest value of n for which there exists a fixed-point-free edge-labeling of K⃡n with functions from [d] to itself? This problem was recently introduced by Chaudhury, Garg, Mehlhorn, Mehta, and Misra and has close connections to fair allocation in social choice theory. We will also discuss the special case where the edges are labeled with permutations of [d], which is related to a problem recently studied by Alon and Krivelevich and by Mészáros and Steiner. This is joint work with Benjamin Aram Berendsohn and László Kozma.
Lei
Xue
University of Washington
A Proof of Grünbaum’s Lower Bound Conjecture for polytopes, lattices, and
strongly regular pseudomanifolds
Abstract.
In 1967, Grünbaum conjectured that any \(d\)-dimensional polytope with \(d +s \leq 2d\) vertices has at least \( \phi_k (d + s, d) = {d +1 \choose k + 1} + { d \choose k + 1} - { d + 1 - s \choose k + 1} \) \(k\)-faces. In the talk, we will prove this conjecture and discuss equality cases. We will then extend our results to lattices with diamond property (the inequality part) and to strongly regular normal pseudomanifolds (the equality part). We will also talk about recent results on \(d\)-dimensional polytopes with \(2d + 1\) or \(2d + 2\) vertices.
Alex
Torzewski
King's College London
Lawrence--Venkatesh bounds for curves in families
Abstract.
We outline how the method of Lawrence-Venkatesh to bound rational points via p-adic period mappings can be used in families. This leads to upper bounds on the number of rational points on curves of genus > 1 depending only on the reduction modulo a well chosen prime and the primes of bad reduction. This was first shown by Faltings as a consequence of the Mordell and Shafarevich Conjectures.
Marie
Brandenburg
Max Planck Institut Leipzig
Intersection Bodies of Polytopes
Abstract.
Intersection bodies are classical objects from convex geometry, that are constructed from given convex body. In the past, they have mainly been studied from the point of view of convex analysis. In this talk we investigate combinatorial and algebraic structures of intersection bodies of polytopes. We consider an algorithm to compute both the radial function and the algebraic boundary of these intersection bodies, and provide an upper bound for their degree. This is joint work with Katalin Berlow, Chiara Meroni and Isabelle Shankar.
Carlos
Amendola
Technische Universität München
Estimating Gaussian mixtures using sparse polynomial moment systems
Abstract.
The method of moments is a statistical technique for density estimation that solves a system of moment equations to estimate the parameters of an unknown distribution. A fundamental question critical to understanding identifiability asks how many moment equations are needed to get finitely many solutions and how many solutions there are. Since the moments of a mixture of Gaussians are polynomial expressions in the means, variances and mixture weights, one can address this question from the perspective of algebraic geometry. With the help of tools from polyhedral geometry, we answer this fundamental question for several classes of Gaussian mixture models. Furthermore, these results allow us to present an algorithm that performs parameter recovery and density estimation, applicable even in the high dimensional case. Based on joint work with Julia Lindberg and Jose Rodriguez (University of Wisconsin-Madison).
Klaus
Künnemann
Regensburg
The Calabi-Yau problem in archimedean and non-archimedean geometry
Abstract.
We discuss the Calabi-Yau problem on complex manifolds and its analog in non-archimedean geometry. The complex Calabi-Yau problem asks for solutions of a PDE of Monge-Ampère type. It was posed by Calabi in 1954 and solved by Yau in 1978. After a short reminder of the complex case we give a brief introduction to non-archimedean analytic geometry and report on recent progress in the non-archimedean case.
Anastasija
Pesic
HU Berlin
What is direct method of calculus of variations?
Abstract.
When looking to show that a minimum of a functional is attained, one usually first turns to the Direct Method of Calculus of Variations. In this lecture, following one concrete example from electrostatics, we will first explore the necessary condition that a minimizer must satisfy — the Euler-Lagrange equation. Then we will move on to the problem of existence of minimizers and present the Direct Method. We will discuss the assumptions involved and explore the method's limits.
Loïc
Dubois
Packness and Tree-Path indices
Paul
Breiding
MPI Leipzig
The Zonoid Algebra
Abstract.
In this talk I will introduce the zonoid algebra. Starting from the monoid structure of zonoids in R^d I will explain how to turn this structure into an algebra, where we can “multiply” zonoids. I will show that every multilinear map between finite dimensional vector spaces has a unique, continuous, Minkowski multilinear extension to the corresponding space of zonoids. Taking the wedge product of vector spaces as the multilinear map, we get a definition of the wedge of zonoids. This is the definition of the product in our algebra. The motivation for this construction comes from probabilistic intersection theory in a compact homogeneous space, where the zonoid algebra plays the role of a probabilistic cohomology ring. This is joint work with Antonio Lerario, Leo Mathis and Peter Bürgisser.
Mara
Belotti
TU Berlin
An enumerative problem for cubic (hyper)surfaces: point and line conditions
Abstract.
The moduli space of smooth cubic hypersurfaces in \(\mathbb{P}^n\) is an open subset of a projective space. We construct a compactification of the latter which allows us to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points. We term it a \(1\)-complete variety of cubic hypersurfaces in analogy to the space of complete quadrics. Paolo Aluffi explored the case of plane cubic curves. Starting from his work, we construct such a space in arbitrary dimensions by a sequence of five blow-ups. The counting problem is then reduced to the computation of five total Chern classes. Finally, we arrive at the desired numbers in the case of cubic surfaces. This is joint work with Alessandro Danelon, Claudia Fevola, Andreas Kretschmer.
Qianyu
Chen
Stony Brook
Limits of Hodge structures via D-modules
Abstract.
It is well-known that each cohomology group of compact Kähler manifold carries a Hodge structure. If we consider a degeneration of compact Kähler manifolds over a disk then it is natural to ask how the Hodge structures of smooth fibers degenerate. When the degeneration only allows a reduced singular fiber with simple normal crossings (i.e. semistable), Steenbrink constructed the limit of Hodge structures algebraically. A consequence of the existence of the limit of Hodge structures is the local invariant cycle theorem: the cohomology classes invariant under monodromy action come from the cohomology classes of the total space. In this talk, I will try to explain a method using D-modules to construct the limit of Hodge structures even when the degeneration is not semistable.
Maximilian
Engel
MATH+ Junior Research Group Leader
Synchronization and Chaos for Random Systems: General insights and application to chemical reaction models
Abstract
Benjamin Aram
Berendsohn
FU Berlin
The Diameter of Graph Associahedra
Abstract.
Graph Associahedra are a family of polytopes that generalize several well-known polytopes such as Associahedra, Permutohedra, and Cyclohedra. The skeleton of a Graph Associahedron corresponds to the rotation graph of the elimination trees on a fixed graph. In this talk, we study the diameter of Graph Associahedra, or, in other words, the maximum rotation distance between two elimination trees on a fixed graph G. We survey several known results and then focus on the case where G is a caterpillar tree, calculating the diameter of each Caterpillar Associahedron up to a constant factor.
Manuel
Radons
Technische Universität Berlin
Nearly flat polytopes in the context of Dürer's problem
Abstract.
Dürer's problem asks whether every 3-polytope P has a net. Is there always a spanning tree T of its edge graph, so that if we cut P along T the resulting surface can be unfolded into the plane without self-overlaps? A common technique in recent works is to fix a spanning tree and then study the deformations of the corresponding unfolding induced by an affine stretching or flattening of P. In the first part of my talk I will highlight landmark results by Ghomi, O'Rourke and Tarasov that emanated from this approach. In the second part I will present my own work on the unfoldability of nested prismatoids, which follows a similar ansatz.
Neil
Olver
London School of Economics and Political Science
Continuity, Uniqueness and Long-Term Behaviour of Nash Flows Over Time
Abstract.
We consider a dynamic model of traffic that has received a lot of attention in the past few years. Users control infinitesimal flow particles aiming to travel from a source to destination as quickly as possible. Flow patterns vary over time, and congestion effects are modelled via queues, which form whenever the inflow into a link exceeds its capacity. We answer some rather basic questions about equilibria in this model: in particular uniqueness (in an appropriate sense), and continuity: small perturbations to the instance or to the traffic situation at some moment cannot lead to wildly different equilibrium evolutions. To prove these results, we make a surprising connection to another question: whether, assuming constant inflow into the network at the source, do equilibria always eventually settle into a "steady state" where all queue delays change linearly forever more? Cominetti et al. proved this under an assumption that the inflow rate is not larger than the capacity of the network - eventually, queues remain constant forever. We resolve the more general question positively. (Joint work with Leon Sering and Laura Vargas Koch).
Janusz
Ginster
Scaling Laws and the Emergence of Complex Patterns in Helimagnetism
Raphael
Steiner
Cycle lengths modulo k in expanders
Raman
Sanyal
Goethe-Universität Frankfurt
Pivot rules and polytopes
Abstract.
The simplex algorithm is the method of choice for solving linear programs but it’s running time crucially depends on the chosen pivot rule. In joint work with Black, De Loera, and Lütjeharms, we introduced a new family of pivot rules, the normalized-weight pivot rules, that has a number of interesting features. In particular, normalized-weight pivot rules can be parametrized in a natural continuous manner and the behavior on a fixed linear program is governed by a polytope, the pivot rule polytope. In this talk I want to give an introduction to normalized-weight pivot rules and their associated polytopes with an emphasis on combinatorics: Among pivot-rule polytopes one finds monotone path polytopes, sweep polytopes, permutahedra, associahedra and many more old and new friends from algebraic and geometric combinatorics.
Sara
Griffith
Brown University
The Torelli theorem, or how to determine a graph from its algebraic data
Abstract.
The Torelli theorem for graphs, due to Caporaso-Viviani and Su-Wagner, shows that with mild hypotheses a graph's matroid can be obtained from the integral lattice in its vector space of flows. New results of the speaker show that the graph itself can be recovered using additional data related to winnable chip firing games on the graph.
Konstantin
Jakob
MIT
Geometric Langlands correspondence for Airy sheaves
Abstract.
The Airy equation is a classic complex ordinary differential equation. An \(\ell\)-adic analogue of this ODE was defined and studied in-depth by N. Katz and collaborators. In this talk I will report on recent work with M. Kamgarpour and L. Yi on the geometric Langlands correspondence for generalizations of the Airy equation to reductive groups. I will explain the construction of a class of \(\ell\)-adic local systems that generalize the Airy equation in a suitable sense. Our approach follows the rigidity method in the geometric Langlands program first applied by J. Heinloth, B.-C. Ngô and Z. Yun in their construction of Kloosterman sheaves for reductive groups. Imposing suitable local conditions on automorphic sheaves leads to a rigid situation in which one can construct a Hecke eigensheaf on the moduli stack of G-bundles. The eigenvalue of this eigensheaf is the sought-after local system.
Yichen
Qin
École Polytechnique
\(L\)-functions of Kloosterman sheaves
Abstract.
Kloosterman sums are exponential sums over finite fields, appearing as traces of Frobenius on some \(\ell\)-adic local systems \(Kl_{n+1}\) on \(G_m\), called Kloosterman sheaves. Fresán, Sabbah, and Yu have constructed a family of motives attached to the symmetric powers of Kloosterman sheaves \(Sym^k Kl_{n+1}\). They prove that the motivic \(L\)-functions of the motives attached \(Sym^k Kl_2\) have meromorphic extensions to the complex plane and satisfy functional equations conjectured by Broadhurst and Roberts. In this talk, I will present some results about the motivic \(L\)-functions of the motives attached to \(Sym^k Kl_3\) and some related motives of dimension 2. In particular, several \(L\)-functions arise from modular forms, which can be determined by the information acquired from motives.
Samuel
Mohr
Masaryk University
Uniform Turán density
Abstract.
In the early 1980s, Erdős and Sós initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph H is the infimum over all d for which any sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at least d contains H. In particular, they raise the questions of determining the uniform Turán densities of K4(3)- and K4(3). The former question was solved only recently in [Israel J. Math. 211 (2016), 349--366] and [J. Eur. Math. Soc. 97 (2018), 77--97], while the latter still remains open for almost 40 years. In addition to K4(3)- , the only 3-uniform hypergraphs whose uniform Turán density is known are those with zero uniform Turán density classified by Reiher, Rödl and Schacht~[J. London Math. Soc. 97 (2018), 77--97] and a specific family with uniform Turán density equal to 1/27. In this talk, we give an introduction to the concept of uniform Turán densities, present a way to obtain lower bounds using color schemes, and give a glimpse of the proof for determining the uniform Turán density of the tight 3-uniform cycle Cℓ(3), ℓ ≥ 5.
January 2022
Klaus
Heeger
Technische Universität Berlin
Stable Matchings Beyond Stable Marriage
Abstract.
Stable matchings are well-studied from computer science, mathematics, and economics. In the most basic setting, called Stable Marriage, there are two sets of agents. Each agent from one set has preferences over the agents from the other set. A matching assigns the agents into groups of two agents. A matching is called stable if there are no two agents preferring each other to the partners assigned to them. In this talk, we will review important parts of my forthcoming PhD thesis concerning the computational complexity of two extensions of this basic model: First, we assume that an instance of Stable Marriage is given, and the aim is to modify the instance (using as few "modifications" as possible) such that a given edge is part of some stable matching. Second, we assume that agents have preferences over sets of d-1 other agents (for some d>2). In this case, a matching matches agents into groups of size d, and a matching is stable if there are no d agents preferring to be matched together to being unmatched. While since 1991 this problem is known to be NP-complete, we study the case that the preferences of all agents are "similar".
George
Mertzios
Durham University, UK
Algorithmic Problems on Temporal Graphs
Abstract.
A temporal graph is a graph whose edge set changes over a sequence of discrete time steps. This can be viewed as a discrete sequence G1, G2, ... of static graphs, each with a fixed vertex set V. Research in this area is motivated by the fact that many modern systems are highly dynamic and relations (edges) between objects (vertices) vary with time. Although static graphs have been extensively studied for decades from an algorithmic point of view, we are still far from having a concrete set of structural and algorithmic principles for temporal graphs. Many notions and algorithms from the static case can be naturally transferred in a meaningful way to their temporal counterpart, while in other cases new approaches are needed to define the appropriate temporal notions. In particular, some problems become radically different, and often substantially more difficult, when the time dimension is additionally taken into account. In this talk we will discuss some natural but only recently introduced temporal problems and some algorithmic approaches to them. This lecture has been recorded.
Henriette
Lipschütz
FU Berlin
What is a regular polytope?
Abstract.
Polytopes in three dimensions are present in several objects used on daily basis: everything having planar sides without dents or holes belongs to this class of objects. Mathematically, they can be described as intersection of half spaces. Next to this definition, we will consider some of their basic properties and special classes of polytopes in 3D such as the Platonic and Archimedean solids.
Paco
Santos
Universidad de Cantabria
Multitriangulations and tropical Pfaffians
Abstract.
Let \(V=\binom{[n]}{2}\) label the possible diagonals among the vertices of a convex \(n\)-gon. A subset of size \(k+1\) is called a \((k+1)\)-crossing if all elements mutually cross, and a general subset is called \((k+1)\)-crossing free if it does not contain a \(k\)-crossing. \((k+1)\)-crossing free subsets form a simplicial complex that we call the \(k\)-associahedron and denote \(Ass_k{n}\) since for \(k=1\) one (essentially) recovers the simplicial associahedron. The \(k\)-associahedron on the \(n\)-gon is known to be (essentially ) a shellable sphere of dimension \(k(n-2k-1)\) and conjectured to be polytopal (Jonsson 2003). It is also a subword complex in the root system of the A.The Pfaffian of an anti-symmetric matrix of size \(2k+2\) is the square root of its determinant, and it is a homogeneous polynomial of degree \(k+1\) with one monomial for each possible complete matching among \(2k+2\) nodes representing the rows and columns. Thus, monomials correspond to certain \((k+1)\)-subsets of \(V\) and among them there is a unique \((k+1)\)-crossing. Calling \(I_k(n)\) the ideal of all Pfaffians of degree \(k+1\) in an antisymmetric matrix of size \(n\), it is known (Jonsson and Welker 2007) that for certain term orders the corresponding initial ideal equals the Stanley-Reisner ideal of the \(k\)-associahedron.In this talk we explore the relation between Pfaffians and the \(k\)-associahedron from the tropical perspective. We show that the part of the tropical Pfaffian variety \(trop(I_k(n))\) lying in the ``four-point positive orthant’’ realises the \(k\)-associahedron as a fan, and that this intersection is contained in (but is not equal to, except for \(k=1\)) the totally positive tropical Pfaffian variety \(trop^+(I_k(n))\). We hope this to be a step towards realising the \(k\)-associahedron as a complete fan, but have only attained this for \(k=1\): we show that for any seed triangulation \(T\), the projection of \(trop^+(I_1(n))\) to the coordinates corresponding to diagonals in T produces a complete polytopal simplicial fan, that is, the normal fan of an associahedron. In fact, the fans we obtain are linearly isomorphic to the \(g\)-vector fans in cluster algebras of type \(A\), as realized by Hoheweg-Pilaud-Stella (2018).
Raju
Krishnamoorthy
Wuppertal
Rank 2 local systems and abelian varieties
Abstract.
Motivated by work of Corlette-Simpson over the complex numbers, we conjecture that all rank 2 \(\ell\)-adic local systems with trivial determinant on a smooth variety over a finite field come from families of abelian varieties. We will survey partial results on a \(p\)-adic variant of this conjecture. Time permitting, we will provide indications of the proofs, which involve the work of Hironaka and Hartshorne on positivity, the answer to a question of Grothendieck on extending abelian schemes via their p-divisible groups, Drinfeld's first work on the Langlands correspondence for \(GL_2\) over function fields, and the pigeonhole principle with infinitely many pigeons. This is joint with Ambrus Pál.
Andrés
Rojas
Uni Bonn
Chern degree functions and applications to abelian surfaces
Abstract.
Given a smooth polarized surface, we will introduce Chern degree functions associated to any object of its derived category. These functions encode the behaviour of the object along the boundary of a certain region of Bridgeland stability conditions. We will discuss their extension to continuous real functions and the meaning of their differentiability at certain points. These functions turn out to be especially interesting for abelian surfaces, as they recover the cohomological rank functions defined by Jiang and Pareschi. In the final part we will apply this equivalence to give new results on the syzygies of abelian surfaces. This is a joint work with Martí Lahoz.
Kalina
Petrova
ETHZ
Resilience of Loose Hamilton Cycles in Random 3-Uniform Hypergraphs
Abstract.
Consider a random 3-uniform hypergraph H ~ H3(np) on n vertices, where each triple of vertices form a hyperedge with probability p. In this work, we prove a resilience result for H with respect to the property of containing a loose Hamilton cycle, that is, a cycle in which consecutive edges overlap in one vertex. More specifically, we show that there is a C s.t. if p >= C n-3/2 log(n), H is with high probability such that any spanning subgraph of H with minimum degree at least (7/16 + o(1)) p (n-1) (n-2) / 2 has a loose Hamilton cycle. This is optimal with respect to the resilience constant, but presumably not with respect to p. We also show a corresponding result about minimum co-degree, which is optimal with respect to both the resilience constant and p. Namely, there is a C s.t. if p >= C n-1 log(n), H is with high probability such that any spanning subgraph of H in which each pair of vertices is in at least (1/4 + o(1)) p (n-2) edges has a loose Hamilton cycle. This is joint work with Miloš Trujić.
Tobias
Kreutz
HU Berlin
Arithmetic and $\ extit{l}$-adic aspects of special subvarieties
Abstract.
We define an $\ell$-adic analog of the Hodge theoretic notion of a special subvariety. The Mumford-Tate conjecture predicts that the two notions are equivalent. In this talk, I want to discuss some properties of these subvarieties and prove this equivalence for subvarieties satisfying a certain monodromy condition. This builds on work of Klingler, Otwinowska and Urbanik on the fields of definition of special subvarieties.
Günter
Rote
Freie Universität Berlin
The maximum number of minimal dominating sets in a tree
Abstract.
A tree with n vertices has at most 95n/13 minimal dominating sets. The growth constant λ=13√95≈1.4194908 is best possible. It is obtained in a semi-automatic way as a kind of "dominant eigenvalue" of a bilinear operation on sextuples that is derived from the dynamic-programming recursion for computing the number of minimal dominating sets of a tree. The core of the method tries to enclose a set of sextuples in a six-dimensional geometric body with certain properties, which depend on some putative value of λ. This technique is generalizable to other counting problems, and it raises interesting questions about the "growth" of a general bilinear operation. We also derive an output-sensitive algorithm for listing all minimal dominating sets with linear set-up time and linear delay between successive solutions.
Stefan
Felsner
Box graphs with large girth and large chromatic number
Antonio
Macchia
FU Berlin
General non-realizability certificates for spheres with linear programming
Abstract.
A classical problem in polytope theory is whether a given combinatorial polytope is realizable as the boundary of a convex polytope. Attempts to answer this question often rely on the theory of oriented matroids, the classification of spheres with a certain dimension and number of vertices and the search for non-realizability certificates based on Grassmann-Plücker relations, called final polynomials. Due to the large number of such relations, it is often necessary to require special assumptions on the structure of the final polynomials. I will present a new method to search for algebraic certificates of non-realizability with no prescribed structure. Our algorithm combines linear programming together with the compact description of the realization space of a polytope given by the so-called reduced slack model, which we developed starting from the notion of slack matrix. This allows us to produce non-realizability certificates for several large simplicial and quasi-simplicial spheres whose realizability was not previously known. This is a joint work with João Gouveia and Amy Wiebe.
Tibor
Szabó
FU Berlin
Improved Integrality Gap in Restricted Max-Min Allocation
Abstract.
In the max-min allocation problem a set P of players are to be allocated disjoint subsets of a set R of indivisible resources, such that the minimum utility among all players is maximized. We study the restricted variant, also known as the Santa Claus problem, where each resource has an intrinsic positive value, and each player covets a subset of the resources. Bezakova and Dani showed that this problem is NP-hard to approximate within a factor less than 2, consequently a great deal of work has focused on approximate solutions. The principal approach for obtaining approximation algorithms has been via the Configuration LP (CLP) of Bansal and Sviridenko. Accordingly, there has been much interest in bounding the integrality gap of this CLP. The existing algorithms and integrality gap estimations are all based one way or another on the combinatorial augmenting tree argument of Haxell for finding perfect matchings in certain hypergraphs. Our main innovation is to introduce the use of topological methods for the restricted max-min allocation problem, to replace the combinatorial ones. This approach yields substantial improvements in the integrality gap of the CLP. In particular we improve the previously best known bound of 3.808 to 3.534. The talk represents joint work with Penny Haxell.
Jonathan
Boretsky
Harvard
The Totally Non-Negative Complete Flag Variety and its Tropicalization
Abstract.
A complete flag in \(\mathbb{R}^n\) is a collection of linear subspaces \(\{0\}=V_0\subsetneq V_1\cdots \subsetneq V_n=\mathbb{R}^n\). The set of all complete flags in \(\mathbb{R}^n\) form a variety called the complete flag variety. We give a combinatorial parameterization of the totally non-negative part of this variety. We also give a simple coordinate-wise characterization of which flags lie in the totally non-negative flag variety. We will also define the notion of a tropical variety and the non-negative part of a tropical variety. There are two tropical varieties that can be described using tropical versions of the equations cutting out the complete flag variety. One, the flag Dressian, can be thought of as describing flags of tropical linear spaces and the second, the tropical flag variety, can be thought of as describing realizable flags of tropical linear space. While these tropical varieties are generally different, we will use our parameterization of the non-negative complete flag variety to show that their non-negative parts coincide.
Kien Huu
Nguyen
KU Leuven
Exponential sums modulo \(p^m\) for Deligne polynomials
Abstract.
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Luca
Battistella
Uni. Heidelberg
Curve singularities, linear series, and stable maps: the case of genus two
Abstract.
I will discuss the geometry of Gorenstein curve singularities, and how these can be leveraged to produce compactifications of moduli spaces of smooth curves, embedded or otherwise. I will focus mainly on the case of genus two, where results have been established by myself with F. Carocci.
Christian
Lehn
TU Chemnitz
Singular varieties with trivial canonical class
Abstract.
We will present recent advances in the field of singular varieties with trivial canonical class obtained in joint work with Bakker and Guenancia building on work by many others. This includes the decomposition theorem which says that such a variety is up to a finite cover isomorphic to a product of a torus, irreducible Calabi-Yau (ICY) and irreducible symplectic varieties (ISV). The proof uses a reduction argument to the projective case which in turn is possible due to advances in deformation theory and a certain result about limits of Kähler Einstein metrics in locally trivial families.
Ji Hoon
Chun
Technische Universität Berlin
The Sausage Catastrophe in dimension 4
Abstract.
The Sausage Catastrophe (Jörg Wills) is the observation that in d = 3 and d = 4, the densest packing of n spheres is a sausage for small n and jumps to a full-dimensional packing for large n without passing through any intermediate dimensions. We denote the smallest value of n for which the densest packing is full-dimensional by k_d. We discuss some upper and lower bounds for k_3 and k_4, including k_3 ≤ 56 by Wills (1985) and k_4 < 375,769 by Gandini and Zucco (1992). We present some initial improvements to the upper bound for k_4 via extending the work of Gandini and Zucco.
María A.
Hernández Cifre
Universidad de Murcia
On discrete Brunn-Minkowski type inequalities
Abstract.
The classical Brunn-Minkowski inequality in the n-dimensional Euclidean space asserts that the volume (Lebesgue measure) to the power 1/n is a concave functional when dealing with convex bodies (non-empty compact convex sets). This result has become not only a cornerstone of the Brunn-Minkowski theory, but also a powerful tool in other related fields of mathematics. In this talk we will make a brief walk on this inequality, as well as on its extensions to the Lp-setting, for non-negative values of p. Then, we will move to the discrete world, either considering the integer lattice endowed with the cardinality, or working with the lattice point enumerator, which provides with the number of integer points contained in a given convex body: we will discuss and show certain discrete analogues of the above mentioned Brunn-Minkowski type inequalities in both cases. This is about joint works with Eduardo Lucas and Jesús Yepes Nicolás.
Benedikt
Preis
Regensburg
Motivic nearby cycle functors and local monodromy
Abstract.
We give a generalisation of Grothendieck's local monodromy theorem. The proof is completely independent of Grothendieck's classical proofs and shows that the unipotence phenomenon is of motivic nature.
Andres
Fernandez Herrero
Cornell
Intrinsic construction of moduli spaces via affine grassmannians
Abstract.
One of the classical examples of moduli spaces in algebraic geometry is the moduli of vector bundles on a smooth projective curve \(C\). More precisely, there exists a quasiprojective variety that parametrizes stable vector bundles on \(C\) with fixed numerical invariants. In order to further understand the geometry of this space, Mumford constructed a compactification by adding a boundary parametrizing semistable vector bundles. If the smooth curve \(C\) is replaced by a higher dimensional projective variety \(X\), then one can compactify the moduli problem by allowing vector bundles to degenerate to an object known as a "torsion-free sheaf". Gieseker and Maruyama constructed moduli spaces of semistable torsion-free sheaves on such a variety \(X\). More generally, Simpson proved the existence of moduli spaces of semistable pure sheaves supported on smaller subvarieties of \(X\). All of these constructions use geometric invariant theory (GIT). In this talk I will explain an alternative GIT-free construction of the moduli space of semistable pure sheaves which is intrinsic to the moduli stack of coherent sheaves. Our main technical tools are the theory of Theta-stability introduced by Halpern-Leistner, and some recent techniques developed by Alper, Halpern-Leistner and Heinloth. In order to apply these results, one needs to prove some monotonicity conditions for a polynomial numerical invariant on the stack. We show monotonicity by defining a higher dimensional analogue of the affine Grassmannian for pure sheaves. If time allows, I will also explain applications of these ideas to some other moduli problems. This talk is based on joint work with Daniel Halpern-Leistner and Trevor Jones.
Manfred
Scheucher
The Big-Line-Big-Clique Conjecture
Olivier
de Gaay Fortman
ENS Paris
The integral Hodge conjecture for one-cycles on Jacobians of curves
Abstract.
In this talk I will report on joint work with Thorsten Beckmann. I will prove that the minimal cohomology class of a principally polarized complex abelian variety of dimension \(g\) is algebraic if and only if all integral Hodge classes in degree \(2g-2\) are algebraic. In particular, this proves the integral Hodge conjecture for one-cycles on the Jacobian of a smooth projective curve over the complex numbers. The idea is to lift the Fourier transform on rational Chow groups to a homomorphism between integral Chow groups. I shall explore such integral lifts of the Fourier transform for an abelian variety over any field, partially answering a question of Moonen-Polishchuk and Totaro. Another corollary is the integral Tate conjecture for one-cycles on the Jacobian of a smooth projective curve over the separable closure over a finitely generated field.
Amedeo
Sgueglia
LSE
Multi-round Maker-Breaker Games
Abstract.
We consider a new procedure, which we call Multi-round Maker-Breaker Game. Maker and Breaker start from G0:=Kn and play several rounds of a usual Maker-Breaker game, where, for i ≥ 1, the i-th round is played as follows. They claim edges of Gi-1 until all edges are distributed, and then they set Gi to be the graph consisting only of Maker's edges. They will then play the next round on Gi. This creates a sequence of graphs G0 ⊃ G1 ⊃ G2 ⊃ ... and, given a monotone graph property, the question is how long Maker can maintain it, i.e. what is the largest k such that Maker has a strategy to guarantee that Gk satisfies such property. We will answer this question for several graph properties. This is joint work with Juri Barkey, Dennis Clemens, Fabian Hamann, and Mirjana Mikalački.
Vera
Traub
ETH Zürich
Better-Than-2 Approximations for Weighted Tree Augmentation
Abstract.
The Weighted Tree Augmentation Problem (WTAP) is one of the most basic connectivity augmentation problems. It asks how to increase the edge-connectivity of a given graph from 1 to 2 in the cheapest possible way by adding some additional edges from a given set. There are many standard techniques that lead to a 2-approximation for WTAP, but despite much progress on special cases, the factor 2 remained unbeaten for several decades. In this talk we present two algorithms for WTAP that improve on the longstanding approximation ratio of 2. The first algorithm is a relative greedy algorithm, which starts with a simple, though weak, solution and iteratively replaces parts of this starting solution by stronger components. This algorithm achieves an approximation ratio of (1 + ln 2 + ε) < 1.7. Second, we present a local search algorithm that achieves an approximation ratio of 1.5 + ε (for any constant ε > 0). This is joint work with Rico Zenklusen.
Jörg
Wolf
Chung-Ang University, Seoul
Compactness Lemma for sequences of divergencefree Bochner measurable functions
Helena
Bergold
Colorings of oriented planar graphs avoiding a monochromatic subgraph
Zuzana
Patáková
Charles University Prague
On Radon and fractional Helly theorems
Abstract.
Radon theorem plays a basic role in many results of combinatorial convexity. It says that any set of d+2 points in R^d can be split into two parts so that their convex hulls intersect. It implies Helly theorem and as shown recently also its more robust version, so-called fractional Helly theorem. By standard techniques this consequently yields an existence of weak epsilon nets and a (p,q)-theorem. We will show that we can obtain these results even without assuming convexity, replacing it with very weak topological conditions. More precisely, given an intersection-closed family F of subsets of R^d, we will measure the complexity of F by the supremum of the first d/2 Betti numbers over all elements of F. We show that bounded complexity of F guarantees versions of all the results mentioned above. Partially based on joint work with Xavier Goaoc and Andreas Holmsen.
Felix
Clemen
UIUC
Max cuts in triangle-free graphs
Abstract.
A well-known conjecture by Erdős states that every triangle-free graph on n vertices can be made bipartite by removing at most n2/25 edges. This conjecture is known to be true for graphs with edge density at least 0.4 and also for graphs with edge density at most 0.172. We present progress on this conjecture; we prove the conjecture for graphs with edge density at most 0.2486 and for graphs with edge density at least 0.3197. Further, we prove that every triangle-free graph can be made bipartite by removing at most n2/23.5 edges improving the previously best bound of n2/18. Time permitting, we will discuss related questions. This is joint work with József Balogh and Bernard Lidický.
Mads
Villadsen
Stony Brook
Generic vanishing and Chen-Jiang decompositions
Abstract.
The Generic Vanishing theorem of Green and Lazarsfeld describes the behaviour of the cohomology of direct images of canonical bundles to abelian varieties when twisted by line bundles with vanishing first Chern class. I will discuss Chen-Jiang decompositions of these direct images, first introduced by J. Chen and Z. Jiang for generically finite morphisms to abelian varieties, which explain their generic vanishing behaviour and certain positivity properties in detail. In particular, I will discuss how to prove the existence of these decompositions using classical variational Hodge theory.
December 2021
Lukas
Kühne
MPI Leipzig
Matroids and Algebra
Abstract.
A matroid is a combinatorial object based on an abstraction of linear independence in vector spaces and forests in graphs. I will discuss how matroid theory interacts with algebra via the so-called von Staudt constructions. These are combinatorial gadgets to encode polynomials in matroids. A main application is concerned with generalized matroid representations over division rings, matrix rings and probability space representations together with their relation to group theory. Based on joint work with Rudi Pendavingh and Geva Yashfe.
André
Uschmajew
Optimization on low-rank manifolds
Abstract.
Low-rank matrix and tensor models are important in many applications for representing and embedding high-dimensional data or functions. They typically lead to non-convex optimization problems on sets of matrices or tensors of given rank. In this talk, we give a basic introduction to the geometry of such sets and how it can be used to derive and study optimization algorithms. Compared to direct optimization of the factors in the model, the geometric approach is more intrinsic and can lead to improved methods. For a class of quadratic cost functions on matrices we also discuss how the geometric viewpoint is useful for studying the non-convex optimization landscape under low-rank constraints.
Sandro
Roch
Algorithms for sampling random pseudoline arrangements
Olaf
Parczyk
FU Berlin
Density of triangles and independent sets of size three
Abstract.
The triangle-density of a graph G is the number of triangles in G divided by the number of possible triangles. In this talk we will characterise all pairs (x,y), where x is the triangle-density in G and y the triangle-density in the complement of G. This is based on two papers of Huang, Linial, Naves, Peled, and Sudakov.
Raphael
Zentner
Universität Regensburg
Instanton gauge theory, the pillowcase, and SU(2)-representations
Abstract
Katharina
Jochemko
KTH
The Eulerian transformation
Abstract.
Many polynomials arising in combinatorics are known or conjectured to have only real roots. One approach to these questions is to study transformations that preserve the real-rootedness property. This talk is centered around the Eulerian transformation which is the linear transformation that sends the i-th standard monomial to the i-th Eulerian polynomial. Eulerian polynomials appear in various guises in enumerative and geometric combinatorics and have many favorable properties, in particular, they are real-rooted and symmetric. We discuss how these properties carry over to the Eulerian transformation. In particular, we disprove a conjecture by Brenti (1989) concerning the preservation of real roots, extend recent results on binomial Eulerian polynomials and provide enumerative and geometric interpretations. This is joint work with Petter Brändén.
Ruijie
Yang
HU Berlin
Higher multiplier ideals
Abstract.
Multiplier ideals of \(\mathbb{Q}\)-divisors are important invariants of singularities, which have had lots of applications in algebraic geometry. In this talk, I will introduce a new series of ideals, arising from the global study of vanishing cycles for D-modules. They can be thought as higher multiplier ideals and capture more refined information. I will discuss some general properties and how they are related to the theory of Hodge ideals developed by Mustata and Popa. This is joint work in progress with Christian Schnell.
Linda
Kleist
TU Braunschweig
Training Neural Networks is even harder
Abstract.
Methods and applications of Machine Learning have made impressive progress in recent years. One approach is neural network, a brain-inspired learning model that can be trained to perform complex and diverse tasks. In order to deliver good results, the training of such a network is crucial. Given a neural network, training data, and a threshold, finding weights for the neural network such that the total error is below the threshold is known to be NP-hard. In this talk, we determine the algorithmic complexity of this fundamental problem precisely, by showing that it is complete for the complexity class _existential theory of the reals_ (ER). This means that, up to polynomial time reductions, the problem is equivalent to deciding whether a system of polynomial equations and inequalities with integer coefficients and real unknowns has a solution. If, as widely expected, ER is strictly larger than NP, our work implies that the problem of training neural networks is not even in NP. Neural networks are usually trained using some variation of backpropagation. The result of this paper offers an explanation why techniques commonly used to solve big instances of NP-complete problems seem not to be of use for this task. Examples of such techniques are SAT solvers, IP solvers, local search, dynamic programming, to name a few general ones. The talk is based on joint work with Mikkel Abrahamsen and Tillman Miltzow.
Tobias
Harks
Uni Augsburg
Machine-Learned Prediction Equilibrium for Dynamic Traffic Assignment
Abstract.
We study a dynamic traffic assignment model, where agents base their instantaneous routing decisions on real-time delay predictions. We formulate a mathematically concise model and derive properties of the predictors that ensure a dynamic prediction equilibrium exists. We demonstrate the versatility of our framework by showing that it subsumes the well-known full information and instantaneous information models, in addition to admitting further realistic predictors as special cases. We complement our theoretical analysis by an experimental study, in which we systematically compare the induced average travel times of different predictors, including a machine-learning model trained on data gained from previously computed equilibrium flows, both on a synthetic and a real road network.
Constantin
Podelski
HU Berlin
The degree of the Gauss map on Theta divisors
Abstract.
The Gauss map attaches to any smooth point of a theta divisor in an abelian variety its tangent space translated to the origin. For indecomposable principally polarized varieties this is a generically finite map. In this talk, I will compute its degree for a generic ppav on some irreducible components of Andreotti-Mayer loci that have been introduced by Debarre in terms of specific Prym constructions. The computation will rely on the technique of Lagrangian specialisation.
Markus
Schmid
Humboldt-Universität zu Berlin
Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality Number
Abstract.
We investigate the locality number, a recently introduced structural parameter for strings (with applications in pattern matching with variables), and its connection to two important graph-parameters, cutwidth and pathwidth. These connections allow us to show that computing the locality number is NP-hard, but fixed-parameter tractable, if parameterised by the locality number or by the alphabet size, which has been formulated as open problems in the literature. Moreover, the locality number can be approximated with ratio O(\sqrt{log(opt)} log(n))$. An important aspect of our work --- that is relevant in its own right and of independent interest --- is that we identify connections between the string parameter of the locality number on the one hand, and the famous graph parameters of cutwidth and pathwidth, on the other hand. These two parameters have been jointly investigated in the literature (with respect to exact, parameterised and approximation algorithms), and are arguably among the most central graph parameters that are based on "linearisations" of graphs. Most importantly, we relate cutwidth with pathwidth via the locality number, which results in an approximation preserving reduction that improves the currently best known approximation algorithm for cutwidth. [This is based on joint work with Katrin Casel, Joel D. Day, Pamela Fleischmann, Tomasz Kociumaka, and Florin Manea, published in Proc. ICALP'19.]
Florin
Manea
Universität Göttingen
Combinatorial String Solving
Abstract.
We consider a series of natural problems related to the processing of textual data, rooted in areas as diverse as information extraction, bioinformatics, algorithmic learning theory, or formal verification, and see how they can all be formalized within the same framework. In this framework, we say that a pattern $\alpha$ (that is, a string of string-variables and letters from a fixed alphabet $\Sigma$) matches another pattern $\beta$ if a text $T$, over $\Sigma$, can be obtained both from $\alpha$ and $\beta$ by uniformly replacing the variables of the two patterns by words over $\Sigma$. In the case when $\beta$ contains no variables, i.e., $\beta=T$ is a text, a match occurs if $\beta$ can be obtained from $\alpha$ by uniformly replacing the variables of $\alpha$ by words over $\Sigma$. The respective matching problems, i. e., deciding whether two given patterns match or a pattern and a text match, are computationally hard, but efficient algorithms exist for classes of patterns with restricted structure. In this talk, we overview a series of recent results in this area.
Joris
Wenzel
Enforceable Drawings of Graphs
Matthias
Köppe
UC Davis
sagemath-polyhedra: A modularized Sage distribution
Abstract.
SageMath is an open-source general-purpose mathematical system based on Python that integrates computer algebra facilities and general computational packages. Sage, initially developed by Stein and first released in 2005, in over 1.5 decades of incubation in its pseudo-distribution comprising over 300 software packages, has grown to provide over 500 Cython and over 2000 of its own Python modules, ranging from sage.algebras.* over sage.geometry.* to sage.tensor.*, with a total of over 2.2 million lines of code.I will give an outline of a project, ongoing since 2020, to modularize the Sage library so that parts of it can be flexibly installed and used by other Python projects. One of the first products of the project is the modularized distribution sagemath-polyhedra, which provides computations and constructions with convex polyhedra with multiple backends, including PPL, Normaliz, and polymake.The project to modularize the Sage library does not conflict with the software integration goals of SageMath. As an example of progress on the latter, I will show the integration of sage.geometry.polyhedron and sage.manifolds, introduced in Sage 9.4.
Daniele
Agostini
MPI Leipzig
The Martens-Mumford Theorem and the Green-Lazarsfeld Secant Conjecture
Abstract.
The Green-Lazarsfeld secant conjecture predicts that the syzygies of a curve of sufficiently high degree are controlled by its special secants. We prove this conjecture for all curves of Clifford index at least two and not bielliptic and for all line bundles of a certain degree. Our proof is based on a classic result of Martens and Mumford on Brill-Noether varieties and on a simple vanishing criterion that comes from the interpretation of syzygies through symmetric products of curves.
Laura
Vargas Koch
ETH Zürich
Convergence of a Packet Routing Model to Flows Over Time
Abstract.
The mathematical approaches for modeling dynamic traffic can roughly be divided into two categories: discrete packet routing models and continuous flow over time models. Despite very vital research activities on models in both categories, the connection between these approaches was poorly understood so far. We build this connection by specifying a (competitive) packet routing model, which is discrete in terms of flow and time, and by proving its convergence to flows over time with deterministic queuing. More precisely, we prove that the limit of the convergence process, when decreasing the packet size and time step length in the packet routing model, constitutes a flow over time with multiple commodities. The convergence result implies the existence of approximate equilibria in the competitive version of the packet routing model. This is of significant interest as exact pure Nash equilibria, similar to almost all other competitive models, cannot be guaranteed in the multi-commodity setting. Moreover, the introduced packet routing model with deterministic queuing is very application-oriented as it is based on the network loading module of the agent-based transport simulation MATSim. As the present work is the first mathematical formalization of this simulation, it provides a theoretical foundation and an environment for provable mathematical statements for MATSim. This is joint work with Leon Sering and Theresa Ziemke.
Alberto
Espuny Díaz
TU Ilmenau
Path decompositions of random directed graphs
Abstract.
In this talk we consider the problem of decomposing the edges of a directed graph into as few paths as possible. The minimum number of paths needed in such an edge decomposition is called the path number of the digraph. The problem of determining the path number is generally NP-hard. However, there is a simple lower bound for the path number of a digraph in terms of its degree sequence, and a conjecture of Alspach, Mason, and Pullman from 1976 states that this lower bound gives the correct value of the path number for any even tournament. The conjecture was recently resolved, and in this talk I will discuss to what extent the conjecture holds for other digraphs. In particular, I will discuss some of the ingredients of a recent result showing that the conjecture holds with high probability for the random directed graph Dn,p for a large range of p. This is joint work with Viresh Patel and Fabian Stroh.
Malte
Renken
Technische Universität Berlin
Connectivity Thresholds in Random Temporal Graphs
Abstract.
We consider a simple model of a random temporal graph, obtained by assigning to every edge of an Erdős–Rényi random graph G_n,p a uniformly random presence time in the real interval [0, 1]. We study several connectivity properties of this random temporal graph model and uncover a surprisingly regular sequence of sharp thresholds at which these different levels of connectivity are reached. Finally, we discuss how our results can be transferred to other random temporal graph models. Based on joint work with Arnaud Casteigts, Michael Raskin, and Viktor Zamaraev.
Juan Carlos
de los Reyes
Escuela Politécnica Nacional, Ecuador
Bilevel learning for inverse problems
Abstract.
In recent years, novel optimization ideas have been applied to several inverse problems in combination with machine learning approaches, to improve the inversion by optimally choosing different quantities/functions of interest. A fruitful approach in this sense is bilevel optimization, where the inverse problems are considered as lower-level constraints, while on the upper-level a loss function based on a training set is used. When confronted with inverse problems with nonsmooth regularizers or nonlinear operators, however, the bilevel optimization problem structure becomes quite involved to be analyzed, as classical nonlinear or bilevel programming results cannot be directly utilized. In this talk, I will discuss on the different challenges that these problems pose, and provide some analytical results as well as a numerical solution strategy.
Arnaud
Casteigts
Université de Bordeaux
Spanners and connectivity problems in temporal graphs
Abstract.
A graph whose edges only appear at certain points in time is called a temporal graph. Such a graph is temporally connected if each ordered pair of vertices is connected by a path which traverses edges in chronological order (i.e., a temporal path). In this talk, I will focus on the concept of a temporal spanner, which is a subgraph of the input temporal graph that preserves temporal connectivity using as few edges (or labels) as possible. In stark contrast with standard graphs, it turns out that linear size spanners, and in fact, even sparse spanners (i.e., spanners with o(n^2) edges) do not always exist in temporal graphs. After presenting basic notions and reviewing these astonishing negative results, I will present some good news as well; namely, sparse spanners always exist in *some* natural classes of temporal graphs. These include the cases when the underlying graph is complete (this talk) or when the labels are chosen at random (subsequent talk). If time permit, I will present two open questions, and discuss some recent attempts at solving them.
Pierre
Clavier
IRIMAS, Université de Haute Alsace
What is resurgence theory?
Abstract.
Resurgence theory was invented over four decades ago by J. Ecalle, but has recently been gaining a lot of interest, in particular because of its applications to Physics. In this introductory talk, I will start to explain why is resugence interesting for physicists, then move on to detail how trans-series arise and can be worked out within the resurgent framework. I will also mention some interesting properties of (accelero-)summation a la Ecalle when applied to quantum field theory.
Felix
Schröder
Asymmetry in Planar Ramsey Graphs
Mariel
Supina
KTH
The Universal Valuation for Coxeter Matroids
Abstract.
Matroids are combinatorial objects that generalize the notion of independence, and their subdivisions have rich connections to geometry. Thus we are often interested in functions on matroids that behave nicely with respect to subdivisions, which are called valuations. Matroids are naturally linked to the symmetric group; generalizing to other finite reflection groups gives rise to Coxeter matroids. I will give an overview of these ideas and then present some work with Chris Eur and Mario Sanchez on constructing the universal valuative invariant of Coxeter matroids. Since matroids and their Coxeter analogues can be understood as families of polytopes with special combinatorial properties, I will present these results from a polytopal perspective.
Benjamin
Schröter
KTH
Ehrhart Polynomials of Rank-Two Matroids
Abstract.
There are many questions that are equivalent to the enumeration of lattice points in convex sets. Ehrhart theory is the systematic study of lattice point counting in dilations of polytopes. Over a decade ago De Loera, Haws and Köppe conjectured that the lattice point enumerator of dilations of matroid basis polytopes is a polynomial with positive coefficients. This intensively studied conjecture has recently been disproved in all ranks greater than or equal to three. However, the questions of what characterizes these polynomials remain wide open.In this talk I will report on my work, with Luis Ferroni and Katharina Jochemko, in which we complete the picture on Ehrhart polynomials of matroid basis polytopes by showing that they have indeed only positive coefficients in low rank. Moreover, we also prove that the closely related $h^∗$-polynomials of sparse paving matroids of rank two are real-rooted, which implies that their coefficients form log-concave and unimodal sequences.
Gavril
Farkas
HU Berlin
The birational geometry of \(M_g\) : new developments via non-abelian Brill-Noether theory and tropical geometry
Abstract.
I will discuss how novel ideas from non-abelian Brill-Noether theory can be used to prove that the moduli space of genus 16 is uniruled and that the moduli space of Prym varieties of genus 13 is of general type. For the much studied question of determining the Kodaira dimension of moduli spaces, both these cases were long understood to be crucial in order to make further progress. I will also explain the use of tropical geometry in order to establish the Strong Maximal Rank Conjecture, necessary to carry out this program.
Valentin
Kirchner
TU Berlin
Optimisation with Squared Lasso Penalty
Abstract.
The lecture offers a concise overview of a master's thesis on the penalised linear regression problem with squared lasso penalty. We derive its dual and optimality conditions and connect it to the well-known lasso problem. Based on the optimality conditions we study the solution path and briefly discuss an exact and an approximate homotopy algorithm.
Simón
Piga
Universität Hamburg
Codegree threshold for cycle decompositions in 3-uniform hypergraphs
Abstract.
We show that 3-graphs on n vertices whose codegree is at least (2/3+o(1))n can be decomposed into tight cycles of fixed length, subject to the trivial necessary divisibility conditions. We also provide a construction showing this result is best possible up to the o(1) term. All together, our results solve a recent conjecture by Glock, Kühn, and Osthus.
November 2021
Igor
Burban
U Paderborn
Tame non-commutative nodal curves, gentle algebras and homological mirror symmetry
Abstract.
In my talk, I am going to introduce a class of finite dimensional associative algebras, which are derived equivalent to certain tame non-commutative nodal curves. Following the approach of Lekili and Polishchuk, the constructed derived equivalence allows to interpret such non-commutative curves as homological mirrors of appropriate graded compact oriented surfaces with non-empty marked boundary. This is a joint work with Yuriy Drozd.
Benjamin
Berendsohn
Freie Universität Berlin
Search trees on graphs
Abstract.
Search trees on graphs (STGs) are a generalization of binary search trees (BSTs). Where the key space of a BST is a totally ordered set, the key space of a STG is a graph. STGs are a relatively recent notion, but have been studied previously under different names, including elimination trees, maximal tubings, and vertex rankings. We survey some results. On the computational side, we consider a model of computation for STGs analagous to the BST model, and study which known results for BSTs can be adapted. On the combinatorial side, we study the diameter of certain polytopes called graph associahedra, which can be defined via STGs.
Henry
Adams
Institute of Science and Technology Austria
Open questions on clique complexes of graphs
Abstract.
The clique complex of a graph is a simplicial complex with a simplex for each clique. Clique complexes are frequently being computed in applications of topology to data, but we do not understand their algorithmic theory or their mathematical theory. I will introduce clique complexes of graphs, explain why applied topologists care about them, and survey open problems about the topology of clique complexes of unit disk graphs, powers of lattice graphs, and powers of hypercube graphs.
Loïc
Dubois
Delaunay flips on flat tori
Henry
Adams
Colorado State University
Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes
Abstract.
The Gromov-Hausdorff distance between two metric spaces is an important tool in geometry, but it is difficult to compute. For example, the Gromov-Hausdorff distance between unit spheres of different dimensions is unknown in nearly all cases. I will introduce recent work by Lim, Memoli, and Okutan that lower bounds the Gromov-Hausdorff distance between spheres using Borsuk-Ulam theorems. We improve these lower bounds by connecting this story to Vietoris-Rips complexes.
Alexander
Stoimenow
GIST College
Exchange moves and non-conjugate braid representatives of links
Abstract.
The braid groups $B_n$ were introduced in the 1930s in the work of Artin. An element in the braid group $B_n$ is called an $n$-braid. Alexander related braids to knots and links in 3-dimensional space, by means of a closure operation. In that realm, it became important to understand the braid representatives of a given link. Markov's theorem relates these representatives by two moves, conjugacy in the braid group, and (de)stabilization, which passes between braid groups. Markov's moves and braid group algebra have become fundamental in Jones' pioneering work and its later continuation towards quantum invariants. Conjugacy is, starting with Garside's, and later many others' work, now relatively well group-theoretically understood. In contrast, the effect of (de)stabilization on conjugacy classes of braid representatives of a given link is in general difficult to control. Only in very special situations can these conjugacy classes be well described.We are concerned with the question when infinitely many conjugacy classes of $n$-braid representatives of a given link occur. Birman and Menasco introduced a move called exchange move, and proved that it necessarily underlies the switch between many conjugacy classes of $n$-braid representatives of a given link.After some very brief general introduction to knots and braids and their applications, we will discuss several results when the exchange move is also sufficient for generating infinitely many conjugacy classes of braid representatives.
Scott
Mullane
HU Berlin
The birational geometry of the moduli space of pointed hyperelliptic curves
Abstract.
The moduli space of pointed hyperelliptic curves is a seemingly simple object with perhaps unexpectedly interesting geometry. I will report on joint work with Ignacio Barros towards a full classification of both the Kodaira dimension and the structure of the effective cone of these moduli spaces.
Arturo
Merino
TU Berlin
Efficient generation of elimination trees and Hamilton paths on graph associahedra
Abstract.
An elimination tree for a connected graph G is a rooted tree on the vertices of G obtained by choosing a root x and recursing on the connected components of G−x to produce the subtrees of x. Elimination trees appear in many guises in computer science and discrete mathematics, and they are closely related to centered colorings and tree-depth. They also encode many interesting combinatorial objects, such as bitstrings, permutations and binary trees. We apply the recent Hartung-Hoang-Mütze-Williams combinatorial generation framework to elimination trees, and prove that all elimination trees for a chordal graph G can be generated by tree rotations using a [simple greedy algorithm](http://www.combos.org/elim). This yields a short proof for the existence of Hamilton paths on graph associahedra of chordal graphs. Graph associahedra are a general class of high-dimensional polytopes introduced by Carr, Devadoss, and Postnikov, whose vertices correspond to elimination trees and whose edges correspond to tree rotations. As special cases of our results, we recover several classical Gray codes for bitstrings, permutations and binary trees, and we obtain a new Gray code for partial permutations. Our algorithm for generating all elimination trees for a chordal graph G can be implemented in time O(m+n) per generated elimination tree, where m and n are the number of edges and vertices of G, respectively. If G is a tree, we improve this to a loopless algorithm running in time O(1) per generated elimination tree. We also prove that our algorithm produces a Hamilton cycle on the graph associahedron of G, rather than just Hamilton path, if the graph G is chordal and 2-connected. Moreover, our algorithm characterizes chordality, i.e., it computes a Hamilton path on the graph associahedron of G if and only if G is chordal. This is joint work with Jean Cardinal (ULB) and Torsten Mütze (Warwick).
Pedro
Araújo
Czech Academy of Sciences
Tight Hamilton cycles in uniformly dense hypergraphs
Abstract.
We study Hamiltonicity in quasirandom hypergraphs. We show that if an n-vertex 3-uniform hypergraph H=(V,E) has the property that for any set of vertices X and for any collection P of pairs of vertices, the number of hyperedges composed by a pair belonging to P and one vertex from X is at least (1/4+o(1))|X||P| - o(|V|³) and H has minimum vertex degree at least Ω(|V|²), then H contains a tight Hamilton cycle. A probabilistic construction shows that the constant 1/4 is optimal in this context. We will also discuss possible extensions to higher uniformities.
Jonathan
Leake
Technische Universität Berlin
Lorentzian polynomials on cones and the Heron-Rota-Welsh conjecture
Abstract.
About 5 years ago, the Heron-Rota-Welsh conjecture (log-concavity of the coefficients of the characteristic polynomial of a matroid) was proven by Adiprasito, Huh, and Katz via the exciting development of a new combinatorial Hodge theory for matroids. In very recent work with Petter Brändén, we have given a new short "polynomial proof" of the Heron-Rota-Welsh conjecture. Our proof uses an extension of the theory of Lorentzian polynomials to convex cones. In this talk, I will briefly discuss the basics of Lorentzian (aka completely log-concave) polynomials, and then I will give an overview of our new proof of the Heron-Rota-Welsh conjecture.
Markus
Brill
Technische Universität Berlin
Approval-Based Apportionment
Abstract.
In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters supporting each party. We study a generalization of this setting, in which voters cast approval ballots over parties, such that each voter can support multiple parties. This approval-based apportionment setting generalizes traditional apportionment and is a natural restriction of approval-based multiwinner elections, where approval ballots range over individual candidates. Using techniques from both apportionment and multiwinner elections, we identify rules that generalize the D'Hondt apportionment method and that satisfy strong axioms which are generalizations of properties commonly studied in the apportionment literature. In fact, the rules we discuss provide representation guarantees that are currently out of reach in the general setting of multiwinner elections: First, we demonstrate that extended justified representation is compatible with committee monotonicity (also known as house monotonicity). Second, we show that core-stable committees are guaranteed to exist and can be found in polynomial time. Joint work with Paul Gölz, Dominik Peters, Ulrike Schmidt-Kraepelin, and Kai Wilker.
Anastasija
Pesic
Membrane models: Variational Analysis and Large Deviations Principle in subcritical dimensions
Jo Andrea
Brüggemann
On the existence of solutions and solution methods for elliptic obstacle-type quasi-variational inequalities with volume constraints
Abstract.
In this talk, an elliptic obstacle-type quasi-variational inequality (QVI) with volume constraints is studied. This type of QVI is motivated by the reformulation of a compliant obstacle problem, where two elastic membranes are subject to external forces while enclosing a constant volume. The existence of solutions to this QVI is established building on fixed-point arguments and partly on the concept of Mosco-convergence. Since Mosco-convergence of the considered feasible sets usually requires complete continuity or compactness properties of the obstacle map, a two-fold approach is explored towards generalising the available existence results for the considered QVI. Based on the analytical findings, the solution of the QVI is approached by solving a sequence of variational inequalities (VIs). Each of these VIs is tackled in function space via a path-following semismooth Newton method. An a posteriori error estimator is derived towards enhancement of the algorithm's numerical performance by using adaptive finite element methods.
Jannik
Peters
TU Berlin
What is fair division?
Abstract.
In this seminar talk, I will give a short introduction to the theory of fair division and present the basic solution concepts and models in fair division and how/if they are achievable. If you are interested in cutting cakes, fairness, and interesting computational problems come join the talk!
Stefan
Felsner
Degrees of freedom and the speed of graph classes
Galen
Dorpalen-Barry
Ruhr-Universität Bochum
Cones of Hyperplane Arrangements through the Varchenko-Gel’fand Ring
Abstract.
The coefficients of the characteristic polynomial of an arrangement in a real vector space have many interpretations. An interesting one is provided by the Varchenko-Gel’fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise multiplication. Varchenko and Gel’fand gave a simple presentation for this ring, along with a filtration whose associated graded ring has its Hilbert function given by the coefficients of the characteristic polynomial. We generalize these results to cones defined by intersections of halfspaces of some of the hyperplanes.
Leticia
Mattos
FU Berlin
Hitting all maximum independent sets
Abstract.
For a graph G on n vertices and independence number a(G) = an, let a'(G)=a'n denote the average value of the independence number of the induced subgraph of G on a uniform random set of vertices. Very recently, Friedgut, Kalai and Kindler (FKK) raised the following conjecture: for any a < 1/2 there is an c(a)>0 so that for every graph G with n vertices and independence number an, we have a'(G) < a - c(a). In this literature seminar, we will show a result of Alon which proves the FKK conjecture in the range where a > 1/4. We will also show his other result which states that the FKK conjecture holds for regular graphs when a > 1/8. If time allows, we will also show Alon's counterexamples for another related conjecture raised by FKK.
Sinai
Robins
Universidade de São Paulo
The null set of a polytope, and the Pompeiu property for polytopes
Abstract.
We study the null set $N(\P)$ of the Fourier transform of a polytope P in $\R^d$, and we find that this null set does not contain (almost all) circles in $\R^d$. As a consequence, the null set does not contain the algebraic varieties $\{ z \in \C^d \mid z_1^2 + \cdots + z_d^2 = \alpha \}$, for each fixed $\alpha \in \C$. In 1929, Pompeiu asked the following question. Suppose we have a convex subset P in R^d, and a function f, defined over R^d, such that the integral of f over P vanishes, and all of the integrals of f, taken over each rigid motion of P, also vanish. Does it necessarily follow that f = 0? If the answer is affirmative, then the convex body P is said to have the Pompeiu property. It is a conjecture that in every dimension, balls are the only convex bodies that do not have the Pompeiu property. Here we get an explicit proof that the Pompeiu property is true for all polytopes, by combining our work with the work of Brown, Schreiber, and Taylor from 1973. Our proof uses the Brion-Barvinok theorem in combinatorial geometry, together with some properties of the Bessel functions. The original proof that polytopes (as well as other bodies) possess the Pompeiu property was given by Brown, Schreiber, and Taylor (1973) for dimension 2. In 1976, Williams observed that the same proof also works for $d>2$ and, using eigenvalues of the Laplacian, gave another proof valid for $d \geq 2$ that polytopes indeed have the Pompeiu property. The null set of the Fourier transform of a polytope has also been used in a different direction, by various researchers, to tackle problems in multi-tiling Euclidean space. Thus, the null set of a polytope is interesting for several applications, including discrete versions of this problem that we will mention, which are generally unsolved. This is joint work with Fabrício Caluza Machado.
Svenja
Griesbach
TU Berlin
Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages
Abstract.
An embedding of a graph in a book, called book embedding, consists of a linear ordering of its vertices along the spine of the book and an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. For planar graphs, a fundamental result is due to Yannakakis, who proposed an algorithm to compute embeddings of planar graphs in books with four pages. Our main contribution is a technique that generalizes this result to a much wider family of nonplanar graphs, which is characterized by a biconnected skeleton of crossing-free edges whose faces have bounded degree. Notably, this family includes all 1-planar and all optimal 2-planar graphs as subgraphs. We prove that this family of graphs has bounded book thickness, and as a corollary, we obtain the first constant upper bound for the book thickness of optimal 2-planar graphs. This is joint work with Michael A. Bekos, Giordano Da Lozzo, Martin Gronemann, Fabrizio Montecchiani, and Chrysanthi Raftopoulou.
Antareep
Mandal
HU Berlin
Uniform sup-norm bounds for Siegel cusp forms
Abstract.
Let $\Gamma\subsetneq \mathrm{Sp}(n,\mathbb{R})$ be a torsion-free arithmetic subgroup of the symplectic group $\mathrm{Sp}(n,\mathbb{R})$ acting on the Siegel upper half-space $\mathbb{H}_n$ of degree $n$. Consider the $d$-dimensional space of Siegel cusp forms $\mathcal{S}_k^n(\Gamma)$ of weight $k$ for $\Gamma$ and let $\{f_j\}_{1\leq j\leq d}$ be a basis of $\mathcal{S}_k^n(\Gamma)$ orthonormal with respect to the Petersson inner product. In this talk we present our work regarding the sup-norm bound of the quantity $S_k^{\Gamma}(Z):=\sum_{j=1}^{d}\det (Y)^{k}\vert{f_j(Z)}\vert^2\,(Z\in\mathbb{H}_n)$. We show using the heat kernel method, for $n=2$ unconditionally and for $n>2$ subject to a conjectural determinant-inequality, that $S_k^{\Gamma}(Z)$ is bounded above by $c_{n,\Gamma} k^{n(n+1)/2}$ when $M:=\Gamma\backslash\mathbb{H}_n$ is compact and by $c_{n,\Gamma} k^{3n(n+1)/4}$ when $M$ is non-compact of finite volume, where $c_{n,\Gamma}$ denotes a positive real constant depending only on the degree $n$ and the group $\Gamma$. Furthermore, we show that this bound is uniform in the sense that if we fix a group $\Gamma_0$ and take $\Gamma$ to be a subgroup of $\Gamma_0$ of finite index, then the constant $c_{n,\Gamma}$ in these bounds depends only on the degree $n$ and the fixed group $\Gamma_0$.
Stefan
Kuhlmann
Technische Universität Berlin
Lattice width of lattice-free polyhedra and height of Hilbert bases
Abstract.
A polyhedron defined by an integral valued constraint matrix and an integral valued right-hand side is lattice-free if it does not contain an element of the integer lattice. In this talk, we present a link between the lattice-freeness of polyhedra, the diameter of finite abelian groups and the height of Hilbert bases. As a result, we will be able to prove novel upper bounds on the lattice width of lattice-free pyramids if a conjecture regarding the height of Hilbert bases holds. Further, we improve existing lattice width bounds of lattice-free simplices. All our bounds are independent of the dimension and solely depend on the maximal minors of the constraint matrix. The second part of the talk is devoted to a study of the above-mentioned conjecture. We completely characterize the Hilbert basis of a pointed polyhedral cone when all the maximal minors of the constraint matrix are bounded by two in absolute value. This can be interpreted as an extension of a well-known result which states that the Hilbert basis elements lie on the extreme rays if the constraint matrix is unimodular, i.e., all maximal minors are bounded by one in absolute value. This is joint work with Martin Henk and Robert Weismantel.
Tim
Netzer
Uni Innsbruck
Free Semialgebraic Geometry and Quantum Information Theory
Abstract.
Quantum information theory studies how quantum information can be represented, stored, processed and sent. On the mathematical side this often involves the study of tensor products and decompositions of positive matrices, as well as positive maps. These are also natural objects in free semialgebraic geometry, where positivity of non-commutative objects are studied from a geometric viewpoint. In this talk I will give an introduction to some important concepts in both areas, and demonstrate how results and methods from either field can be successfully employed in the other. This is joint ongoing work between the group of Gemma De las Cuevas and my own research group in Innsbruck.
Torsten
Mütze
Star transposition Gray codes for multiset permutations
Eleni
Tzanaki
U Crete
Upper bounds on the number of faces of the Minkowski sum of polytopes
Abstract.
Given two convex polytopes P, Q, their Minkowski sum, which is again a polytope, is defined as P + Q = {p + q : p ∈ P, q ∈ Q}. In this talk I will present tight expressions for the maximum number of k-dimensional faces, 0 ≤ k ≤ d − 1, of the Minkowski sum P_1 + P_2 of two convex d-dimensional polytopes in R_d. These upper bounds are proved making use of basic notions such as f - and h-vector calculus, stellar-subdivisions and shellings, and generalize the steps used by McMullen to prove the Upper Bound Theorem for polytopes. The key idea behind the approach is to express the Minkowski sum P_1 + P_2 as a section of the Cayley polytope C of the summands; bounding the k-faces of P_1 + P_2 reduces to bounding the subset of the (k + 1)-faces of C that contain vertices from each of the two summands. I will briefly explain how these steps should be adjusted when one considers the Minkowski sum of three or more polytopes. This is joint work with M. Karavelas.
Austin
Alderete
UT Austin
A polymake extension for computing the homology groups of minor-closed classes of matroids
Abstract.
We present a polymake extension for performing computations in the intersection ring of matroids. Addition in the ring is carried out on the indicator vectors of maximal chains of flats while the product is the usual matroid intersection when the intersection is loopfree and zero otherwise. The primary feature of this extension is the ability to compute the homology groups of the intersection ring induced by deletion and contraction as well as those of any subring generated by a minor-closed class of matroids.
Christian
Schnell
Stony Brook, visiting Bonn
A new look at degenerating variations of Hodge structure
Abstract.
In the early 1970s, Schmid published a detailed analysis of variations of Hodge structure on the punctured disk. All subsequent developments in Hodge theory (including Saito's theory of Hodge modules), and most applications of Hodge theory to questions about families of algebraic varieties, ultimately depend on Schmid's results. For that reason, I think one should try to understand this topic as well as possible. In the talk, I will present a new take on Schmid's work that greatly simplifies the existing proofs; works in the more general setting of complex variations of Hodge structure; and, most importantly, makes it much clearer what is going on.
David
Weckbecker
TU Darmstadt
Fractionally Subadditive Maximization under an Incremental Knapsack Constraint
Abstract.
We consider the problem of maximizing a fractionally subadditive function under a knapsack constraint that grows over time. An incremental solution to this problem is given by an order in which to include the elements of the ground set, and the competitive ratio of an incremental solution is defined by the worst ratio over all capacities relative to an optimum solution of the corresponding capacity. We present an algorithm that finds an incremental solution of competitive ratio at most max{3.293*sqrt{M},2M}, under the assumption that the values of singleton sets are in the range [1,M], and we give a lower bound of max{2.449,M} on the attainable competitive ratio. In addition, we establish that our framework captures potential-based flows between two vertices, and we give a tight bound of 2 for the incremental maximization of classical flows with unit capacities.
Kevin
Schewior
Uni Köln
Stochastic Probing with Increasing Precision
Abstract.
We consider a selection problem with stochastic probing. There is a set of items whose values are drawn from independent distributions. The distributions are known in advance. Each item can be tested repeatedly. Each test reduces the uncertainty about the realization of its value. We study a testing model, where the first test reveals if the realized value is smaller or larger than the median of the underlying distribution. Subsequent tests allow to further narrow down the interval in which the realization is located. There is a limited number of possible tests, and our goal is to design near-optimal testing strategies that allow to maximize the expected value of the chosen item. We study both identical and non-identical distributions and develop polynomial-time algorithms with constant approximation factors in both scenarios. This is joint work with Martin Hoefer and Daniel Schmand.
Michael
Anastos
FU Berlin
Determining the Hamiltonicity of sparse graphs in polynomial expected time
Abstract.
The Hamilton cycle problem asks to determines whether a given graph G has a Hamilton cycle. It belongs to the list of Karp’s 21 NP-complete problems and if P ≠ NP then there does not exist an algorithm that determines the Hamiltonicity of G in poly(n) time for every graph G on n vertices. On the other hand Gurevich and Shelah gave an algorithm that determines the Hamiltonicity of an n-vertex graph in poly(n) time on average. Equivalently the expected running time of their algorithm over the input distribution G ∼ G(n, p) is polynomial in n when p = 1/2. The last statement raises the question for which values of p there exist an algorithm that solves the Hamilton cycle problem in polynomial expected running time over the input distribution G ∼ G(n, p). Gurevich and Shelah, Thomason and Alon and Krivelevich gave such an algorithm for p ∈ [0, 1] being constant, p ≥ 12n−1/3 and p ≥ 70n-1/2 respectively. In this talk we present an algorithm which solves the Hamilton cycle problem in polynomial expected running time over the input distribution G ∼ G(n, p) for p ≥ 500/n.
Ruijie
Yang
HU Berlin
Vanishing cycles for divisors
Abstract.
Given a holomorphic function $f$ on $\mathbb{C}^n$ with an isolated critical point at the origin, the vanishing cycles associated to $f$ are homology cycles of the level set $\{f=t\}$ for $|t|\ll 1$. If the singularity is non-isolated, the vanishing cycles glue to complexes of vector spaces. In algebraic geometry, divisors arise more frequently than functions. A natural question is, how do the vanishing cycles associated to local defining functions glue together? In this talk I will explain a global construction using $\mathscr{D}$-modules and discuss its relation to singularities of divisors. This is based on joint work in progress with Christian Schnell.
Davide
Lofano
Technische Universität Berlin
Hadamard Matrix Torsion
Abstract.
After a brief overview about torsion in homology and examples of complexes with small torsion and few vertices we will focus our attention on a particular series of examples. In particular, we will construct a series HMT(n) of 2-dimensional simplicial complexes with huge torsion, |H_1(HMT(n))|=|det(H(n))|=n^(n/2), where the construction is based on the Hadamard matrices H(n) and they have linearly many vertices in n. Our explicit series improves a previous construction by Speyer, narrowing the gap to the highest possible asymptotic torsion growth achieved by Newman via a randomized construction.
Manfred
Scheucher
Technische Universität Berlin
Erdős-Szekeres-type Problems on Planar Point Sets
Abstract.
In 1935, Erdős and Szekeres proved that, for every positive integer k, every sufficiently large point set contains a "k-gon", that is, a subset of k points which is the vertex set of a convex polygon. Their theorem is a classical result in both, combinatorial geometry and Ramsey theory, and motivated a lot of further research including numerous modifications and extensions of the theorem. In this talk we discuss some results and methods that played an essential role in the study of k-gons and the variant of "k-holes".
Raphael
Steiner
Two notes on relatives of Hadwiger's conjecture
Jesus
De Loera
UC Davis
Stochastic Tverberg-type theorems and their relevance in Machine Learning and Statistical Inference
Abstract.
Discrete geometry can play a role in foundations of data science. Here I present concrete examples. In statistical inference we wish to find the properties or parameters of a distribution or model through sufficiently many samples. A famous example is logistic regression, a popular non-linear model in multivariate statistics and supervised learning. Users often rely on optimizing of maximum likelihood estimation, but how much training data do we need, as a function of the dimension of the covariates of the data, before we expect an MLE to exist with high probability? Similarly, for unsupervised learning and non-parametric statistics, one wishes to uncover the shape and patterns from samples of a measure or measures. We use only the intrinsic geometry and topology of the sample. A famous example of this type of method is the $k$-means clustering algorithm. A fascinating challenge is to explain the variability of behavior of $k$-means algorithms with distinct random initializations and the shapes of the clusters. In this talk we present new stochastic combinatorial theorems, direct new variations of Tverberg’s theorem, that give bounds on the probability of existence of maximum likelihood estimators in multinomial logistic regression and also quantify to the variability of clustering initializations. Along the way we will see fascinating connections to the coupon collector problem, topological data analysis, and to the computation of Tukey centerpoints of data clouds (a high-dimensional generalization of median). This is joint work with (in various papers) with T. Hogan, R. D. Oliveros, E. Jaramillo-Rodriguez, A. Torres-Hernandez, and Dominic Yang.
Rahul
Pandharipande
ETH Zürich
Gromov-Witten theory of hypersurfaces
Abstract.
I will explain how to think about the GW theory of hypersurfaces in projective space (and more generally complete intersections). The interesting new aspect is the control of the primitive cohomology. Full use of monodromy, degeneration, and nodal relative geometry, leads to an inductive solution. A consequence is that all GW cycles for hypersurfaces (and complete intersections) lie in the tautological ring of the moduli space of curves. Joint work with Argüz, Bousseau, and Zvonkine.
Salvatore
Floccari
U Hannover
Isogenous hyper-Kähler varieties
Abstract.
The Torelli theorem for hyper-Kähler varieties explains to which extent such a variety can be recovered from its integral second cohomology, together with its pairing and Hodge structure. I will address a variant of this problem: How much of a hyper-Kähler variety is determined by its rational second cohomology? Work of Huybrechts and Fu-Vial provides the answer for K3 surfaces. In higher dimension we expect this rational cohomology group to control the full motive of the variety; the main result of the talk confirms this in the realm of André motives.
Letícia
Mattos
Freie Universität Berlin
Singularity of random symmetric matrices
Abstract.
Let M_n be a uniformly-chosen random symmetric n x n matrix with entries in {-1,1}. What is the probability for det(M_n)=0? A wellknown conjecture states that the probability of this event is asymptotically equal to the probability that two of the rows or columns of M_n are equal (up to a factor of +-1) and hence is equal to \Theta(n^2 2^{-n}). We developed an inverse Littlewood-Offord theorem in Z^n_p that applies under very mild conditions and made progress towards this conjecture, showing that the probability is bounded by exp(-c\sqrt{n}). Joint work with Marcelo Campos, Robert Morris and Natasha Morrison.
Florian
Frick
Carnegie Mellon University and FU Berlin
Topological methods in graph theory
Abstract.
When we study the structure of a graph, we encounter parameters that are local (such as the clique number) and parameters that are global (such as the chromatic number). Topology provides tools to measure global phenomena. I will explain the hidden topology of the global structure of graphs for problems such as chromatic numbers, covering and matching problems, and partitions into independent sets with various constraints.
October 2021
Manfred
Scheucher
Gons and Holes in Projective Point Sets (Part 1)
Martin
Skrodzki
TU Delft
Single-sized spheres on surfaces (S4)
Abstract.
Surface representations play a major role in a variety of applications throughout a diverse collection of fields, such as biology, chemistry, physics, or architecture. From a simulation point of view, it is important to simulate the surface as good as possible, including the usage of a wide range of different approximating elements. However, when it comes to manufacturing, it is desirable to have as few different building blocks as possible, as these can then be produced cost-efficiently. The talk discusses a procedure to be used in the simulation of natural phenomena as well as within the designers' creative toolbox. It represents a surface via a collection of equally sized spheres. In the first part of the talk, we investigate mathematical challenges of the problem. These include estimating the maximal intersection area of a sphere with a surface of bounded curvature as well as counting the minimal and maximal number of spheres to be placed on a surface. Following these theoretical results, in the second part of the talk, we compare a depth-first, a breadth-first, and a heuristic algorithm for the generation of surface coverings by single-sized spheres. We prove the applicability of our algorithm by a multitude of experiments and compare our procedure to ellipsoidal and multi-sized sphere methods.
Davide
Lofano
TU Berlin
Hadamard Matrix Torsion
Abstract.
After a brief overview about torsion in homology and examples of complexes with small torsion and few vertices we will focus our attention on a particular series of examples. In particular, we will construct a series HMT(n) of 2-dimensional simplicial complexes with huge torsion, |H_1(HMT(n))|=|det(H(n))|=n^(n/2), where the construction is based on the Hadamard matrices H(n) and they have linearly many vertices in n. Our explicit series improves a previous construction by Speyer, narrowing the gap to the highest possible asymptotic torsion growth achieved by Newman via a randomized construction.
Mohammed Majthoub
Almoghrabi
TU Berlin
Evaluating the Potential of Reinforcement Learning for Stochastic Machine Scheduling Problems
Simona
Boyadzhiyska
Freie Universität Berlin
Ramsey simplicity of random graphs
Abstract.
We say that a graph G is q-Ramsey for another graph H if any q-coloring of the edges of G yields a monochromatic copy of H. Much of the research related to Ramsey graphs is concerned with determining the smallest possible number of vertices in a q-Ramsey graph for a given H, known as the q-color Ramsey number of H. In the 1970s, Burr, Erdős, and Lovász initiated the study of another graph parameter in the context of Ramsey graphs: the minimum degree. A straightforward argument shows that, if G is a minimal q-Ramsey graph for H, then we must have δ(G) >= q(δ(H) - 1) + 1, and we say that H is q-Ramsey simple if this bound can be attained. In this talk, we will ask how typical Ramsey simplicity is; more precisely, we will discuss for which pairs p and q the random graph G(n,p) is almost surely q-Ramsey simple. This is joint work with Dennis Clemens, Shagnik Das, and Pranshu Gupta.
Penny
Haxell
University of Waterloo
Tashkinov trees
Abstract.
The technique of Tashkinov trees is an important tool that has been used to obtain many significant results on edge colouring of multigraphs. We describe the method and its main motivation, the famous Goldberg-Seymour conjecture from the 1970's. We also outline several results from the last decade that have used the Tashkinov trees technique. This talk is intended as an introduction, overview and survey to accompany the upcoming short course "Edge colouring of multigraphs and the Tashkinov trees method" (Oct 26-29).
Alessandro
Verra
Università Roma Tre
A series of hyperkähler varieties and their Pfaffian counterparts
Simon
Breneis
WIAS Berlin
What is rough volatility?
Abstract.
Starting with an introduction of Brownian motion, we discuss the general goals of mathematical finance and explain the intuition behind the Black-Scholes model. After discussing option pricing in this standard framework, we observe some of the shortcomings of the Black-Scholes model. Finally, to overcome these deficiencies, we introduce stochastic volatility and rough volatility models.
Laura
Merker
A Sublinear Bound on the Page Number of Upward Planar Graphs
Leticia
Mattos
FU Berlin
Clique packings in random graphs
Abstract.
Let u(n,p,k) be the k-clique packing number of the random graph G(n,p), that is, the maximum number of edge-disjoint k-cliques in G(n,p). In 1992, Alon and Spencer conjectured that for p = 1/2 we have u(n,p,k) = Ω(n²/k²) when k = k(n,p) - 4, where k(n,p) is minimum t such that the expected number of t-cliques in G(n,p) is less than 1. This conjecture was disproved a few years ago by Acan and Kahn, who showed that in fact u(n,p,k) = O(n²/k³) with high probability, for any fixed p ∈ (0,1) and k = k(n,p) - O(1). In this talk, we show a lower bound which matches Acan and Kahn's upper bound on u(n,p,k) for all p ∈ (0,1) and k = k(n,p) - O(1). To prove this result, we follow a random greedy process and use the differential equation method. This is a joint work with Simon Griffiths and Rob Morris.
Sylvain
Spitz
TU Berlin
Generalized Permutahedra and Optimal Auctions
Abstract.
We study a family of convex polytopes, called SIM-bodies, which were introduced by Giannakopoulos and Koutsoupias (2018) to analyze so-called Straight-Jacket Auctions. First, we show that the SIM-bodies belong to the class of generalized permutahedra. Second, we prove an optimality result for the Straight-Jacket Auctions among certain deterministic auctions. Third, we employ computer algebra methods and mathematical software to explicitly determine optimal prices and revenues.
Javier
Cembrano
TU Berlin
Multidimensional Apportionment through Discrepancy Theory
Abstract.
Deciding how to allocate the seats of a house of representatives is one of the most fundamental problems in the political organization of societies, and has been widely studied over already two centuries. The idea of proportionality is at the core of most approaches to tackle this problem, and this notion is captured by the divisor methods, such as the Jefferson/D'Hondt method. In a seminal work, Balinski and Demange extended the single-dimensional idea of divisor methods to the setting in which the seat allocation is simultaneously determined by two dimensions, and proposed the so-called biproportional apportionment method. The method, currently used in several electoral systems, is however limited to two dimensions and the question of extending it is considered to be an important problem both theoretically and in practice. In this work we initiate the study of multidimensional proportional apportionment. We first formalize a notion of multidimensional proportionality that naturally extends that of Balinski and Demange. By means of analyzing an appropriate integer linear program we are able to prove that, in contrast to the two-dimensional case, the existence of multidimensional proportional apportionments is not guaranteed and deciding its existence is NP-complete. Interestingly, our main result asserts that it is possible to find approximate multidimensional proportional apportionments that deviate from the marginals by a small amount. The proof arises through the lens of discrepancy theory, mainly inspired by the celebrated Beck-Fiala Theorem. We finally evaluate our approach by using the data from the recent 2021 Chilean Constitutional Convention election. This is joint work together with José Correa (UChile) and Victor Verdugo (UOH).
Petter
Brändén
KTH
Lorentzian polynomials on cones and the Heron-Rota-Welsh conjecture
Abstract.
We extend the theory of Lorentzian polynomials to convex cones. This is used to give a short proof of the log-concavity of the coefficients of the reduced characteristic polynomial of a matroid, and reprove the Hodge-Riemann relations of degree one for the Chow ring of a matroid. This is joint work with Jonathan Leake.
Moritz
Buchem
Maastricht University
Additive approximation schemes for load balancing problems
Abstract.
We formalize the concept of additive approximation schemes and apply it to load balancing problems on identical machines. Additive approximation schemes compute a solution with an absolute error in the objective of at most εh for some suitable parameter h and any given ε > 0. We consider the problem of assigning jobs to identical machines with respect to common load balancing objectives like makespan minimization, the Santa Claus problem (on identical machines), and the envy-minimizing Santa Claus problem. For these settings we present additive approximation schemes for h = p<sub>max</sub>, the maximum processing time of the jobs. Our technical contribution is two-fold. First, we introduce a new relaxation based on integrally assigning slots to machines and fractionally assigning jobs to the slots. We refer to this relaxation as the slot-MILP. While it has a linear number of integral variables, we identify structural properties of (near-)optimal solutions, which allow us to compute those in polynomial time. The second technical contribution is a local-search algorithm which rounds any given solution to the slot-MILP, introducing an additive error on the machine loads of at most εpmax. This is joint work together with Lars Rohwedder (EPFL), Tjark Vredeveld (UM) and Andreas Wiese (UChile).
Helena
Bergold
Higher Dimensional Signotopes
September 2021
Lilli
Leifheit
Rechteckzerlegungen und Baxterpermutationen
Boro
Sofranac
IOL
An Algorithm-Independent Measure of Progress for Linear Constraint Propagation
Abstract.
Propagation of linear constraints has become a crucial sub-routine in modern Mixed-Integer Programming (MIP) solvers. In practice, iterative algorithms with tolerance-based stopping criteria are used to avoid problems with slow or infinite convergence. However, these heuristic stopping criteria can pose difficulties for fairly comparing the efficiency of different implementations of iterative propagation algorithms in a real-world setting. Most significantly, the presence of unbounded variable domains in the problem formulation makes it difficult to quantify the relative size of reductions performed on them. In this work, we develop a method to measure—independently of the algorithmic design—the progress that a given iterative propagation procedure has made at a given point in time during its execution. Our measure makes it possible to study and better compare the behavior of bounds propagation algorithms for linear constraints. We apply the new measure to answer two questions of practical relevance: 1. We investigate to what extent heuristic stopping criteria can lead to premature termination on real-world MIP instances. 2. We compare a GPU-parallel propagation algorithm against a sequential state-of-the-art implementation and show that the parallel version is even more competitive in a real-world setting than originally reported.
Sophie
Huiberts
Combinatorial Diameter of Random Polyhedra
Abstract.
The long-standing polynomial Hirsch conjecture asks if the combinatorial diameter of any polyhedron can be bounded by a polynomial of the dimension and number of facets. Inspired by this question, we study the combinatorial diameter of two classes of random polyhedra. We prove nearly-matching upper and lower bounds, assuming that the number of facets is very large compared to the dimension.
August 2021
Georg
Loho
University of Twente
Generalized permutahedra and complete classes of valuated matroids
Abstract.
Generalized permutahedra form an important class of polytopes occurring explicitly or implicitly in several branches of mathematics and beyond. I start with an overview of concepts related with generalized permutahedra from optimization and Discrete Convex Analysis. Valuated generalized matroids give rise to a particularly interesting subclass. I show some fundamental constructions for these objects. This leads to an answer to two open questions from auction theory and discrete convex analysis.
Khai Van
Tran
TU Berlin
A Faster Algorithm for Quickest Transshipments via an Extended Discrete Newton Method
Abstract.
The Quickest Transshipment Problem is to route flow as quickly as possible from sources with supplies to sinks with demands in a network with capacities and transit times on the arcs. It is of fundamental importance for numerous applications in areas such as logistics, production, traffic, evacuation, and finance. More than 25 years ago, Hoppe and Tardos presented the first (strongly) polynomial-time algorithm for this problem. Their approach, as well as subsequently derived algorithms with strongly polynomial running time, are hardly practical as they rely on parametric submodular function minimization via Megiddo‘s method of parametric search. In this talk we present a considerably faster algorithm for the Quickest Transshipment Problem that instead employs a subtle extension of the Discrete Newton Method. This improves the previously best known running time of Õ(m<sup>4</sup>k<sup>14</sup>}) to Õ(m<sup>2</sup>k<sup>5</sup>+m<sup>3</sup>k<sup>3</sup>+m<sup>3</sup>n), where n is the number of nodes, m the number of arcs, and k the number of sources and sinks.
Chiara
Meroni
MPI MiS
Fiber convex bodies
Abstract.
Given a convex polytope P and a projection map it is possible to construct the associated fiber polytope, that intuitively is the average of the fibers of P under this projection and has interesting combinatorial properties. In this joint work with Léo Mathis we study this construction for all convex bodies. The aim is to relate aspects of the original convex body with those of its fiber body. I will discuss theoretical results as well as some concrete examples.
July 2021
Tobias
Boege
Universität Magdeburg
The Gaussian conditional independence inference problem
Abstract.
Conditional independence is a ternary relation on subsets of a finite vector of random variables \xi. The CI statement \xi_i \CI \xi_j \mid \xi_K asserts that ``whenever the outcome of all the variables \xi_k, k \in K, is known, learning the outcome of \xi_i provides no further information on \xi_j''. These relations are highly structured, in particular under assumptions about the joint distribution. The goal is to describe this by CI inference rules: given that certain CI statements hold, which other (disjunctions of) CI statements are implied under the distribution assumption? This talk is about regular Gaussian distributions where conditional independence has an algebraic characterization in terms of subdeterminants of the covariance matrix and inference becomes a geometric problem concerning the vanishing of certain polynomials on varieties inside the cone of positive-definite matrices. I first show that the inference problem for Gaussians is just as difficult as deciding whether a polynomial system with integer coefficients has a solution over the real numbers. Then, I present some approximations to the inference problem which exploit the special structure of the polynomials which are relevant for conditional independence. In these formulations, SAT solvers and linear programming are able to prove some (but not all) valid inference rules and they terminate much faster than a general method.
Frieder
Smolny
TU Berlin
Computational experiments and multiscale optimization
Abstract.
In this talk we will see how to solve a network flow problem by a coarsening technique, and speculate if this technique can be carried over to other optimization problems. Then we will witness the struggles of experimentally evaluating an algorithm with many parameters.
Paramjit
Singh
HU Berlin
What is Floer homology?
Abstract.
We shall introduce Morse theory and the notion of Floer homology for a finite dimensional orientable closed manifold with an example. Then we will outline how this is generalized for infinite dimensional manifolds (like loop spaces, curves in a manifold, etc) and in the symplectic/contact settings.
Felix
Schröder
Density of Fan-Planar Graphs
Helmut
Hofer
IAS Princeton
The Floer Jungle: 35 Years of Floer Theory - in cooperation with the IAS/Princeton/Montreal/Paris/Tel-Aviv symplectic geometry zoominar
Alexey
Garber
University of Texas Rio Grande Valley
The Voronoi conjecture: convex polytopes that tile space with translations
Abstract.
In this talk I am going to discuss a well-known connection between lattices in $\mathbb{R}^d$ and convex polytopes that tile $\mathbb{R}^d$ with translations only. My main topic will be the Voronoi conjecture, a century old conjecture which is, while stated in very simple terms, is still open in general. The conjecture states that every convex polytope that tiles $\mathbb{R}^d$ with translations can be obtained as an affine image of the Voronoi domain for some lattice. I plan to survey several known results on the Voronoi conjecture and give an insight on a recent proof of the Voronoi conjecture in the five-dimensional case. The talk is based on a joint work with Alexander Magazinov (Skoltech, Russia).
Tássio
Naia
Universidade de São Paulo
Tree-like structures in dense digraphs
Abstract.
What large structures appear in every dense digraph? A celebrated result of Komlós, Sárközy and Szemerédi, answering a conjecture of Bollobás, states that for all ε > 0, every sufficiently large graph of order n and minimum degree n(1/2 + ε) contains every spanning tree of bounded maximum degree. This was later strengthened by the same authors to allow trees of degree c_ε n/log n (where c_ε is a constant depending on ε). The result has also been extended in a large number of directions. In this talk, we discuss a generalization of this result for digraphs: for all ε > 0, every sufficiently large digraph G of order n and minimum semidegree n(1/2 + ε) contains every spanning tree of bounded degree (the minimum semidegree of G is the minimum of in- and out-degrees over all vertices of G). In fact, our method establish the presence of a much more general family of arbitrarily oriented spanning subdigraphs, such as large subdivisions of a small graph, or collections of o(n^1/4) vertex-disjoint cycles. Ths is joint work with Richard Mycroft.
Claudia
Yun
Brown University
The S_n-equivariant rational homology of the tropical moduli spaces \Delta_{2,n}
Abstract.
Fix non-negative integers g and n such that 2g-2+n>0, the tropical moduli space \Delta_{g,n} is a topological space that parametrizes isomorphism classes of n-marked stable tropical curves of genus g with total volume 1. These spaces are important because their reduced rational homology is identified equivariantly with the top weight cohomology of the algebraic moduli spaces M_{g,n}, with respect to the S_n-action of permuting marked points. In this talk, we focus on the genus 2 case and compute the characters of \tilde{H}_i(\Delta_{2,n};Q) as S_n-representations for n up to 8. To carry out the computations, we use a cellular chain complex for symmetric \Delta-complexes, a technique developed by Chan, Galatius, and Payne. We will also briefly mention work in progress that extends the computations to n=10, in collaboration with Christin Bibby, Melody Chan, and Nir Gadish. All computations are done in SageMath.
Modelling and Optimization of Semiconductor Lasers for Quantum Metrology Applications
Abstract
Raphael
Steiner
Technische Universität Berlin
The dichromatic number-a survey
Abstract.
The dichromatic number of a digraph is the smallest number of acyclic subsets of vertices (not spanning a directed cycle) that can be used to cover its vertex-set. This parameter is a natural extension of the chromatic number to directed graphs and has been introduced in 1980 by Erdös and Neumann-Lara. Since 2000, many groups of authors have studied this parameter intensively and in various settings. In this talk, I will give a small survey of this topic, which has concerned me quite a bit in the last 3-epsilon years. I will also mention own results obtained during my PhD at appropriate places.
Frédéric
Havet
INRIA Sophia-Antipolis
Unavoidability and universality of digraphs
Abstract.
A digraph F is n-unadoidable} (resp. n-universal) if it is contained in every tournament of order n (resp. n-chromatic digraph). Well-known theorems imply that there is an nF such that F is nF-unavoidable (resp. nF-universal) if and only if F is acyclic, (resp. an oriented forest). However, determining the smallest nF for which it occurs is a challenging question. In this talk, we survey the results on unavoidability and universality and detail some recent recults obtained with various co-authors.
Clemens
Sirotenko
WIAS Berlin
What is variational image processing?
Abstract.
We will briefly introduce the concept of inverse problems in the context of image reconstruction tasks and take a look on practical examples. Subsequently, we discuss the variational approach to solve these problems and point out existing advantages, disadvantages and potential challenges. After a small glimpse at some theoretical results, we turn to more modern methods that aim to integrate information from existing data into the solution process.
Carola-Bibiane
Schönlieb
U Cambridge
Mathematical imaging - together with an outlook on the upcoming TES "Mathematics of Imaging on Real-World Challenges"
Shagnik
Das
Isomorphic bisections of cubic graphs
Laith
Rastanawi
FU Berlin
Towards a geometric understanding of 4-dimensional point groups
Abstract.
The 4-dimensional point groups can be roughly classified as follows. There is a finite list of polyhedral groups, which are related to the regular polytopes. There are torodial groups, which are characterized by having a torus that remains invariant. They come in several infinite families. There are axial groups, which leave a line through the origin invariant. Finally, there are eleven classes of tubular groups. We propose a new, geometric classification for the torodial groups, and we make effort to understand the action of the tubular groups in terms of orbit polytopes. Joint work with Günter Rote.
Alp
Müyesser
FU Berlin
Transversals in multiplication tables of abelian groups
Abstract.
Given an n x n matrix M filled in with arbitrary symbols, a transversal is a selection of n entries of M which do not share a row, a column, or a symbol. When M is the multiplication table of a finite group G, Hall and Paige conjectured in 1955 that M has a transversal if and only if the Sylow 2-subgroups of G are trivial or non-cyclic. In 2009, Wilcox, Evans, and Bray gave a proof of this conjecture using the classification of finite simple groups. Giving a characterisation of sub-matrices of M with transversals is currently open, and even the abelian case is not fully understood. In this talk, we will characterise sub-matrices of multiplication tables of abelian groups which admit transversals. In particular, we will address a conjecture of Snevily from 1999. This is joint work with Alexey Pokrovskiy.
Ilia
Itenberg
Sorbonne Université
Planes in cubic fourfolds
Abstract.
We discuss possible numbers of 2-planes in a smooth cubic hypersurface in the 5-dimensional projective space. We show that, in the complex case, the maximal number of planes is 405, the maximum being realized by the Fermat cubic. In the real case, the maximal number of planes is 357. The proofs deal with the period spaces of cubic hypersurfaces in the 5-dimensional complex projective space and are based on the global Torelli theorem and the surjectivity of the period map for these hypersurfaces, as well as on Nikulin's theory of discriminant forms. Joint work with Alex Degtyarev and John Christian Ottem.
Daniel
Schmidt
TU Berlin
Restricted Adaptivity in Stochastic Scheduling
Abstract.
We consider the stochastic scheduling problem of minimizing the expected makespan on m parallel identical machines. While the (adaptive) list scheduling policy achieves an approximation ratio of 2, any (non-adaptive) fixed assignment policy has performance guarantee 𝛺(log m/loglog m). Although the performance of the latter class of policies are worse, there are applications, e.g. in surgery scheduling, in which rather non-adaptive policies are desired. We introduce two classes of policies, namely 𝛿-delay and 𝜏-shift policies, whose degree of adaptivity can be controlled by a parameter. Using the fixed assignment policy induced by LEPT as a basis, we present a policy - belonging to both classes - for which we show that it is an O(loglog m)-approximation for reasonably bounded parameters. In other words, an improvement on the performance of any fixed assignment policy can be achieved when allowing a small degree of adaptivity. Moreover, we provide a matching lower bound for any 𝛿-delay and 𝜏-shift policy when both parameters, respectively, are in the order of the expected makespan of an optimal non-anticipatory policy. This is joint work with Guillaume Sagnol.
Ansgar
Freyer
Technische Universität Berlin
Shaking a convex body in order to count its lattice points
Abstract.
We prove inequalities on the number of lattice points inside a convex body K in terms of its volume and its successive minima. The successive minima of a convex body have been introduced by Minkowski and since then, they play a major role in the geometry of numbers. A key step in the proof is a technique from convex geometry known as Blascke's shaking procedure by which the problem can be reduced to anti-blocking bodies, i.e., convex bodies that are "located in the corner of the positive orthant". As a corollary of our result, we will obtain an upper bound on the number of lattice points in K in terms of the successive minima, which is equivalent to Minkowski's Second Theorem, giving a partial answer to a conjecture by Betke et al. from 1993. This is a joint work with Eduardo Lucas Marín.
Patrick
Farrell
University of Oxford, UK
Computing disconnected bifurcation diagrams of partial differential equations
Abstract.
Computing the distinct solutions $u$ of an equation $f(u, \lambda) = 0$ as a parameter $\lambda \in \mathbb{R}$ is varied is a central task in applied mathematics and engineering. The solutions are captured in a bifurcation diagram, plotting (some functional of) $u$ as a function of $\lambda$. In this talk I will present a new algorithm, deflated continuation, for this task. Deflated continuation has three advantages. First, it is capable of computing disconnected bifurcation diagrams; previous algorithms only aimed to compute that part of the bifurcation diagram continuously connected to the initial data. Second, its implementation is very simple: it only requires a minor modification to an existing Newton-based solver. Third, it can scale to very large discretisations if a good preconditioner is available; no auxiliary problems must be solved. We will present applications to hyperelastic structures, liquid crystals, and Bose-Einstein condensates, among others.
Papa
Sissokho
Illinois State University
Geometry of the minimal solutions of a linear Diophantine Equation
Abstract.
Let a1,...,an and b1,...,bm be fixed positive integers, and let S denote the set of all nonnegative integer solutions of the equation x1a1+...+xnan=y1b1+...+ymbm. A solution (x1,...,xn,y1,...,ym) in S is called minimal if it cannot be expressed as the sum of two nonzero solutions in S. For each pair (i,j), with 1 ≤ i ≤ n and 1 ≤ j ≤ m, the solution whose only nonzero coordinates are xi = bj and yj = ai is called a generator. We show that every minimal solution is a convex combination of the generators and the zero-solution. This proves a conjecture of Henk-Weismantel and, independently, Hosten-Sturmfels.
Ivan
Spirandelli
U Potsdam
What is persistent homology?
Abstract.
In the talk I give a practical introduction to alpha complexes and persistent homology. Persistent homology is a formalism that enables us to compute topological features of a space at different spatial resolutions. It is often used on simplicial complexes that were constructed on point clouds. Alpha complexes are one popular way of constructing these simplicial complexes.
Raphael
Steiner
Polynomial bound for divisible subdivisions
Kathryn
Hess
EPFL Lausanne
Topological insights in neuroscience
Thomas
Kahle
OvGU Magdeburg
Symmetric chains in algebra and discrete geometry
Abstract.
We describe a framework to study chains of symmetric objects from algebra (e.g. ideals) and discrete geometry (e.g. cones or monoids). Each such chain has a natural limit object in countable-dimensional space and we are interested in finite generation in the limit (which is often equivalent to stablization of the chain). While in the algebraic situation one has Noetherianity up to symmetry, that is any suitably symmetric chain stabilizes this fails for cones and monoids. Our framework yields characterizations of stabilization in terms of properties of the chain. Joint work with Dinh Van Le and Tim Römer.
June 2021
Holger
Eble
TU Berlin
Enumerating Chambers of Hyperplane Arrangements with Symmetry
Abstract.
We introduce a new algorithm for enumerating chambers of hyperplane arrangements which exploits their underlying symmetry groups. Our algorithm counts the chambers of an arrangement as a byproduct of computing its characteristic polynomial. We showcase our julia implementation, based on OSCAR, on examples coming from hyperplane arrangements with applications to physics and computer science. This is joint work with Taylor Brysiewicz and Lukas Kuehne.
Adaptive Algorithms through Machine Learning: Exploiting Interactions in Integer Programming
Abstract
Igor
Balla
Tel Aviv University
Equiangular lines and graph spectra
Abstract.
A set of lines passing through the origin in Rr is called equiangular if every pair of lines makes the same angle. In 1973, Lemmens and Seidel asked to determine the maximum number Nα(r) of equiangular lines in Rr with common angle arccos(α). Recently, B., Dräxler, Keevash, and Sudakov showed that Nα(r) ≤ 2r-2 for any fixed α ∈ (0,1) when r is exponentially large in 1/α, with equality if and only if α = 1/3. Building on their work, Jiang, Tidor, Yao, Zhang, and Zhao were able to determine Nα(r) completely for all α such that there exists a graph whose spectral radius is (1/α-1)/2 and when r is at least doubly exponentially large in 1/α. In both cases, the approach crucially relies on using Ramsey's theorem in order to bound the maximum degree of a corresponding graph. In this talk we will discuss how we can use orthogonal projections of matrices with respect to the Frobenius inner product in order to overcome this limitation and thereby prove new bounds on Nα(r) which effectively bridge the gap between 2r(1+o(1)) when α = Θ(1) and (1-o(1))r2/2 when α = Θ(1/r1/2), as well as significantly improving on the only previously known universal bound Nα(r) ≤ 2/3 · r/α2 · (1+o(1)), due to Glazyrin and Yu. Using the connection to real equiangular lines, our methods can also be used to obtain bounds on eigenvalues of the adjacency matrix of a regular graph. In particular, we show that for any k-regular graph on n vertices whose adjacency matrix has second and last eigenvalue λ2 and λn such that the spectral gap satisfies k - λ2 = o(n), we have λ2 ≥ (1 - o(1)) max(k1/3, (-λn)1/2). In fact, our bounds work up to O(1) provided that the spectral gap is slightly smaller than n/2, and since we do not need an assumption on the diameter, this can be seen as the first generalization of the Alon-Boppana theorem to dense graphs. Finally, we note that our method provides new inequalities involving the corresponding Gram matrix, which become equalities whenever there exist r·(r+1)/2 equiangular lines in Rr. We also derive analogous inequalities for complex equiangular lines and time permitting, we will discuss how our results generalize to this setting.
Gorav
Jindal
Technische Universität Berlin
Arithmetic Circuit Complexity of Division and Truncation
Abstract.
Given n-variate polynomials f,g,h such that f=g/h, where both g and h are computable by arithmetic circuits of size s, we show that f can be computed by a circuit of size poly(s, deg(h)). This solves a special case of division elimination for high-degree circuits (Kaltofen’87 & WACT’16). This result is an exponential improvement over Strassen’s classic result (Strassen’73) when deg(h) is poly(s) and deg(f) is exp(s), since the latter gives an upper bound of poly(s, deg(f)).The second part of this work deals with the complexity of computing the truncations of uni-variate polynomials or power series. We first show that the truncations of rational functions are easy to compute. We also prove that the truncations of even very simple algebraic functions are hard to compute,unless integer factoring is easy. This is a joint work with Pranjal Dutta, Anurag Pandey and Amit Sinhababu. A pre-print can be found athttps://eccc.weizmann.ac.il/report/2021/072/ .
Xavier
Allamigeon
Inria and Ecole Polytechnique
Formalizing the theory of polyhedra in a proof assistant
Abstract.
In this talk, I will present the project Coq-Polyhedra that aims at formalizing the theory of polyhedra as well as polyhedral computations in the proof assistant Coq. I will explain how the intuitionistic nature of the logic of a proof assistant like Coq requires to define basic properties of polyhedra in a quite different way than is usually done, by relying on a formal proof of the simplex method. I will also focus on the formalization of the faces of polyhedra, and present a mechanism which automatically introduces an appropriate representation of a polyhedron or a face, depending on the context of the proof. I will demonstrate the usability of this approach by establishing some of the most important combinatorial properties of faces, namely that they constitute a family of graded atomistic and coatomistic lattices closed under interval sublattices, as well as Balinski’s theorem on the d-connectedness of the graph of d-polytopes. Finally, I will discuss recent progress on the formal computation of the graph of a polytope directly within the proof assistant, thanks to a certified algorithm that checks a posteriori the output of Avis’ vertex enumeration library lrslib. Joint work with Quentin Canu, Ricardo D. Katz and Pierre-Yves Strub.
Jörg
Wolff
Existence of weak solutions to the equations of incompressible power law fluids with variable exponent
Tim
Bastian
Listenfärbbarkeit von Arrangement-Graphen
Julian
Pfeifle
UPC Barcelona
Positive Plücker Tree Certificates for Non-Realizability
Abstract.
In 2020, Hailun Zheng constructed a balanced, 2-neighborly combinatorial 3-sphere Z on 16 vertices whose graph is the complete 4-partite graph K_{4,4,4,4}. However, it was unknown whether Z is realizable as the boundary of a convex polytope. If so, Z would provide the first example of a polytope with such a graph aside from the cross-polytope. However, known techniques for proving or disproving the realizability of Z could not cope with this example. In the present talk, we level up the old idea of Plücker relations, and assemble them using integer programming into a new and more powerful structure, called 'positive Plücker trees'. They certify the non-realizability of Z and many other previously inaccessible families of simplicial spheres, such as several combinatorial prismatoids by Criado & Santos, and all centrally symmetric neighborly d-spheres recently constructed by Novik and Zheng.
Manuel
Radons
TU Berlin
Edge-unfolding nested prismatoids
Abstract.
A 3-Prismatoid is the convex hull of two convex polygons A and B which lie in parallel planes H_A, H_B in R^3. Let A' be the orthogonal projection of A onto H_B. A prismatoid is called nested if A' is properly contained in B, or vice versa. We show that every nested prismatoid has an edge-unfolding to a non-overlapping polygon in the plane.
Olaf
Parczyk
LSE
Resilience for Hamiltonicity in random hypergraphs
Abstract.
Sudakov and Vu introduced the concept of local resilience of graphs for measuring robustness with respect to satisfying a given property. A classical result of Dirac states that any subgraph G of the complete graph on n vertices of minimum degree n/2 contains a Hamilton cycle. In the binomial random graph G(n,p) the threshold for the appearance of a Hamilton cycle is log(n)/n. Lee and Sudakov generalised Dirac’s result to random graphs by showing that with p > C log(n)/n asymptotically almost surely any subgraph G of G(n,p) with minimum degree (1/2+ε)n contains a Hamilton cycle, where C depends only on ε>0. These kind of resilience problems in random graphs received a lot of attention. In this talk we discuss a generalisation of the result of Lee and Sudakov to tight Hamilton cycles in random hypergraphs. This is joint work with Peter Allen and Vincent Pfenninger.
Max-Georg
Schorr
IOL
Evolution of Boosting
Abstract.
The question whether we can convert a bunch of weak learners into a strong learner arose in the late 80s. It was confirmed by Schapire and led to the most famous approach: AdaBoost. This algorithm is well known for it's good performance in practice and there are a lot of variants. In this talk, we will look at the application of linear and mixed-integer optimization and we will derive corresponding LP and MIP formulations. In particular, we will look at: - Properties of AdaBoost including possible drawbacks and approaches to overcome them - Column generation as an efficient way to solve LP formulations - LPBoost and IPBoost
Patrick
Morris
Freie Universität Berlin
Triangle factors in pseudorandom graphs
Abstract.
An (n, d, λ)-graph is an n vertex, d-regular graph with second eigenvalue in absolute value λ. When λ is small compared to d, such graphs have pseudo-random properties. A fundamental question in the study of pseudorandom graphs is to find conditions on the parameters that guarantee the existence of a certain subgraph. A celebrated construction due to Alon gives a triangle-free (n, d, λ)-graph with d = Θ(n^2/3) and λ = Θ(d^2/n). This construction is optimal as having λ = o(d^2/n) guarantees the existence of a triangle in a (n, d, λ)-graph. Krivelevich, Sudakov and Szab ́o (2004) conjectured that if n ∈ 3N and λ = o(d^2/n) then an (n, d, λ)-graph G in fact contains a triangle factor: vertex disjoint triangles covering the whole vertex set. In this talk, we discuss a solution to the conjecture of Krivelevich, Sudakov and Szab ́o. The result can be seen as a clear distinction between pseudorandom graphs and random graphs, showing that essentially the same pseudorandom condition that ensures a triangle in a graph actually guarantees a triangle factor. In fact, even more is true: as a corollary to this result and a result of Han, Kohayakawa, Person and the author, we can conclude that the same condition actually guarantees that such a graph G contains every graph on n vertices with maximum degree at most 2.
Alexey
Pokrovskiy
University College London
Linear size Ramsey numbers of hypergraphs
Abstract.
The size-Ramsey number of a hypergraph H is the minimum number of edges in a hypergraph G whose every 2-edge-colouring contains a monochromatic copy of H. This talk will be about showing that the size-Ramsey number of r-uniform tight path on n vertices is linear in n. Similar results about hypergraph trees and their powers will also be discussed. This is joint work with Letzter and Yepremyan.
Sebastian
Wiederrecht
Even Circuits in Oriented Matroids
Felix
Otto
MPI Leipzig
Aram
Dermenjian
York University
Sign Variations and Descents
Abstract.
In this talk we consider a poset structure on projective sign vectors. We show that the order complex of this poset is partitionable and give an interpretation of the h-vector using type B descents of the type D Coxeter group.
Christoph
Hertrich
TU Berlin
Towards Lower Bounds on the Depth of ReLU Neural Networks
Abstract.
We contribute to a better understanding of the class of functions that is represented by a neural network with ReLU activations and a given architecture. Using techniques from mixed-integer optimization, polyhedral theory, and tropical geometry, we provide a mathematical counterbalance to the universal approximation theorems which suggest that a single hidden layer is sufficient for learning tasks. In particular, we investigate whether the class of exactly representable functions strictly increases by adding more layers (with no restrictions on size). We also present upper bounds on the sizes of neural networks required for exact function representation.
Julian
Sahasrabudhe
University of Cambridge
The singularity probability of random symmetric matrices
Abstract.
Let Mn be drawn uniformly from all {+1,-1}-symmetric n by n matrices. We consider the following basic problem: what is the probability Mn is singular? While the analogous problem for matrices with all entries independent is now well understood, the case of symmetric matrices has remained somewhat more mysterious, despite the attention it has received. In this talk I will discuss our recent result which shows that the singularity probability is exponentially small. This talk is based on joint work with Marcelo Campos, Matthew Jenssen and Marcus Michelen.
Alex
Postnikov
MIT
Polypositroids
Abstract.
Polypositroids is a class of convex polytopes defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and alcoved polytopes. Whereas positroids are the matroids arising from the totally nonnegative Grassmannian,polypositroids are "positive" polymatroids. We parametrize polypositroids using Coxeter necklaces and balanced graphs, and describe the cone of polypositroids by extremal rays and facet inequalities. We generalize polypositroids to an arbitrary finite Weyl group W, and connect them to cluster algebras and to generalized associahedra. We also discuss membranes, which are certain triangulated surfaces. They extend the notion of plabic graphs from positroids to polypositroids. The talk is based on a joint work with Thomas Lam.
Ozan
Öktem
KTH, Sweden
Data driven large-scale convex optimisation
Abstract.
This joint work with Jevgenjia Rudzusika (KTH), Sebastian Banert (Lund University) and Jonas Adler (DeepMind) introduces a framework for using deep-learning to accelerate optimisation solvers with convergence guarantees. The approach builds on ideas from the analysis of accelerated forward-backward schemes, like FISTA. Instead of the classical approach of proving convergence for a choice of parameters, such as a step-size, we show convergence whenever the update is chosen in a specific set. Rather than picking a point in this set through a handcrafted method, we train a deep neural network to pick the best update. The method is applicable to several smooth and non-smooth convex optimisation problems and it outperforms established accelerated solvers.
Davide
Lofano
Technische Universität Berlin
Random Simple-Homotopy Theory
Abstract.
A standard task in topology is to simplify a given finite presentation ofa topological space. Bistellar flips allow to search for vertex-minimaltriangulations of surfaces or higher-dimensional manifolds, and elementarycollapses are often used to reduce a simplicial complex in size andpotentially in dimension. Simple-homotopy theory, as introduced byWhitehead in 1939, generalizes both concepts.We take on a random approach to simple-homotopy theory and present aheuristic algorithm to combinatorially deform non-collapsible, butcontractible complexes (such as triangulations of the dunce hat, Bing'shouse or non-collapsible balls that contain short knots) to a point.The procedure also allows to find substructures in complexes, e.g.,surfaces in higher-dimensional manifolds or subcomplexes with torsion inlens spaces.(Joint work with Bruno Benedetti, Crystal Lai, and Frank Lutz.)
Niklas
Affolter
Triple Crossing Diagrams (TCDs) and some Posets
Isabel
Hubard
UNAM
Twisted honeycombs revisited: chirality in polytope-like structures
Abstract.
In the 70´s Coxeter considered the 4-dimensional regular polytopes and used the so-called Petrie Polygons to obtain quotients of the polytopes that, while having all possible rotational symmetry, lack reflectional symmetry. He called these objects Twisted Honeycombs. Nowadays, objects with such symmetry properties are often called chiral. In this talk I will review Coxeter´s twisted honeycombs and explore a natural way to extend Coxeter´s work to polytope-like structures, in particular, we shall see some properties of so-called chiral skeletal polyhedra.
Simona
Boyadzhiyska
FU Berlin
Ramsey simplicity of random graphs
Abstract.
We say that a graph G is q-Ramsey for another graph H if any q-coloring of the edges of G yields a monochromatic copy of H. Much of the research related to Ramsey graphs is concerned with determining the smallest possible number of vertices in a q-Ramsey graph for a given H, known as the q-color Ramsey number of H. In the 1970s, Burr, Erdős, and Lovász initiated the study of another graph parameter in the context of Ramsey graphs: the minimum degree. A straightforward argument shows that, if G is a minimal q-Ramsey graph for H, then we must have δ(G) ≥ q(δ(H) - 1) + 1, and we say that H is q-Ramsey simple if this bound can be attained. In this talk, we will ask how typical Ramsey simplicity is; more precisely, we will discuss for which pairs p and q the random graph G(n,p) is almost surely q-Ramsey simple. This is joint work with Dennis Clemens, Shagnik Das, and Pranshu Gupta.
Matteo
Gallet
RICAM
Mobile Icosapods
Abstract.
Pods are mechanical devices constituted of two rigid bodies, the base and the platform, connected via spherical joints by a number of other rigid bodies, called legs. The maximal number, when finite, of legs of a mobile pod is 20. In 1904, Borel designed a technique to construct examples of such 20-pods over the complex numbers. We show that recent results about spectrahedra provide a way to determine real solutions for Borel’s construction. This is a joint work with Georg Nawratil, Jon Selig, and Josef Schicho.
Arturo
Merino
TU Berlin
Greedy strategies for exhaustive generation
Abstract.
Day to day we frequently encounter different kinds of combinatorial objects, e.g., permutations, binary strings, binary trees, spanning trees of a graph, bases of a matroid. Usually, there are three fundamental tasks for these objects: counting, random sampling, and exhaustive generation. Even though for the first two there are powerful well-understood general approaches like generating functions or Markov chains; this has been lacking for exhaustive generation. In this talk, we will discuss greedy strategies for exhaustive generation. This is a very natural attempt to provide a general approach for exhaustive generation as greedy strategies are a widely used algorithmic paradigm. In particular, we will discuss: - Various classical examples of Greedy based approaches in exhaustive generation. - An introduction to the zig-zag framework by Hartung, Hoang, Mütze, and Williams, which provides a common lense for many greedy strategies. - Some greedy strategies that appear to not fit into the zig-zag framework, including a new greedy approach for generating spanning trees of any given connected graph.
Michael
Joswig
Technische Universität Berlin
Tropical bisectors and Voronoi diagrams
Abstract.
We consider norms in real vector spaces where the unit ball is an arbitrary convex polytope, possibly centrally symmetric. In contrast with the Euclidean norm, the topological shape of bisectors may be complicated. Our first main result is a formula for the Betti numbers of bisectors of three points in sufficiently general position. Specializing our results to the tropical polyhedral norm then yields structural results and algorithms for tropical Voronoi diagrams. The tropical distance function plays a key role in current applications of tropical geometry. Joint work with Francisco Criado and Francisco Santos.
Stefan
Felsner
Counting Plane Partitions - Kuo's proof of the MacMahon formula
Martina
Juhnke-Kubitzke
U Osnabrück
On γ-vectors for symmetric edge polytopes
Abstract.
Symmetric edge polytopes (SEP) are a class of lattice polytopes that can be constructed from a graph. Recently, their h*-vectors have been studied by several authors and it was shown that the corresponding γ-vectors are non-negative for several classes of graphs. The goal of this talk is twofold: First, we will show that γ2 is non-negative for any SEP and characterize those graphs for which γ2=0. Second, we will take an asymptotic point of view and show that, for graphs with n vertices and cn edges (for some c>1), not containing any even cycle, γk grows as 2^k(c-1)^k/k!n^k if n is large enough. If time permits, I will also show that a similar statement (without excluding the existence of even cycles) holds with high probability if we consider graphs constructed by the probabilistic Erdös-Renyi model. This is joint work with Alessio D'Alí, Daniel Köhne and Lorenzo Venturello.
Marieke
van der Wegen
Utrecht University
Computing stable gonality is hard
Abstract.
Based on analogies between algebraic curves and graphs, there are several graph parameters defined. In this talk we will study the so-called stable gonality. The stable gonality is a measure for the complexity of a graph and is defined using morphisms from a graph to a tree. We show that computing the stable gonality of a graph is NP-hard. This is joint work with Dion Gijswijt and Harry Smit.
Giulia
Maesaka
Universität Hamburg
Embedding spanning subgraphs in uniformly dense and inseparable graphs
Abstract.
In this talk we consider sufficient conditions for the existence of k-th powers of Hamiltonian cycles in n-vertex graphs G with minimum degree μn for arbitrarily small μ>0. About 20 years ago Komlós, Sarközy, and Szemerédi resolved the conjectures of Pósa and Seymour and obtained optimal minimum degree conditions for this problem by showing that μ=k/(k+1) suffices for large n. For smaller values of μ, the given graph G must satisfy additional assumptions. We show that inducing subgraphs of density d>0 on linear subsets of vertices and being inseparable, in the sense that every cut has density at least μ>0, are sufficient assumptions for this problem and, in fact, for a variant of the bandwidth theorem. This generalises recent results of Staden and Treglown. Joint work with O. Ebsen, C. Reiher, M. Schacht and B. Schülke
May 2021
Arturo
Merino
Generation of elimination trees and Hamilton paths on graph associahedra
Fernando
Codá Marques
Princeton U
Morse Theory for the Area
Isabella
Novik
University of Washington
Many neighborly spheres
Abstract.
A simplicial complex on $n$ vertices is $s$-neighborly if it has the same $(s−1)$-skeleton as the $(n − 1)$-simplex on the same vertex set. While a $(d-1)$-dimensional sphere with $n>d+1$ vertices cannot be more than $loor{d/2}$-neighborly, $loor{d/2}$-neighborly $(d-1)$-dimensional spheres with $n$ vertices do exist. How many such neighborly spheres are there? We will present a recent construction showing that for $d ext{≥}5$, there are at least $2^{ exts{ extOmega}(n^{loor{(d-1)/2}})}$ distinct combinatorial types of neighborly spheres. Joint work with Hailun Zheng.
Guido
Montúfar
MPI Leipzig and UCLA
Sharp bounds for the number of regions of maxout networks and
vertices of Minkowski sums
Abstract.
We present results on the number of linear regions of the functions that can be represented by artificial feedforward neural networks with maxout units. A rank-k maxout unit is a function computing the maximum of k linear functions. For networks with a single layer of maxout units, the linear regions correspond to the upper vertices of a Minkowski sum of polytopes. We obtain face counting formulas in terms of the intersection posets of tropical hypersurfaces or the number of upper faces of partial Minkowski sums, along with explicit sharp upper bounds for the number of regions for any input dimension, any number of units, and any ranks, in the cases with and without biases. Based on these results we also obtain asymptotically sharp upper bounds for networks with multiple layers. This is joint work with Yue Ren and Leon Zhang.
Leticia
Mattos
IMPA
Asymmetric Ramsey properties of random graphs for cliques and cycles
Abstract.
We say that G ⟶ (F,H) if in every red and blue colouring of the edges of G we can find a red copy of F or a blue copy of H. In 1997, Kohayakawa--Kreuter conjectured the value of the threshold for this property when G is a random graph G(n,p). In this talk, we sketch the proof of Kohayakawa--Kreuter conjecture for the case that F is a clique and H is a cycle. Our main tool is a structural characterisation of Ramsey graphs for pairs of cliques and cycles. Joint work with Anita Liebenau, Walner Mendonça and Jozef Skokan.
Paco
Criado
IOL
Borsuk's problem
Abstract.
In 1932, Borsuk proved that every 2-dimensional convex body can be divided in three parts with strictly smaller diameter than the original. He also asked if the same would hold for any dimension d: is it true that every d-dimensional convex body can be divided in d+1 pieces of strictly smaller diameter? The answer to this question is positive in 3 dimensions (Perkal 1947), but in 1993, Kahn and Kalai constructed polytopal counterexamples in dimension 1325. Nowadays, we know that the answer is “no” for all dimensions d≥64 (Bondarenko and Jenrich 2013). It remains open for every dimension between 4 and 63. In this talk we introduce the problem, and we show the proofs for dimensions 2 and 3 plus computational ideas on how to extend these solutions to dimension 4.
Kıvanç
Ersoy
FU Berlin
What is a finite simple group?
Abstract.
In this introductory talk, finite simple groups and their classification will be introduced. In particular, we will talk about some examples of simple groups of Lie type.
Raphael
Steiner
Disjoint cycles with length constraints in digraphs
Martin R.
Bridson
U Oxford
Chasing finite shadows of infinite groups through geometry
Henri
Mühle
TU Dresden
Refined Face Enumeration in v-Associahedra
Abstract.
Chapoton defined for each finite root system three remarkable polynomials, the F-, H- and M-triangle. The F-triangle is a refined face generating polynomial of the cluster complex, the H-triangle is a refined generating function of nonnesting partitions, and the M-triangle is a bivariate generating function of the Möbius function in the noncrossing partition lattice. He observed that these three polynomials are related by an invertible substitution of variables. These observations were proven uniformly by Athanasiadis and Thiel, respectively. In this talk, we explain how the F- and the H-triangle arise as refined versions of the f- and the h-polynomial of a polytope, and give a simple, combinatorial proof of Chapoton's relation in the context of v-associahedra. The relation with the M-triangle is much less clear. We conjecture a characterization of the northeast paths for which Chapoton's relation generalizes to this setting. Computational evidence suggests a close connection to the pureness of v-associahedra. This is based on joint work with Cesar Ceballos from TU Graz.
Simon
Telen
MPI MiS Leipzig
Likelihood equations and scattering amplitudes
Abstract.
We identify the scattering equations from particle physics as the likelihood equations for a particular statistical model. The scattering potential plays the role of the log-likelihood function. We employ recent methods from numerical nonlinear algebra to solve challenging instances of the scattering equations. We revisit the theory of stringy canonical forms proposed by Arkani-Hamed, He and Lam, introducing positive statistical models and their amplitudes. This is joint work with Bernd Sturmfels.
Stochastic Scheduling with Restricted Adaptivity
Hong
Liu
University of Warwick
Sublinear expander and embeddings in sparse graphs
Abstract.
A notion of sublinear expander has played a central role in the resolutions of a couple of long-standing conjectures in embedding problems in graph theory, including e.g. the odd cycle problem of Erdos and Hajnal that the harmonic sum of odd cycle length in a graph diverges with its chromatic number, Komlos's conjecture on the number of Hamilton subsets, average degree forcing dense/sparse graphs as minors etc. I will survey some of these developments.
Aniket
Shah
Ohio State University
A q-analogue of Brion's identity
Abstract.
Rogers-Szego polynomials are a family of orthogonal polynomials on the circle. We introduce a generalization of these polynomials which depend on the data of a polytope and prove a vertex sum formula for them when the polytope is smooth. This formula recovers Brion's formula when the parameter q is set to 0.
Denes
Stolte
On the Locality of the \Gamma-Limit of a Nonlocal Energy in Molecular Solvation in the Presence of the Ionic Effect
Sophie
Rehberg
FU Berlin
Combinatorial reciprocity theorems for generalized permutahedra, hypergraphs, and pruned inside-out polytopes
Abstract.
Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck–Zaslavsky (2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We study the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart–Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar–Ardila (2017) and Billera–Jia–Reiner (2009). Applying this reciprocity theorem to hypergraphic polytopes allows us to give an arguably simpler proof of a recent combinatorial reciprocity theorem for hypergraph colorings by Aval–Karaboghossian–Tanasa (2020). Our proof relies, aside from the reciprocity for generalized permutahedra, only on elementary geometric and combinatorial properties of hypergraphs and their associated polytopes.
Andrea
Freschi
University of Birmingham
On Deficiency Problems for Graphs
Abstract.
Nenadov, Sudakov and Wagner recently introduced the notion of graph deficiency: given a global spanning property P (eg Hamiltonicity) and a graph G, the deficiency def(G) of G with respect to P is the smallest non-negative integer t such that the join G*Kt has property P, where G*Kt is the graph obtained by adding t new vertices to G and adding all edges incident to at least one of the new vertices. In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an n-vertex graph G needs to ensure G*Kt contains a Kr-factor (for any fixed r >= 3).In this talk we present a solution to this problem. We also briefly discuss about an analogous result which forces G*Kt to contain any fixed bipartite (n+t)-vertex graph of bounded degree and small bandwidth. This is joint work with Joseph Hyde and Andrew Treglown.
Melanie
Koser
Asymptotic self-similarity and local bounds in a model of shape-memory alloys
Christoph
Hertrich
TU Berlin
Tackling Neural Network Expressivity via (virtual Newton) polytopes
Abstract.
I will talk about how to translate two open problems in the context of ReLU neural network (NN) expressivity into the polytope world: 1. Can constant-depth NNs compute any continuous piecewise linear (CPWL) function? Smallest open special case: Can 3-layer NNs compute the maximum of 5 numbers? 2. Can polynomial-size NNs precisely compute the objective value of the Minimum Spanning Tree problem (or other combinatorial optimization problems) for arbitrary real-valued edge weights? To achieve these translations, I will introduce an isomorphism between the two semigroups (convex CPWL functions, pointwise addition) and (polytopes, Minkowski sum), known as “support functions” (in convex geometry) and “Newton polytopes” (in tropical geometry). Extending this concept to arbitrary (not necessarily convex) CPWL functions leads to the (in my opinion beautiful) construction of “virtual” polytopes, that is, the group of formal Minkowski differences of two polytopes.
Max
Klimm
Technische Universität Berlin
Complexity and Parametric Computation of Equilibria in Atomic Splittable Congestion Games
Abstract.
We settle the complexity of computing an equilibrium in atomic splittable congestion games with player-specific affine cost functions as we show that the computation is PPAD-complete. To prove that the problem is contained in PPAD, we develop a homotopy method that traces an equilibrium for varying flow demands of the players. A key technique for this method is to describe the evolution of the equilibrium locally by a novel block Laplacian matrix where each entry of the Laplacian is a Laplacian again. These insights give rise to a path following formulation eventually putting the problem into PPAD. For the PPAD—hardness, we reduce from computing an approximate equilibrium for bimatrix win-lose games. As a byproduct of our analyse, we obtain that also computing a multi-class Wardrop equilibrium with class-dependent affine cost functions is PPAD-complete as well. As another byproduct, we obtain an algorithm that computes a continuum of equilibria parametrised by the players’ flow demand. For games with player-independent costs, this yields an output-polynomial algorithm. (Joint work with Philipp Warode)
Elias
Wirth
TU Berlin/Zuse-Institut Berlin
What is conditional gradients?
Abstract.
We introduce the Frank-Wolfe Algorithm, also known as Conditional Gradients. Recently, the algorithm has been revisited for various applications in machine learning. We motivate the Frank-Wolfe Algorithm and discuss the properties that make it appealing for practitioners.
Sebastian
Pokutta
TU Berlin/ZIB
Conditional Gradients in Machine Learning and Optimization
Maximilian
Gorsky
k-Outerplanarity and Poset Dimension
Sara
Faridi
Dalhousie University
Breaking up cycles in simplicial complexes and applications to subadditivity of degrees of syzygies of monomial ideals
Abstract.
The motivation for this work is finding methods to prove the subadditivity conjecture for the maximal shifts in the betti diagram of a monomial ideal. Hochster's work in the 1970's showed that the betti diagram encodes dimensions of homology modules of subcomplexes of a simplicial complex known as the Stanley-Reisner complex. In this talk we will show, using lattice complementation, how the subadditivity question can be extended to a stronger question about cycles of a simplicial complex breaking into smaller ones. We will also discuss what is known for the stronger question. This talk is based on joint work with Mayada Shahada.
Sandra
Di Rocco
KTH
Iterated discriminants
Abstract.
The Hyperdeterminant is a celebrated tool in tensor theory, geometrically it represents the discriminant of a Segre embedding. The Segre embedding is a toric map associated to a Cayley sum, probably the basic example of Cayley polytopes. Unlike other discriminants the hyperdeterminant is at times accessible to computation due to a decomposition method developed by Schäfli. We propose a generalisation of the Schäfli method to discriminants associated to general Cayley sums and show how this can be treated as a discriminant of a system of polynomials. This is joint work with A. Dickenstein and R. Morrison.
Direct Reconstruction of Biophysical Parameters Using Dictionary Learning and Robust Regularization
Abstract
Raphael
Steiner
TU Berlin
Zero sum cycles in complete digraphs
Abstract.
Given a non-trivial finite Abelian group (A,+), let n(A)≥2 be the smallest integer such that for every labelling of the arcs of the bidirected complete graph of order n(A) with elements from A there exists a directed cycle for which the sum of the arc-labels is zero. The problem of determining n(Zq) for integers q≥2 was recently considered by Alon and Krivelevich, who proved that n(Zq)=O(qlogq). Here we improve their bound and show that n(Zq) grows linearly. More generally we prove that for every finite Abelian group A we have n(A)≤8|A|, while if |A| is prime then n(A)≤32|A|. As a corollary we also obtain that every K16q-minor contains a cycle of length divisible by q for every integer q≥2, which improves a result by Alon and Krivelevich. This is joint work with Tamás Mészáros.
Chris
Eur
Stanford University
Tautological classes of matroids
Abstract.
We introduce certain torus-equivariant classes on permutohedral varieties which we call "tautological classes of matroids" as a new geometric framework for studying matroids. Using this framework, we unify and extend many recent developments in matroid theory arising from its interaction with algebraic geometry. We achieve this by establishing a Chow-theoretic description and a log-concavity property for a 4-variable transformation of the Tutte polynomial, and by establishing an exceptional Hirzebruch-Riemann-Roch-type formula for permutohedral varieties that translates between K-theory and Chow theory. This is a joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.
Lars
Ruthotto
Emory University, USA
A Machine Learning Framework for Mean Field Games and Optimal Control
Abstract.
We consider the numerical solution of mean field games and optimal control problems whose state space dimension is in the tens or hundreds. In this setting, most existing numerical solvers are affected by the curse of dimensionality (CoD). To mitigate the CoD, we present a machine learning framework that combines the approximation power of neural networks with the scalability of Lagrangian PDE solvers. Specifically, we parameterize the value function with a neural network and train its weights using the objective function with additional penalties that enforce the Hamilton Jacobi Bellman equations. A key benefit of this approach is that no training data is needed, e.g., no numerical solutions to the problem need to be computed before training. We illustrate our approach and its efficacy using numerical experiments. To show the framework's generality, we consider applications such as optimal transport, deep generative modeling, mean field games for crowd motion, and multi-agent optimal control.
April 2021
Felix
Schröder
Two extensions of the Erdös–Szekeres problem
Federico
Castillo
MPI Leizig
Lineup polytopes
Abstract.
Motivated by an instance of the quantum marginal problem in physics, we define the r-lineup polytope of P as a polytope parametrizing all possible linear orders on the vertices of P. We focus on the concrete case when P is a hypersimplex. This example sits in between the sweep polytopes of A.Padrol and E.Philippe and the theory of symmetric polytopes. This is based on joint work with JP. Labbe, J.Liebert, A.Padrol, E.Philippe and C.Schilling.
Daniel
Corey
University of Wisconsin-Madison
Initial degenerations of Grassmannians and spinor varieties
Abstract.
We construct closed immersions from initial degenerations of Gr_0(d,n)---the open cell in the Grassmannian Gr(d,n) given by the nonvanishing of all Plücker coordinates---to limits of thin Schubert cells associated to diagrams induced by the face poset of the corresponding tropical linear space. These are isomorphisms in many cases, including (d,n) equal to (2,n), (3,6) and (3,7). As an application, Gr_0(3,7) is schön, and the Chow quotient of Gr(3,7) by the maximal torus in PGL(7) is the log canonical compactification of the moduli space of 7 points in P^2 in linear general position, making progress on a conjecture of Hacking, Keel, and Tevelev. Finally, I will discuss recent work on extending these results to the Lie-type D setting.
Peleg
Michaeli
Tel Aviv University
Discrepancies of Spanning Trees
Abstract.
Discrepancy theory is concerned with colouring elements of a ground set so that each set in a given set system is as balanced as possible. In the graph setting, the ground set is the edge set of a given graph, and the set system is a family of subgraphs. In this talk, I shall discuss the discrepancy of the set of spanning trees in general graphs, a notion that has been recently studied by Balogh, Csaba, Jing and Pluhár. More concretely, for every graph G and a number of colours r, we look for the maximum D such that in any r-colouring of the edges of G one can find a spanning tree with at least (n-1+D)/r edges of the same colour. As our main result, we show that under very mild conditions (for example, if G is 3-connected), D is equal, up to a constant factor, to the minimal integer s such that G can be separated into r equal parts by removing s vertices. This strong and perhaps surprising relation between the extremal quantity D and a geometric quantity allows us to estimate the spanning-tree discrepancy for many graphs of interest. In particular, we reprove and generalize results of Balogh et al., as well as obtain new ones. For certain graph families, we also obtain tight asymptotic results. The talk is based on joint work with Lior Gishboliner and Michael Krivelevich.
Matthieu
Rosenfeld
LIRMM
Bounding the number of sets defined by a given MSO formula on trees
Abstract.
Monadic second order logic can be used to express many classical notions of sets of vertices of a graph as for instance: dominating sets, induced matchings, perfect codes, independent sets, or irredundant sets. Bounds on the number of sets of any such family of sets are interesting from a combinatorial point of view and have algorithmic applications. Many such bounds on different families of sets over different classes of graphs are already provided in the literature. In particular, Rote recently showed that the number of minimal dominating sets in trees of order n is at most 95^(n/13) and that this bound is asymptotically sharp up to a multiplicative constant. We build on his work to show that what he did for minimal dominating sets can be done for any family of sets definable by a monadic second-order formula.I will first illustrate the general technique with a really simple concrete example ( Dominating independent sets). Then I will explain how to generalize this into a more general technique. I will end my talk by mentioning a few of the results obtained with this technique.
Kolja
Knauer
Universitat de Barcelona
On sensitivity in Cayley graphs
Abstract.
Recently, Huang proved the Sensitivity Conjecture, by showing that every set of more than half the vertices of the $d$-dimensional hypercube $Q_d$ induces a subgraph of maximum degree at least $\sqrt{d}$. This is tight by a result of Chung, F\"uredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs. In this lecture we study Huang's question on Cayley graphs of groups.We show that high symmetry alone does not guarantee similar behavior and present three infinite families of Cayley graphs of unbounded degree that contain induced subgraphs of maximum degree $1$ on more than half the vertices. In particular, this refutes a conjecture of Potechin and Tsang, for which first counterexamples were shown recently by Lehner and Verret. The first family consists of dihedrants. The second family are star graphs, these are edge-transitive Cayley graphs of the symmetric group. All members of the third family are $d$-regular containing an induced matching on a $\frac{d}{2d-1}$-fraction of the vertices. This is largest possible and answers a question of Lehner and Verret.On the positive side, we consider Cayley graphs of Coxeter groups, where a lower bound similar to Huang's can be shown. A generalization of the construction of Chung, F\"uredi, Graham, and Seymour shows that this bound is tight for products of Coxeter groups of type $\mathbf{A_n}$, $\mathbf{I_n}(2k+1)$, most exceptional cases and not far from optimal in general.Then, we show that also induced subgraphs on more than half the vertices of Levi graphs of projective planes and of the Ramanujan graphs of Lubotzky, Phillips, and Sarnak have unbounded degree. This yields more classes of Cayley graphs with properties similar to the ones provided by Huang's results. However, in contrast to Coxeter groups these graphs have no large subcubes.Joint with Ignacio Garcia-Marco.
Matthias
Kirchhart
RWTH Aachen
Div-Curl Problems and H^1-regular Stream Functions in 3D Lipschitz Domains
Lisa
Schottstädt
Unique Sink Orientierungen und Anwendungen
Mara
Belotti
TU Berlin
Algebraic degrees of 3-dimensional polytopes
Abstract.
Results of Koebe (1936), Schramm (1992), and Springborn (2005) yield realizations of 3-polytopes with edges tangent to the unit sphere. We introduce two notions of algebraic degree for such constrained realizations and we compute them for some classes of polytopes. This is joint work with Michael Joswig and Marta Panizzut.
Shagnik
Das
FU Berlin
Isomorphic bisections of cubic graphs
Abstract.
Graph partitioning is the division of a graph into two or more parts based on certain combinatorial conditions, and problems of this kind of have been studied extensively. In the 1990s, Ando conjectured that the vertices of every cubic graph can be partitioned into two parts that induce isomorphic subgraphs. Using probabilistic methods together with delicate recolouring arguments, we prove Ando's conjecture for large connected graphs.This is joint work with Alexey Pokrovskiy and Benny Sudakov.
Mario
Kummer
TU Dresden
Iso Edge Domains
Abstract.
A Conjecture by Conway and Sloane from the 90s can be phrased as ''Every tropical abelian variety is determined by its tropical theta constants''. It was proved by Conway and Sloane in dimension up to 3 and by Vallentin in dimension 4. We prove it in dimension 5. The prove relies on the classification of iso-edge domains in dimension 5: These are a variant of the iso-Delaunay decomposition introduced by Baranovskii and Ryshkov. We further characterize the so-called matroidal locus in terms of the tropical theta constants in dimension up to 5 and point out connections to the tropical Schottky problem. The talk is based on a joint work with Mathieu Dutour Sikirić.
EF4-2: Understanding Tipping and Other Dynamical Transitions in Systems Containing Human Agents
Abstract
Tamara
Fastovska
Qualitative behavior of solutions to a transmission problem with full von Karman nonlinearity
Raphael
Steiner
TU Berlin
What is a tree-decomposition?
Abstract.
In the talk I explain the definition of a tree decomposition of a graph, and using simple examples I illustrate some of the intuition behind this important combinatorial concept, which is useful in computer science for designing efficient algorithms as well as in structural graph theory as a tool for proving theoretical results.
Stefan
Felsner
Triangles and Lenses in Arrangements of Pseudocircles
Maria
Chudnovsky
Induced subgraphs and tree decompositions
Alessio
D'Alì
U Osnabrück
A tale of quadratic algebras and flag simplicial complexes
Abstract.
Koszul algebras are quadratic algebras satisfying desirable homological properties and arising naturally in many geometric and combinatorial contexts: for instance, the coordinate rings of Veronese, Segre and Grassmannian varieties (in their natural embeddings) are all Koszul, and so is the Stanley-Reisner ring of any flag simplicial complex. However, the Koszul property is hard to control and to check in general, and many conjectures about the general behaviour of Koszul algebras are currently open. Starting from a flag simplicial complex Delta, we propose a construction of a (non-monomial) quadratic Gorenstein ring R_Delta which is Koszul if and only if Delta is Cohen-Macaulay, thus providing a rather unexpected bridge between these two worlds. On a more combinatorial level, the very same correspondence also yields that R_Delta has a Gröbner basis of quadrics if and only if Delta is shellable. As an application, we provide counterexamples to an algebraic generalization of a conjecture by Charney and Davis about flag homology spheres. This is joint work with Lorenzo Venturello (KTH Stockholm).
Antony
Della Vecchia
One Relator Groups and Regular Sectional Curvature
Abstract.
We present the notion of regular sectional curvature for angled 2-complexes. It is known that fundamental groups of angled 2-complexes with negative or nonpositive regular section curvature are coherent and, in certain cases, locally quasi-convex. We state some open problems for one-relator groups, describe a construction of angled 2-complexes for one-relator groups and provide some computer generated results supporting the claims.
March 2021
Serge
Gratton
ENSEEIHT, Toulouse, France
On a multilevel Levenberg-Marquardt method for the training of artificial neural networks and its application to the solution of partial differential equations
Abstract.
We propose a new multilevel Levenberg-Marquardt optimizer for the training of artificial neural networks with quadratic loss function. When the least-squares problem arises from the training of artificial neural networks, the variables subject to optimization are not related by any geometrical constraints and the standard interpolation and restriction operators cannot be employed any longer. A heuristic, inspired by algebraic multigrid methods, is then proposed to construct the multilevel transfer operators. We test the new optimizer on an important application: the approximate solution of partial differential equations by means of artificial neural networks. The learning problem is formulated as a least squares problem, choosing the nonlinear residual of the equation as a loss function, whereas the multilevel method is employed as a training method. Numerical experiments show encouraging results related to the efficiency of the new multilevel optimization method compared to the corresponding one-level procedure in this context.
Eugenia
Sinatti
Zirkuläre chromatische Zahl von zirkulierenden Graphen
Martin
Skutella
TU Berlin
A simple proof of the Moore-Hodgson Algorithm for minimizing the number of late jobs
Abstract.
The Moore–Hodgson Algorithm minimizes the number of late jobs on a single machine. That is, it finds an optimal schedule for the classical problem where jobs with given processing times and due dates have to be scheduled on a single machine so as to minimize the number of jobs completing after their due date. Several proofs of the correctness of this algorithm have been published. We present a new short proof.
Maximilian
Stahlberg
TU Berlin
Robust conic optimization in Python
Abstract.
I present an extension of the Python-embedded optimization modeling language PICOS that makes key results from the fields of robust and distributionally robust optimization available to application developers. While the Python ecosystem already offers multiple frameworks concerned with mathematical optimization, library support for optimization under uncertainty has been sparse in comparison. My talk opens with a brief introduction to the modeling interface provided by PICOS, then leads into a discussion of the recent additions that allow uncertainty in the data to be modelled explicitly and accounted for, assuming worst-case outcomes, during the solution process. In particular, I present an illustrative application in linear signal estimation where distributionally robust optimization can be seen to perform quite well.
Max
Klimm
TU Berlin
Static and dynamic pricing of identical items
Abstract.
We consider the problem of maximizing the expected revenue from selling k identical items to n unit-demand buyers who arrive sequentially with independent and identically distributed valuations. We show bounds on the gap between the revenue achievable with dynamic prices and that achievable with a static price. These bounds are tight for all values of k and n and vary depending on a regularity property of the underlying distribution. The bound for regular distributions increases in n and tends to 1/(1-k<sup>k</sup>/(e<sup>k</sup> k!)). Our upper bounds are obtained via a classic relaxation of the pricing problem known as the ex-ante relaxation, and a novel use of Bernstein polynomials to approximate the revenue from static pricing. This technique allows for a uniform and lossless analysis across all parameter values, both in the finite regime and in the limit. (Joint work with Paul Dütting and Felix Fischer)
Felix
Schröder
Actual coloring numbers
Thomas
Kerdreux
IOL
Set Curvature in Machine Learning
Abstract.
From the global strong convexity assumptions to local Łojasiewicz gradient inequalities, functional structures are ubiquitous in Machine Learning. They successfully describe convergence properties of optimization methods, regret bounds of online methods, or generalization bounds. However, much less attention has been dedicated to similar structures in the optimization constraint sets or online learner's action sets. Here, we first motivate such an issue from the perspective of projection-free methods (e.g., Conditional Gradients in offline optimization). We then describe how global and local set uniform convexity, i.e., a natural quantitative notion for upper curvature, can help obtain faster convergence rates or regret bounds for online or bandit algorithms. By doing so, we explore some useful characterization of these sets leveraging tools developed to study Banach Spaces. We finally provide some connection between uniform convexity and upper-bound on some Rademacher constants.
Valentin
Hartmann
EPFL
Differential Privacy for Machine Learning
Abstract.
Since its invention in 2006, differential privacy (DP) has become the de-facto standard for releasing aggregate information about data in a privacy-preserving way. It has not only found its way into companies such as Google, Microsoft and Apple, but is also used by government agencies such as the U.S. Census Bureau. In this talk, I will give an introduction into the concept of DP. I will further describe the most important building blocks that can be used to construct differentially private machine learning models that are safe against, e.g., training data extraction attacks.
Nicholas
Williams
University of Leicester
New interpretations of the higher Stasheff–Tamari orders
Abstract.
The higher Stasheff–Tamari orders are two partial orders introduced in the 1990s by Kapranov, Voevodsky, Edelman and Reiner on the set of triangulations of a cyclic polytope. Edelman and Reiner conjectured these two orders to coincide, which remains an open problem today. In this talk we give new combinatorial interpretations of the orders. These interpretations differ depending on whether the cyclic polytope is 2d-dimensional or (2d + 1)-dimensional, but are expressed in each case in terms of the d-skeleton of the triangulation. This naturally leads us to also characterise the d-skeletons of triangulations of (2d + 1)-dimensional cyclic polytopes, complementing the description of the d-skeleton of a 2d-dimensional triangulation given by Oppermann and Thomas. Our results make the conjectured equivalence of the orders a more tractable problem, and we use them to run computational experiments checking the conjecture, to which we find no counter-examples. Our results also have interpretations in the representation theory of algebras, but we will not discuss these connections.
Raphael
Steiner
Zero sum cycles in group-labelled complete digraphs
Lena
Walter
FU Berlin
Toric Newton-Okounkov functions and polytopes
Abstract.
We initiate a combinatorial study of Newton-Okounkov functions on toric varieties with an eye on the rationality of asymptotic invariants of line bundles. In the course of our efforts we identify a combinatorial condition which ensures a controlled behavior of the appropriate Newton-Okounkov function on a toric surface. Our approach yields the rationality of many Seshadri constants that have not been settled before. This is joint work with Christian Haase and Alex Küronya. In the upcoming talk I will explain the setup from a discrete geometer's point of view and how polyhedral methods can be used to study the above problem.
Daniel
Schmidt
TU Berlin
Scheduling under Contact Restrictions - A Problem Arising in Pandemics
Abstract.
In pandemic times, prohibiting close proximity between people is crucial to control the spread of the virus. This is particularly interesting in test and vaccination centers where each person should keep sufficient distance to other people during their transits to and from their designated rooms to prevent possible infections. To model this as a machine scheduling problem, each job has a transit time before and after its processing time. The task is to find a conflict-free schedule of minimum makespan in which the transit times of no two jobs on machines in close proximity (captured by an undirected graph) intersect. In this talk, we discuss hardness as well as approximation results for the case of identical jobs.
Kaie
Kubjas
Aalto University
Identifying 3D Genome Organization in Diploid Organisms via Euclidean Distance Geometry
Abstract.
The 3D organization of the genome plays an important role for gene regulation. Chromosome conformation capture techniques allow one to measure the number of contacts between genomic loci that are nearby in the 3D space. In this talk, we study the problem of reconstructing the 3D organization of the genome from whole genome contact frequencies in diploid organisms, i.e. organisms that contain two indistinguishable copies of each genomic locus. In particular, we study the identifiability of the 3D organization of the genome and optimization methods for reconstructing it. This talk is based on joint work with Anastasiya Belyaeva, Lawrence Sun and Caroline Uhler.
February 2021
Carolina
Benedetti
U de los Andes
Quotients of lattice path matroids
Abstract.
In this talk we will focus on a particular class of matroids called Lattice Path Matroids (LPMs). We will determine when a collection M_1,...,M_k of LPMs are a flag matroid, using their combinatorics. Part of our work will show that the polytope associated to such a flag can be thought as an interval in the Bruhat order, and thus provides a partial understanding of flags of LPMs from a polytopal point of view. We will not assume previous knowledge on quotients. This is joint work with Kolja Knauer.
Malte
Renken
TU Berlin
What does the Cyclic Polytope have to do with Visibility Graphs?
Abstract.
In the 90's the study of visibility graphs prompted the definition of "persistent graphs", a class of vertex-ordered graphs defined by three simple combinatorial properties. We reveal a surprising connection of these persistent graphs to the triangulations of the 3-dimensional cyclic polytope via a natural bijection.(Joint work with Vincent Froese)
Joanna
Lada
Merton College Oxford
On colour-bias Hamilton cycles in dense graphs
Peter
Bürgisser
Technische Universität Berlin
Optimization, Complexity and Invariant Theory
Abstract.
Invariant and representation theory studies symmetries by means of group actions and is a well established source of unifying principles in mathematics and physics. Recent research suggests its relevance for complexity and optimization through quantitative and algorithmic questions. The goal of the lecture is to give an introduction to new algorithmic and analysis techniques that extend convex optimization from the classical Euclidean setting to a general geodesic setting. We also point out surprising connections to a diverse set of problems in different areas of mathematics, statistics, computer science, and physics. The lecture is mainly based on this joint article with Cole Franks, Ankit Garg, Rafael Oliveira, Michael Walter and Avi Wigderson: http://arxiv.org/abs/1910.12375
Irina
Kmit
Global Classical Solvability of Nonautonomous Quasilinear Hyperbolic Systems
Manfred
Scheucher
Tight bounds on the expected number of holes in random point sets
Afshin
Goodarzi
KTH Stockholm
Mixed connectivity for cell complexes
Abstract.
The classical Steinitz's theorem (1922) asserts that a graph is the 1-skeleton of a convex 3-polytope if and only if it is 3-connected and planar. In 1961, Balinski extended the 'only if' direction of Steinitz's theorem by showing that the 1-skeleton of a convex d-polytope is d-connected. In this talk, which is based on a joint work with Anders Björner, some results that combine k-connectivity of graphs and topological k-connectivity of spaces will be presented. In particular, it will be shown that if we remove some d-k vertices (and all faces containing them) from the k-skeleton of the boundary of a convex d-dimensional polytope, then the remaining complex is topologically (k-1)-connected. This extends the result of Balinski (the k = 1 case).
Guillaume
Sagnol
TU Berlin
Greedy Batch-Scheduling
Abstract.
In this short talk, we will discuss some ongoing work on the Min Weighed Sum Bin Packing problem. The goal is to assign n items (with weight w<sub>j</sub> and size s<sub>j</sub>) to bins, without exceeding the bin capacities, when assigning item j to the i'th bin incurs a cost of i * w<sub>j</sub>. This can also be understood as a (batch)-scheduling problem, in which all jobs have unit processing time, the machine can process batches of jobs satisfying a knapsack constraint, and the goal is to minimize the weighted sum of completion time. A PTAS is known for this problem, but its astronomic running time makes it useless in the practice. Apart from that, the First-Fit algorithm that considers items in non-increasing order of w<sub>j</sub>/s<sub>j</sub> is known to have a tight performance guarantee of 2, and there is a greedy algorithm with a performance guarantee of 4 for a slight generalization of this problem, in which the feasible batches are given explicitly by a family of subsets of items. We will show that in the considered case (batches defined implicitly by a knapsack constraint), the performance guarantee of the greedy algorithm can be improved to 2.
Dömötör
Pálvögyi
ELTE
At most 4.47n stable matchings
Stefan
Felsner
Flip structures and the 4 color theorem
Jorge
Olarte
TU Berlin
The tropical symplectic Grassmannian
Abstract.
The symplectic Grassmannian SpGr(k,2n) is the space of a all linear subspaces of dimension k of a vector space of dimension 2n which are isotropic with respect to a symplectic form. We look at several equivalent characterizations of isotropic linear subspaces and formulate a tropical analog for each, such as, for example, being in the tropicalization of the symplectic Grassmannian. It turns out that in the tropical world these characterizations are no longer equivalent and we will see exactly for which k and n does one characterization imply another. This is joint work with George Balla.
Oguzhan
Yürük
TU Berlin
Detecting the Regions of Multistaionarity in CRNT via Symbolic
Nonnegativity Certificates
Abstract.
Parameterized ordinary differential equation systems are crucial for modeling in biochemical reaction networks under the assumption of mass-action kinetics. Various questions concerning the signs of multivariate polynomials in positive orthant arise from studying the solutions' qualitative behavior with respect to parameter values, in particular the existence of multistationarity. In this work, we revisit a method of detecting multistationarity, in which we utilize the circuit polynomials to find symbolic certificates of nonnegativity for 2-site phosphorylation cycle in a recent work of the speaker joint with Elisenda Feliu, Nidhi Kaihnsa and Timo de Wolff. Moreover, we will give a description of all possible certificates that can arise from 2-site phosphorylation cycle, and provide further insight into the number of positive steady states of the n-site phosphorylation cycle model.
Tuan
Tran
IBS
Minimum saturated families of sets
Abstract.
A family F of subsets of [n] is called s-saturated if it contains no s pairwise disjoint sets, and moreover, no set can be added to F while preserving this property. More than 40 years ago, Erdős and Kleitman conjectured that an s-saturated family of subsets of [n] has size at least (1 – 2-(s-1))2n. It is easy to show that every s-saturated family has size at least 2n-1, but, as was mentioned by Frankl and Tokushige, even obtaining a slightly better bound of (1/2 + ε)2n, for some fixed ε > 0, seems difficult. We prove such a result, showing that every s-saturated family of subsets of [n] has size at least (1 – 1/s)2n. In this talk, I will present two short proofs. This is joint work with M. Bucic, S. Letzter and B. Sudakov.
Helena
Bergold
Fern Universität Hagen
Topological Drawings meet Classical Theorems of Convex Geometry
Abstract.
In this talk we discuss classical theorems from Convex Geometry such as Carathéodory's Theorem in a more general context of topological drawings of complete graphs. In a (simple) topological drawing the edges of the graph are drawn as simple closed curves such that every pair of edges has at most one common point. Triangles of topological drawings can be viewed as convex sets. This gives a link to convex geometry. Our main result is a generalization of Kirchberger's Theorem that is of purely combinatorial nature. For this we introduce a structure called ''generalized signotopes'' which are a combinatorial generalization of topological drawings. We discuss further properties of generalized signotopes. Joint work with Stefan Felsner, Manfred Scheucher, Felix Schröder and Raphael Steiner.
Gaëtan
Borot
HU Berlin
What is a Coulomb gas?
Abstract.
I will explain some physical and mathematical motivations underlying the study of Coulomb gases (and more generally, of repulsive particle systems) and set up natural questions about the macroscopic and microscopic properties of such models. In the way we will introduce the notion of equilibrium measure, large deviation principles, and Gaussian free field. Prof. Borot allowed us to share his slides; you can find them here.
Matthias
Täufer
Group testing, Steiner Systems and Reed-Solomon Codes
Eva
Philippe
Sorbonne Université Paris
Sweep polytopes and sweep oriented matroids
Abstract.
Consider a configuration of n labeled points in a Euclidean space. Any linear functional gives an ordered partition of these points. We call it a sweep, because we can imagine its parts as the sets of points successively hit by a sweeping hyperplane. The set of all such sweeps forms a poset which is isomorphic to the face lattice of a polytope, called the sweep polytope. I will present several constructions of the sweep polytope, related to zonotopes, projections of permutahedra and monotone path polytopes of zonotopes. In a second part, I will present sweep oriented matroids, an abstract generalization of this structure in terms of oriented matroids. Besides their intrinsic interest, they also appear in connection to the Generalized Baues Problem for cellular strings of some oriented matroids beyond the realizable case.
Mathieu
Besançon
IOL
Differentiable Optimization & Integration within Differentiable Programming
Abstract.
Differentiable optimization aims at bringing constrained convex optimization to differentiable programming, allowing the computation of solution derivatives with respect to arbitrary input parameters. After an overview of some of the advances in that direction, we will see how to integrate differentiable optimization as a layer in differentiable models and its interplay with automatic differentiation, and conclude with some comparisons with other approaches.
Siddarth
Kannan
Brown University
Symmetries of tropical moduli spaces of curves
Abstract.
I will briefly introduce the moduli space of n-marked stable tropical curves of genus g, and then discuss the combinatorial calculation of its automorphism group. I will conclude by talking about natural follow-up questions to this result, motivated by connections to the Deligne-Mumford moduli stack of marked stable algebraic curves and the mapping class group.
Michael
Anastos
FU Berlin
On a k-matching algorithm and finding k-factors in cores of random graphs
Abstract.
We prove that for k+1 ≥ 3 w.h.p. the (k+1)-core of Gn,p, if non empty, spans a (near) spanning k-regular subgraph. This improves upon a result of Chan and Molloy and completely resolves a conjecture of Bollobas, Kim and Verstraëte. In addition, we show that w.h.p. such a subgraph can be found in linear time. A substantial element of the proof is the analysis of a randomized algorithm for finding k-matchings in random graphs with minimum degree k+1.
Melanie
Koser
A model of low-hysteresis shape-memory alloys: Properties of minimizers
January 2021
Margaret
Bayer
U Kansas
Lattice Polytopes from Schur Polynomials
Abstract.
Associated with a polynomial (in any number of variables) is the Newton polytope, the convex hull of the exponent vectors occurring in the polynomial. Information about Newton polytopes can shed light on families of polynomials. In this talk I report on a study of some combinatorial properties of the Newton polytopes of Schur polynomials, which enumerate certain tableaux. This work was the result of a project at the Graduate Research Workshop in Combinatorics, and is joint with Goeckner, Hong, McAllister, Olsen, Pinckney, Vega and Yip.
Sven
Jäger
TU Berlin
Online Scheduling of Deterministic and Stochastic Jobs on Unrelated Machines
Abstract.
We consider the problem of scheduling jobs on parallel machines so as to minimize the sum of weighted job completion times. We assume that each job must be completely processed on one machine and that the time required may depend arbitrarily on the job and the machine. Furthermore, the jobs arrive online over time. In the first part of the talk, we consider preemptive scheduling of deterministic jobs. I will use dual fitting to prove that a simple greedy algorithm for this problem is 4-competitive. The second part is about non-preemptive scheduling of stochastic jobs. Here a similar greedy algorithm for the assignment of jobs to machines, combined with known policies for scheduling the jobs assigned to each machine (Schulz, 2008), leads to stochastic online policies with performance guarantees depending on an upper bound 𝛥 for the squared coefficients of variation of the random variables. These improve upon the previously best known guarantees (Gupta, Moseley, Uetz, & Xie) when 𝛥 is large enough (for the simplest variant considered when 𝛥 ≥ 0.815). This includes the important case of exponentially distributed processing times.
Julian
Schick
Der Flipgraph von Split und Merge auf Schnyder Woods
Abhishek
Rathod
TU Munich
Parameterized inapproximability of Morse matching
Abstract.
In this talk, we look at the problem of minimizing the number of critical simplices (Min-Morse) from the point of view of inapproximability and parameterized complexity. Let n denote the size of a simplicial complex. We first show that Min-Morse can not be approximated within a factor of 2^{\log^{(1-\epsilon)}n}- \delta, for any \epsilon,\delta>0, unless NP \subseteq QP. Our second result shows that Min-Morse is WP-hard with respect to the standard parameter. Next, we show that Min-Morse with standard parameterization has no FPT-approximation algorithm for any approximation factor \rho, unless WP = FPT. Since the gadgets involved in our reduction are 2-complexes, the above hardness results are applicable for all complexes of dimension \geq 2.
David
Fabian
FU Berlin
The running time of Q3-bootstrap percolation
Abstract.
The bootstrap process of a graph H on a graph G is the sequence (Gi)i≥0, where G0 := G and Gi is obtained from Gi-1 by adding every edge which completes a copy of H. Let MH(n) be the smallest integer such that Gi+1 = Gi for all i ≥ MH(n) and every graph G on n vertices. It was shown by Bollobás, Przykucki, Riordan and Sahasrabudhe and, independently, by Matzke that MK_4(n) = n-3. In 2019 Balogh, Kronenberg, Pokrovskiy and Szabó found a construction that gives MK_r(n) = Θ(n2) for r ≥ 6. In this talk we consider the problem of determining MH(n) when H = Q3 and show that there exist constants c,C > 0 such that cn3/2 ≤ MQ_3(n) ≤ Cn8/5 for all n ∈ N. This is joint work with Patrick Morris and Tibor Szabó.
Michaela
Borzechowski
One-Permutation-Discrete-Contraction is UEOPL-hard
Abstract.
The complexity class Unique End of Potential Line (UEOPL) was introduced in 2018 by Fearnley et al. and contains many interesting search problems. UEOPL captures problems that have either by definition a unique solution, like the Arrival problem, or that are promised to have a unique solution by some property, like the P-Matrix linear complementary problem. Furthermore the problems in UEOPL have the property that the candidate solutions can be interpreted as an exponentially large graph which form a line, i.e. each node has in and out degree at most 1. The solution of each problem is at the end of that line. In 2017, Daskalakis, Tzamos and Zampetakis proved the problem of finding a fixpoint of a contraction map in a continuous space whose existence is guaranteed by the Banach fixed point theorem to be CLS-complete. A discrete version of the contraction problem, called One-Permutation-Discrete-Contraction, is proven to be the first UEOPL-complete problem. This proof is particularly interesting because it is currently the only one of its kind and lays the groundwork for future UEOPL-completeness proofs. This talk will provide an overview of the reduction from the problem Unique-End-of-Potential-Line to One-Permutation-Discrete-Contraction as well as correcting some errors that were done in the original paper.
Jörg
Wolf
Chung-Ang University, Seoul
On the epsilon-regularity condition for viscosity solutions to the Navier-Stokes equations
Felix
Baumann
Zuse-Institut Berlin
What is a multigrid method?
Abstract.
The discretization of PDEs leads to linear systems with a very large number of unknowns. While direct solvers fail due to the large scale, Multigrid Methods provide a powerful solution technique. In this talk we present the core ideas behind the Multigrid Method and discuss mesh-independent convergence and its optimal complexity.
Anastasia
Chavez
UC Davis
Characterizing quotients of positroids
Abstract.
We characterize quotients of specific families of positroids. Positroids are a special class of representable matroids introduced by Postnikov in the study of the nonnegative part of the Grassmannian. Postnikov defined several combinatorial objects that index positroids. In this talk, we make use of one of these objects called a decorated permutation to combinatorially characterize when certain positroids form quotients. Furthermore, we conjecture a general rule for quotients among arbitrary positroids on the same ground set. This is joint work with Carolina Benedetti and Daniel Tamayo
Philipp
Warode
TU Berlin
Parametric Computation of Minimum Cost Flows
Abstract.
Minimum cost flows have numerous applications, e.g. electrical or gas flows or equilibrium flows in traffic networks. In the classical minimum cost flow problem we are given some flow-dependent cost functions on the edges and demand values on the vertices. In this setting we want to find a flow with minimal costs satisfying the demands. However, in practice, demand and supply values are not constant but may vary or be uncertain. Therefore, we want to analyze the minimum cost flows as a function of the demand values. More specifically, we consider a setting where the demands in the network are given as a function depending on a one-dimensional parameter 𝜆. We analyze these functions theoretically and use the insights to develop an output-polynomial algorithm that computes the minimum cost flow function explicitly for piecewise-quadratic edge costs and demands that are piecewise-linear functions of 𝜆. Further, we can generalize this approach to compute approximate solutions for arbitrary convex cost functions. The algorithm is tested on real-world gas and traffic networks. This is joint work with Max Klimm and Per Joachims.
Kaja
Wille
Queue and Stack Layouts of partial orders
Christian
Steinert
RWTH Aachen University
A Diagrammatic Approach to String Polytopes
Abstract.
It is a common principle of mathematics to translate problems from one area of research to another and solve them there. A very particular example of this approach is the study of string polytopes, originally introduced and studied by Berenstein and Zelevinsky as well as Littelmann. Their original motivation stems from representation theory but they also provide fruitful tools in the algebraic geometry of flag varieties and more as studied by Caldero as well as Alexeev and Brion. Although these string polytopes have been around for more than 30 years, results on their combinatorial properties are scarce.After a brief introduction to the history and motivation behind those polytopes, we will present a recent integrality criterion for a special family of string polytopes. For its proof we will explain a diagrammatic way to classify the vertices of a given string polytope. As a corollary one can show that each partial flag variety admits a toric degeneration to a Gorenstein Fano variety.
Raphael
Steiner
TU Berlin
Complete minors via dichromatic number
Abstract.
The dichromatic number of a directed graph is the smallest integer k for which the vertex-set of the graph can be partitioned into k acyclic sets (not spanning a directed cycle). Inspired by the famous Hadwiger conjecture for undirected graphs, several groups of authors have recently studied the containment of complete digraph minors in directed graphs with given dichromatic number. We present a very short argument which improves the existing exponential upper bounds to almost linear bounds by reducing the problem to a recent result of Postle on Hadwiger's conjecture. The talk represents recent joint work with Tamás Mészáros (FU Berlin).
Pavle
Blagojević
Freie Universität Berlin
Ten years in one lecture
Abstract.
Ten years ago, in February 2011, I joined the group of Günter M. Ziegler at Freie Universität Berlin. Now, ten years later, I will show you some of the problems in Geometric and Topological Combinatorics that intrigued us, some of which we managed to solve, and sketch some of the crucial ideas, methods, and the tools we had to develop in order to answer them. We will see how -- work on the Bárány-Larman conjecture on colored point sets in the plane gave birth to the Optimal colored Tverberg theorem, -- the constraint method collected all classical Tverberg type results under one roof and opened a door towards counter-examples to the topological Tverberg conjecture. Furthermore, we will illustrate how the search for convex partitions of a polygon into pieces of equal area and equal perimeter forced us to -- study the topology of the classical configuration spaces, -- construct equivariant cellular models for them, -- prove a new version of an equivariant Goresky-MacPherson formula for complements of arrangements, -- revisit a classical vanishing theorem of Frederick Cohen, and explain why these answers are related to the existence of highly regular embeddings and periodic billiard trajectories. Finally, we will talk about -- equi-partitions of convex bodies by affine hyperplanes, and -- greedy convex partitions of many measures. These results are joint work with, in chronological order, Günter M. Ziegler, Benjamin Matschke, Florian Frick, Albert Haase, Nevena Palić, Günter Rote, and Johanna K. Steinmeyer.
Florian
Frick
Carnegie Mellon University
New applications of the Borsuk--Ulam theorem
Abstract.
The classical Borsuk--Ulam theorem states that any continuous map from the d-sphere to d-space identifies two antipodal points. Over the last 90 years numerous applications of this result across mathematics have been found. I will survey some recent progress, such as results about the structure of zeros of trigonometric polynomials, which are related to convexity properties of circle actions on Euclidean space, a proof of a 1971 conjecture that any closed spatial curve inscribes a parallelogram, and finding well-behaved smooth functions to the unit circle in any closed finite codimension subspace of square-intergrable complex functions.
Anne-Sophie
Sur
NTNU
A variational approximation of fracture in elasto-plastic materials
Alp
Müyesser
FU Berlin
What is an expander graph?
Abstract.
A graph has good expansion if it is sparse yet very well-connected. In this talk, we will discuss several equivalent ways to quantify the expansion of a given graph. Further, we will do a couple of case studies demonstrating why graphs with good expansion are useful in various areas of mathematics.
Juan Camilo
Torres
Universidad de los Andes
An algebraic approach to projective uniqueness with an application to order polytopes
Abstract.
A polytope is said to be projectively unique if it has a single realization up to projective transformations. Projective uniqueness is a geometrically compelling property but is difficult to verify. In this talk, I will present two approaches to projective uniqueness in the literature. One is primarily geometric and is due to McMullen, who showed that certain natural operations on polytopes preserve projective uniqueness. The other is more algebraic and is due to Gouveia, Macchia, Thomas, and Wiebe. They use certain ideals associated to a polytope to verify a property called graphicality that implies projective uniqueness. I will show that McMullen's operations preserve not only projective uniqueness but also graphicality. As an application, I will show that order polytopes from finite ranked posets with no 3-antichain are graphic, and therefore projectively unique. These results were obtained in collaboration with professors Tristram Bogart and Joao Gouveia.
Kartikey
Sharma
IOL
Robust Optimization and Learning
Abstract.
Robust Optimization is a popular paradigm for optimisation under uncertainty. It assumes that the uncertainty is specified by a set and aims to optimise the worst case objective over all elements in the set. However in many problems the uncertainty changes over time as more information comes in. For this project our goal is to develop an adaptive learning algorithm that can simultaneously learn the uncertainty set and solve the optimisation problem. We achieve this by statistical updates and leveraging online learning algorithms. This is joint work with Kevin-Martin Aigner, Kristin Braun, Frauke Liers and Sebastian Pokutta.
Shpresim
Sadiku
IOL
Neural Network Approximation Theory
Abstract.
We review classical and modern results in approximation theory of neural networks. First, the density of neural networks within different function spaces under various assumptions on the activation function is considered. Next, lower and upper bounds on the order of approximation with neural networks are given based on the input dimension, the number of neurons and a parameter quantifying the smoothness of the target function. Lastly, a family of compositional target functions for which the curse of dimensionality can be overcome using deep neural networks is examined.
Raphael
Steiner
Complete minors via dichromatic number
Dhruv
Ranganathan
University of Cambridge
Donaldson-Thomas theory and the secondary polytope
Abstract.
Donaldson-Thomas theory extracts numerical invariants from an algebraic variety X by examining the geometry of the Hilbert schemes of X. While the sister subject of Gromov-Witten theory has had a long and rich interaction with tropical and polyhedral methods, the relationship between the Hilbert scheme and tropical geometry is less developed. I will give an introduction to tropical and logarithmic aspects of Donaldson-Thomas theory, from the point of view of the GKZ secondary polytope construction, and generalizations thereof. This is based on joint work with Davesh Maulik (MIT).
Yannick
Mogge
TU Hamburg
r-cross t-intersecting families via necessary intersection points
Abstract.
Given integers r ≥ 2 and n,t ≥1 we call families ℱ1,...,ℱr ⊆ ℘([n]) r-cross t-intersecting if for all Fi ∈ ℱi, i ∈ [r], we have |⋂i ∈ [r] Fi| ≥ t. We obtain a strong generalisation of the classic Hilton-Milner theorem on cross intersecting families. In particular, we determine the maximum of ∑j ∈ [r] |ℱj| for r-cross t-intersecting families in the cases when these are k-uniform families or arbitrary subfamilies of ℘([n]). Only some special cases of these results had been proved before. We obtain the aforementioned theorems as instances of a more general result that considers measures of r-cross t-intersecting families. This also provides the maximum of ∑j ∈ [r] |ℱj| for families of possibly mixed uniformities k1,...,kr. This is a joint work with Pranshu Gupta, Simón Piga and Bjarne Schülke.
Dante
Luber
TU Berlin
Generalized permutahedra and positive flag Dressians
Abstract.
A well known result relates matroid subdivisions of hypersimplices to tropical linear spaces. We continue the study of subdivisions of generalized permutahedra into cells that are generalized permutahedra. This talk will focus on subdivisions of the regular permutahedron into cells that correspond to flag matroids and intervals in a partial ordering on the symmetric group. This is joint work with Michael Joswig, Georg Loho, and Jorge Olarte
Maria
Dostert
Royal Institute of Technology
Exact semidefinite programming bounds for packing problems
Abstract.
In the first part of the talk, I present how semidefinite programming methods can provide upper bounds for various geometric packing problems, such as kissing numbers, spherical codes, or packings of spheres into a larger sphere. When these bounds are sharp, they give additional information on optimal configurations, that may lead to prove the uniqueness of such packings. For example, we show that the lattice E8 is the unique solution for the kissing number problem on the hemisphere in dimension 8. However, semidefinite programming solvers provide approximate solutions, and some additional work is required to turn them into an exact solution, giving a certificate that the bound is sharp. In the second part of the talk, I explain how, via our rounding procedure, we can obtain an exact rational solution of a semidefinite program from an approximate solution in floating point given by the solver. This is a joined work with David de Laat and Philippe Moustrou.
Sophie
Rehberg
FU Berlin
Combinatorial reciprocity theorems for generalized permutahedra, hypergraphs, and pruned inside-out polytopes
Abstract.
Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of concepts introduced by Beck-Zaslavsky (2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We study the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart-Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar-Ardila (2017) and Billera-Jia-Reiner (2009). Applying this reciprocity theorem to hypergraphic polytopes allows us to give an arguably simpler proof of a recent combinatorial reciprocity theorem for hypergraph colorings by Aval-Karaboghossian-Tanasa (2020). Our proof relies, aside from the reciprocity for generalized permutahedra, only on elementary geometric and combinatorial properties of hypergraphs and their associated polytopes.
Davide
Lofano
TU Berlin
Contractibility vs Collapsibility
Abstract.
Since the beginning of Topology, one of the most used approach to study an object was to triangulate it. Many invariants have been introduced in the years, the two most important surely being homotopy and homology. However, the direct computation of these invariants is infeasible for triangulations with a large number of faces. This is why methods to reduce the number of faces of a complex without changing its homology and homotopy type are particularly important. In this talk, we will focus on various ways to see if a complex is contractible, starting from collapsibility and then trying to generalize the concept.
Felix
Schröder
A characterization of always solvable trees in Lights Out game and the activation numbers of vertices
Alp
Müyesser
FU Berlin
A proof of Ringel's Conjecture
Abstract.
A typical decomposition question asks whether the edges of some graph G can be partitioned into disjoint copies of another graph H. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with n edges packs 2n+1 times into the complete graph K2n+1. Here we present a recent breakthrough result of Montgomery, Pokrovskiy and Sudakov that proves this conjecture for large n.
Alp
Müyesser
Rainbow factors and trees
Abstract.
Generalizing a conjecture of Aharoni, Joos and Kim asked the following intriguing question. Let H be a graph on m edges, and let G_i (1<=i<=m) be a sequence of m graphs on the common vertex set [n]. What is the weakest minimum degree restriction we can impose on each G_i to guarantee a rainbow copy of H? Joos and Kim answered this question when H is a Hamilton cycle or a perfect matching. We provide an asymptotic answer when H is an F-factor, or a spanning tree with maximum degree o(n/log n). This is joint work with Richard Montgomery and Yani Pehova.
Sampada
Kolhatkar
affiliation not specified
Bivariate chromatic polynomials of mixed graphs
Abstract.
For a graph G=(V,E), the chromatic polynomial X_G counts the number of vertex colourings as a function of number of colours. Stanley’s reciprocity theorem connects the chromatic polynomial with the enumeration of acyclic orientations of G. One way to prove the reciprocity result is via the decomposition of chromatic polynomials as the sum of order polynomials over all acyclic orientations. From the Discrete Geometry perspective, the decomposition is as the sum of Ehrhart polynomials through real braid arrangement. Beck, Bogart, and Pham proved the analogue of this reciprocity theorem for the strong chromatic polynomials for mixed graph. Dohmen–Pönitz–Tittmann provided a new two variable generalization of the chromatic polynomial for undirected graphs. We extend this bivariate chromatic polynomial to mixed graphs, provide a deletion-contraction like formula and study the colouring function geometrically via hyperplane arrangements.
December 2020
Jorge
Olarte
RWTH Aachen University
The tropical symplectic Grassmannian
Abstract.
Given a vector space V with a bilinear, alternating, non-degenerate form w, we call a linear subspace L isotropic if every two vectors v and u in L are orthogonal with respect to w, i.e. w(u,v) = 0. The symplectic Grassmannian SpGr(V,k) is the space of all isotropic subspaces of V of dimension k. It is a subvariety of the usual Grassmannian Gr(V,k), and its ideal is generated by the usual Plücker relations plus some additional linear relations called the symplectic relations. Our main object of study is the tropicalization of this space, the tropical symplectic Grassmannian TSpGr_p(2n,k) where 2n is the dimension of V and p is the characteristic of the field. In this talk we show when does the usual basis of SpGr(V,k) form a tropical basis. Then we will look at several examples that exhibit different pathologies, such as why the characteristic of the field makes a difference. We end the talk by proposing several research directions. This is work in progress with George Balla.
Arturo
Merino
TU Berlin
Efficient generation of rectangulations via permutation languages
Abstract.
Generic rectangulations are tilings of the unit square by rectangles such that no 4 rectangles share a corner. Generic rectangulations are a fundamental object in computer science: not only appearing in practice (e.g. in VLSI circuit design) and theory (e.g. being closely related to binary search trees); but also within many other fundamental combinatorial objects such as permutations. In this work, we present a versatile algorithmic framework for exhaustively generating a large variety of different classes of generic rectangulations. Our algorithms work under very mild assumptions and apply to a considerable number of interesting rectangulation classes known from the literature, such as generic rectangulations, diagonal rectangulations, S-equivalent rectangulations, 1-sided/area-universal rectangulations, and their guillotine variants. The resulting generation algorithms are efficient, in most interesting cases even loopless or constant amortized time, i.e., each new rectangulation is generated in constant time in the worst case or on average, respectively. In this talk we will show: - A simple introduction to rectangulations and some of its interesting classes, - Our algorithmic generation framework for rectangulation classes and - Correctness proof (sketch) of the framework by relating rectangulations to permutations. This is joint work with Torsten Mütze.
Raphael
Steiner
Towards the Directed Gyarfas-Sumner Conjecture
Patrick
Morris
FU Berlin
A robust Corrádi-Hajnal Theorem
Abstract.
The Corrádi-Hajnal Theorem states that any graph G with n ∈ 3ℕ vertices and minimum degree δ(G) ≥ 2n/3, contains a triangle factor, i.e. a collection of n/3 pairwise vertex-disjoint triangles. We show that any such G is `robust' with respect to this property, in that almost all (sufficiently dense) subgraphs of G contain a triangle factor. In detail, we define p*(n):=n-2/3 log1/3n and for a graph G and p=p(n), we define Gp to be random sparsification of G: the graph obtained from G keeping each edge independently with probability p. Our main result then states that there exists a C>0 such that for any G satisfying the Corrádi-Hajnal condition and any p=p(n) ≥ Cp*(n), Gp contains a triangle factor with high probability. The minimum degree condition is tight due to the existence of graphs of minimum degree 2n/3 - 1 that do not contain a triangle factor. The bound on the probability is also tight up to the value of C. Indeed, as shown in seminal work of Johansson, Kahn and Vu, p* is the threshold for containing a triangle factor in G(n,p) (that is, taking G=Kn above). Our result can thus be seen as a common strengthening of the theorems of Corrádi-Hajnal and Johansson-Kahn-Vu.This represents joint work with Peter Allen, Julia Böttcher, Jan Corsten, Ewan Davies, Matthew Jenssen, Barnaby Roberts and Jozef Skokan.
Jannik
Peters
Efficiency and Stability in Euclidean Network Design
Abstract.
We study the recently proposed Euclidean Generalized Network Creation Game by Bilò et al.[SPAA 2019] and investigate the creation of (beta,gamma)-networks, which are in beta-approximate Nash equilibrium and have a total cost of at most gamma times the optimal cost. In our model we have n agents corresponding to points in Euclidean space create costly edges among themselves to optimize their centrality in the created network. Our main result is a simple O(n^2)-time algorithm that computes a (beta,beta)-network with low beta for any given set of points. Along the way, we significantly improve several results from Bilò et al. and we asymptotically resolve a conjecture about the Price of Anarchy.
Dante
Luber
Boundary Complexes for Moduli Spaces of Curves
Abstract.
In 2016, Noah Giansiracua showed that a collection of boundary divisors in the moduli space of genus-0 n-pointed curves has nonempty intersection if and only if all pairwise intersections are nonempty. This result is equivalent to showing that the boundary complex associated to such a moduli space is a flag complex. Kyla Quillin extended Giansiracusa's result to most moduli spaces of genus-g n-pointed curves. We give a complete classification of all (g,n) pairs for which the boundary complex is a flag complex.
Matthias
Himmelmann
Generalized Principal Component Analysis for Algebraic Varieties
Abstract.
The Buchberger-Möller algorithm is a famous symbolic method for finding all polynomials that vanish on a point cloud. It has even been extended to noisy samples. However, the resulting variety does not necessarily represent the topological or geometric structure of the data well. By making use of the Vandermonde matrix, it is possible to find polynomials of a prescribed degree vanishing on the samples. As this matrix is severely ill-conditioned, modifications are necessary. By making use of statistical and algebro-geometric techniques, an algorithm for learning a vanishing ideal that represents the data points‘ geometric properties well is presented. It is investigated that this method -- among various other desirable properties -- is more robust against perturbations in the data than the original algorithm.
Niklas
Martensen
HU Berlin
What is Noether's theorem?
Abstract.
Noether's theorem is one of the most important results in classical mechanics. It shows that any continuous symmetry of a mechanical system leads to a conserved quantity such as energy or momentum. In order to understand this result we give a brief introduction into Lagrangian mechanics and show applications of the theorem.
Giulia
Codenotti
Goethe-Universität Frankfurt
Beyond flatness: maximizing width of convex bodies with forbidden subpolytopes
Abstract.
Recently, Averkov, Hofscheier and Nill introduced the so-called generalized flatness constants. These generalize usual flatness by substituting hollow convex bodies with convex bodies avoiding a certain fixed subset up to a chosen class of transformations (for example unimodular transformations followed by real translations). Such convex bodies are called X-free, if X is the subset we avoid. In joint work with Hall and Hofscheier, we prove that inclusion-maximal X-free convex bodies are actually polytopes. Even in dimension 2 and with X the unimodular triangle, the family of such maximal X-free bodies is very rich and complex. However, in this case we are able to determine the flatness constants by other means. The goal of this talk is to present these results starting from basic definitions of width and flatness, and to give plenty of motivation and highlight applications of the generalized flatness constant
Boro
Sofranac
IOL
Accelerating Domain Propagation: an Efficient GPU-Parallel Algorithm over Sparse Matrices
Abstract.
Fast domain propagation of linear constraints has become a crucial component of today's best algorithms and solvers for mixed integer programming and pseudo-boolean optimization to achieve peak solving performance. Irregularities in the form of dynamic algorithmic behaviour, dependency structures, and sparsity patterns in the input data make efficient implementations of domain propagation on GPUs and, more generally, on parallel architectures challenging. This is one of the main reasons why domain propagation in state-of-the-art solvers is single thread only. In this paper, we present a new algorithm for domain propagation which (a) avoids these problems and allows for an efficient implementation on GPUs, and is (b) capable of running propagation rounds entirely on the GPU, without any need for synchronization or communication with the CPU. We present extensive computational results which demonstrate the effectiveness of our approach and show that ample speedups are possible on practically relevant problems: on state-of-the- art GPUs, our geometric mean speed-up for reasonably-large instances is around 10x to 20x and can be as high as 195x on favorably-large instances.
Simona
Boyadzhiyska
FU Berlin
Hyperplane coverings with multiplicities
Abstract.
It is known that the minimum number of hyperplanes needed to cover all points of F2n \ {0} while leaving the origin uncovered is n. In this talk, we will consider a variant of this problem in which the points have to be covered with certain multiplicities. More specifically, we will discuss the problem of determining the smallest number of hyperplanes required to cover all points of F2n \ {0} at least k times while covering the origin at most k-1 times and give tight bounds for certain ranges of the parameters n and k. This talk is based on joint work with Anurag Bishnoi, Shagnik Das, and Tamás M\észáros.
Michael
Joswig
TU Berlin
Moduli of tropical curves - data and algorithms
Abstract.
In joint work with Sarah Brodsky, Ralph Morrison and Bernd Sturmfels in 2015 we computed the moduli of tropical plane curves of genus $\leq 4$. After a brief summary of the mathematical results the focus of this presentation is about the computational methods and the 400MB of data produced. We will see how this is being used for further research (ongoing joint work with Dominic Bunnett). On the way we will touch upon general questions concerning data in mathematics.
Filippos
Christodoulou
Design and Analysis of Combinatorial Problems in Random Intersection Graphs
Kemal
Rose
Hussein
Houdrouge
Subquadratic High-Dimensional Hierarchical Clustering
Deniz
Kus
Ruhr-University Bochum
Polytope parametrization of bases of representations
Abstract.
The representation theory of finite-dimensional complex Lie algebras is well-understood. There are dimension formulas, character Formulas and many parametrizations of their bases, e.g. using Young tableaux or GT pattern. The goal of this talk is to overview some known results and to present new parametrizations by convex polytopes. At the end of the talk we will discuss how these results can be extended to the representation theory of Lie superalgebras or quantum groups which is less understood
Alejandro
Carderera
Georgia Tech
Local Acceleration of Conditional Gradients
Abstract.
Conditional gradients (CG) constitute a class of projection-free first-order algorithms for smooth convex optimization. As such, they are frequently used in solving smooth convex optimization problems over polytopes, for which the computational cost of projections is prohibitive. We will discuss two topics in this talk. First of all, CG algorithms do not enjoy the optimal convergence rates achieved by projection-based first-order accelerated methods; achieving such globally-accelerated rates is information-theoretically impossible. To address this issue, we present Locally Accelerated Conditional Gradients -- an algorithmic framework that couples accelerated steps with conditional gradient steps to achieve local acceleration on smooth strongly convex problems. Our approach does not require projections onto the feasible set, but only on (typically low-dimensional) simplices, thus keeping the computational cost of projections at bay. Further, it achieves optimal accelerated local convergence. The second topic in this talk deals with the setting where first and second-order information about the function is costly to compute, and so we would like to make as much primal progress as possible per oracle call. Constrained second-order convex optimization algorithms are the method of choice when a high accuracy solution to a problem is needed, due to their local quadratic convergence. These algorithms require the solution of a constrained quadratic subproblem at every iteration. In this setting, we present the Second-Order Conditional Gradient Sliding (SOCGS) algorithm, which uses a projection-free algorithm to solve the constrained quadratic subproblems inexactly and uses inexact Hessian oracles (subject to an accuracy requirement). When the feasible region is a polytope, and a lower bound on the primal gap is known, the algorithm converges quadratically in primal gap after a finite number of linearly convergent iterations. Once in the quadratic regime the SOCGS algorithm requires O(log log (1/ε)) first-order and inexact Hessian oracle calls and O(log (1 / ε) ⋅ log log (1/ε)) linear minimization oracle calls to achieve an ε-optimal solution.
Ander
Lamaison
Masaryk University Brno
Quasirandom Latin squares
Abstract.
In this talk we present the notion of quasirandom Latin squares, analogous to similar definitions in other discrete structures such as graphs and permutations. Proving a conjecture of Garbe et al., we will show that a sequence of Latin squares is quasirandom if and only if every $2\times 3$ pattern has density $1/720+o(1)$, and that $2\times 2$ or $1\times k$ patterns are not enough to guarantee quasirandomness. The main tool that we will use will be Latinons, a limit structure for Latin squares introduced by Garbe et al. Joint work with Jacob Cooper, Dan Král’ and Samuel Mohr.
Rahul
Jain
Space-efficient algorithm for the grid graph reachability
Melody
Chan
Brown University
Tropical abelian varieties
Abstract.
I'll give an introduction to the moduli space of tropical abelian varieties, assuming no background in tropical geometry. Lots of different combinatorics arises, including the beautiful century-old combinatorics of Voronoi reduction theory, perfect quadratic forms, regular matroids, and metric graphs. On the geometric side, it relates to toroidal compactifications of the classical moduli space A_g of abelian varieties. I'll explain how, and, time permitting, I'll report on work-in-progress in which we use tropical techniques to find new rational cohomology classes in A_g in a previously inaccessible range. Joint work with Madeline Brandt, Juliette Bruce, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.
November 2020
Stefan
Mengel
CNRS
A Biased Introduction to Decomposable Negation Normal Forms
Abstract.
Decomposable Negation Normal Forms (DNNF) are a class of Boolean circuits first introduced by Darwiche in 2001 in the context of artificial intelligence. Since then they have found applications as a flexible framework for encoding Boolean functions in other areas of computer science like theoretical computer science and database theory. In this talk, I will introduce DNNF, discuss some uses and sketch how one can show bounds on their size.
Ilse
Fischer
U Wien
Where are the bijections? Plane Partitions and Alternating Sign Matrices
Abstract.
For about 35 years, combinatorialists have not been able to find bijections between three classes of objects that are all counted by the product formula $\prod\limits_{i=0}^{n-1} \frac{(3i+1)!}{(n+i)!}$. These objects are $n \times n$ alternating sign matrices, totally symmetric self-complementary plane partitions in a $2n \times 2n \times 2n$ box, and cyclically symmetric rhombus tilings of a hexagon of side lengths $n+2,n,n+2,n,n+2,n$ with a central hole of size $2$. Recently, we have added a fourth class of objects to this list, namely alternating sign triangles, and, even more recently, we have extended this class to alternating sign trapezoids, and have shown that they are equinumerous with cyclically symmetric rhombus tilings of a hexagon with a central hole of size $k$. At about the same time, we have constructed the first (complicated) bijection relating alternating sign matrices to cyclically symmetric rhombus tilings with a central hole of size two. In my talk, I shall tell the fascinating story of this search for explicit bijections. I will also report on joint work with Arvind Ayyer, Roger Behrend and Matjaz Konvalinka.
Rico
Raber
TU Berlin
Multidimensional Packing under Convex Quadratic Constraints
Abstract.
We consider a general class of binary packing problems with multiple convex quadratic knapsack constraints. We present three constant-factor approximation algorithms based upon three different algorithmic techniques: (1) a rounding technique tailored to a convex relaxation in conjunction with a non-convex relaxation; (2) a greedy strategy; and (3) a randomized rounding method. The practical performance is tested by a computational study of the empirical approximation of these algorithms for problem instances arising in the context of real-world gas transport networks.
Stefan
Felsner
Crossings in star-simple drawings
Anurag
Bishnoi
TU Delft
Lower bounds for diagonal Ramsey numbers
Abstract.
In a recent breakthrough Conlon and Ferber gave an exponential improvement in the lower bounds on diagonal Ramsey number R(t, t, \dots, t), when the number of colours is at least 3. We discuss their construction, along with the further improvement of Wigderson, and the finite geometry behind it.
Kemal
Rose
TU Berlin
Certifying roots of polynomial systems using interval arithmetic
Abstract.
We report on an implementation of the Krawczyk method which establishes interval arithmetic as a practical tool for certification in numerical algebraic geometry. Our software HomotopyContinuation.jl now has a built-in function certify, which proves the correctness of an isolated solution to a square system of polynomial equations. This implementation dramatically outperforms earlier approaches to certification.
Hailun
Zheng
U Copenhagen
The stresses on centrally symmetric complexes
Abstract.
A simplicial complex is called centrally symmetric, or cs, if it possesses a free simplicial involution. In 1987, Stanley conjectured that if a cs Cohen-Macaulay (d-1)-dimensional simplicial complex satisfies h_i = {d choose i} for some i > 0, then h_j = {d choose j} for all j > i. More recently, a similar conjecture on the g-numbers of cs simplicial d-polytopes was proposed by Klee, Nevo, Novik and Zheng. In this talk, I will give a brief introduction to the theory of stress spaces developed by Lee. Then I will prove the above two conjectures using this machinery. This is joint work with Isabella Novik.
Elias
Wirth
IOL
Learning Relations From Data With Conditional Gradients
Abstract.
The learning of relations, dynamics, and governing equations from (sampled) data is a prevalent problem in various scientific fields. In the absence of noise, this task can often be solved using the Buchberger-Möller Algorithm, see Möller and Buchberger (1982). In general, however, when we do not have direct access to aforementioned equations, we have access to them indirectly only via noisy sampling. Known methods which extract relations from noisy data are the Approximate Vanishing Ideal Algorithm (AVI) and Vanishing Component Analysis (VCA), see Heldt et al. (2009) and Livni et al. (2013), respectively. These algorithms are often computationally inefficient and only allow for restricted recovery of relations from data. We present a new AVI algorithm and two variants utilizing Conditional Gradient Descent or the Frank-Wolfe Algorithm, see, e.g., Frank and Wolfe (1956) or Levitin and Polyak (1966). Our method is more general, easily regularizable, and can also extract non-homogeneous relations. We then present the efficacy and versatility of our approach for three learning problems: reconstruction of governing equations from data, classification, and regression.
Charlotte
Lenz
Zeichnungen vollständiger Graphen und ihre Gap-Planarität
Benjamin
Schröter
KTH Stockholm
Reconstructing Matroid Polytopes
Abstract.
This talk deals with two fundamental objects of discrete mathematics that are closely related - (convex) polytopes and matroids. Both appear in many areas of mathematics, e.g., algebraic geometry, topology and optimization.A classical question in polyhedral combinatorics is 'Does the vertex-edge graph of a d dimensional polytope determine its face lattice?'. In general the answer is no, but a famous result of Blind and Mani, and later Kalai is a positive answer of that question for simple polytopes. In my talk I discuss this reconstructability question for the special class of matroid (base) polytopes. Matroids encode an abstract version of dependency and independency, and thus generalize graphs, point configurations in vector spaces and algebraic extensions of fields. Maximal independent sets in a matroid satisfy a basis-exchange axiom. The exchanges correspond to edges in the basis-exchange graph which is the vertex-edge graph of the matroid polytope.This is joint work with Guillermo Pineda-Villavicencio
Raphael
Steiner
TU Berlin
Disjoint cycles of distinct lengths in directed graphs of large connectivity or large minimum degree
Abstract.
A classical result by Thomassen from 1983 states that for every k ≥1 there is an integer f(k) such that every digraph with minimum out-degree f(k) contains k vertex-disjoint directed cycles. The known proof methods for Thomassen's result however do not give any information concerning the lengths of the k disjoint cycles. In undirected graphs, it is true that sufficiently large minimum degree guarantess k disjoint cycles of equal lengths, as shown by Alon (1996), and also k disjoint cycles of distinct lengths, as shown by Bensmail et al (2017). Alon also gave a construction showing that there are digraphs of unbounded minimum out- and in-degree containing no k disjoint directed cycles of the same length. In 2014 Lichiardopol made the following conjecture: For every k there exists an integer g(k) such that every digraph of minimum out-degree g(k) contains k vertex-disjoint directed cycles of pairwise distinct lengths. This conjecture seems quite challenging, as already the existence of g(3) is unknown. For general k the conjecture is only proved in some special cases such as tournaments and regular digraphs by Bensmail et al. (2017). In my talk I will present some recent ideas for finding disjoint cycles of distinct lengths in digraphs based on a new tool from structural digraph theory. I have the following partial results. For every k there exists an integer s(k) such that every strongly s(k)-connected digraph contains k vertex-disjoint directed cycles of pairwise distinct lengths. There exists an integer K such that every digraph of minimum out- and in-degree at least K contains 3 vertex-disjoint directed cycles of pairwise distinct lengths.
Stephan
Wäldchen
TU Berlin
Understanding Neural Network Decisions is Hard - From Probabilistic Prime Implicants to Arc Bending
Abstract.
This talk is a basic overview over my PhD projects. I focus on a project on making neural networks interpretable. If a neural network outputs a classification result for a high-dimensional input, we would like to know whether a small set of the input parameters was already decisive for the classification. This problem is formalised by introducing a probabilistic variant of prime implicants. We show that this problem is hard for the class NP<sup>PP</sup> and remains NP-hard for any non-trivial approximation. We present a heuristic solution strategy for real-world applications that is build on Assumed Density Filtering: A technique to propagate multivariate normal distributions through neural networks. To justify the use of this heuristic we show that in fact no sensible family of distributions can be invariant under ReLu-neural network layers.
Lisa
Sauermann
Institute for Advanced Study, Princeton
On the extension complexity of low-dimensional polytopes
Abstract.
It is sometimes possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope P is defined to be the minimum number of facets in a (possibly higher-dimensional) polytope from which P can be obtained as a (linear) projection. In this talk, we discuss some results on the extension complexity of random d-dimensional polytopes (obtained as convex hulls of random points on either on the unit sphere or in the unit ball), and on the extension complexity of polygons with all vertices on a common circle. Joint work with Matthew Kwan and Yufei Zhao.
Daniel
Tamayo Jiménez
Université Paris-Saclay
Permutree sorting, weak order quotients, and automata
Abstract.
We define permutree sorting which generalizes the stack sorting and coxeter sorting algorithms due to Knuth and Reading respectively. Given two disjoint subsets U and D of {2,...,n}, (U, D)-permutree sorting consists of an algorithm that fails for a permutation if and only if there are integers i<j<k in [n] such that the permutation contains the subword jki (resp. kij) if j is in U (resp. in D). We present this algorithm through automata that read reduced expressions and accept only those that form a special structure within the weak order. This is joint work with Vincent Pilaud and Viviane Pons.
Khai Van
Tran
TU Berlin
Improved Bounds on the Competitive Ratio for Symmetric Rendezvous-on-the-Line with Unknown Initial Distance
Abstract.
During the COGA Retreat 2019 we started research on the Symmetric Rendezvous-on-the-Line-Problem with Unknown Initial Distance: Two robots are placed on the real line with unknown initial distance. They do not know the location of the other robot and do not necessarily share the same sense of “left” and “right” on the line. They move with constant speed over the line and try to achieve rendez-vous using the same (randomized) search strategy in as little time as possible. In a March 2020 talk (given in the COGA Research Seminar) we presented a family of search strategies that performed well in simulations without being able to determine their exact competitive ratio. During this talk we will present a novel family of search strategies whose precise competitive ratio can be computed using generating functions. This is WIP by Guillaume, Khai Van, Martin, and Max.
Thomas
Kerdreux
IOL
The artification of the so-called A.I. art and the creative industry
Abstract.
In Computer Vision, algorithms are now capable of "intelligent" automatic processing of the images. And these possibilities have been recently recycled in many creative processes by A.I. artists, which outputs sometimes reached hammer prizes in art auctions. This emergence of A.I. art looks like an *artification*, i.e., a societal process by which a practice becomes commonly acknowledged as an art form. Art history abounds with such examples, e.g., the transitioning from craftmanship to solo painters in the Renaissance. Hence, rather than assessing the artist's narratives, this point of view allows understanding A.I. art as a figurehead of the broader ecosystem of the creative industry. The latter indeed highly benefited from recent advances in Computer Vision. We discuss here the possible intertwinements between A.I. art, the use of Computer Vision in the next generation of creative software, the creative industry, and broader societal consequences. This is ongoing work with Louis Thiry and Léa Saint-Raymond at Ecole Normale Supérieure in Paris.
Tibor
Szabó
FU Berlin
Mader-perfect digraphs
Abstract.
We investigate the relationship of dichromatic number and subdivision containment in digraphs. We call a digraph Mader-perfect if for every (induced) subdigraph F, any digraph of dichromatic number at least v(F) contains an F-subdivision. We show that, among others, arbitrary orientated cycles, bioriented trees, and tournaments on four vertices are Mader-perfect. The first result settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura, and Thomassé, while the last one extends Dirac's Theorem about 4-chromatic graphs containing a K4-subdivision to directed graphs. The talk represents joint work with Lior Gishboliner and Raphael Steiner.
Florian
Oberender
Drawings of acyclic digraphs: Algorithmic aspects
Nick
Early
Perimeter Institute for Theoretical Physics
Combinatorial Geometries: From Polyhedral Subdivisions to Scattering Amplitudes
Abstract.
In 1948, Richard Feynman introduced a formalism in Quantum Field Theory to organize the expansion of a scattering amplitude as a sum of elementary rational functions, labeled by graphs. These graphs, now called Feynman diagrams, specify which singularities of the amplitude are compatible and can appear simultaneously.In the past year this question of compatibility has been given the following interpretation: given two matroid subdivisions of the second hypersimplex $\Delta_{2,n}$, when is their common refinement also a matroid subdivision?Motivated in part by recent joint works with Cachazo, and Cachazo Guevara and Mizera, I will discuss how basic constructions in tropical geometry allow this question to be reformulated -- and generalized to arbitrary hypersimplices $\Delta_{k,n}$ -- as a statement about weakly separated collections, in the case when the maximal cells in the subdivisions are positroid polytopes.
Alex
Goeßmann
Uniform Concentration of Random Tensors
Viviane
Pons
Paris-Sud University
Permutrees
Abstract.
We present the construction of permutrees: a combinatorial family which includes permutations, binary trees and binary words. We show how the permutree lattices describe certain quotient of the weak order while permutreehedra are generalized permutahedron.
Christoph
Hertrich
TU Berlin
Computing the Maximum Function with ReLU Neural Networks
Abstract.
This talk deals with an open problem concerning the expressivity of feedforward neural networks (NNs) with rectified linear unit (ReLU) activations. While it has been shown that every continuous, piecewise linear (CPWL) function can be exactly represented by such an NN with logarithmic depth (with respect to the input dimension), it is still unknown whether there exists any CPWL function that cannot be represented with three layers. Based on a yet unproven theoretical assumption, we show a computer-aided proof of non-existence of a 3-layer ReLU NN that computes the maximum of five scalar input numbers. We achieve this partial result by converting the existence question into a linear programming feasibility problem. This is joint work in progress with Amitabh Basu, Marco Di Summa, and Martin Skutella.
Jonathan
Rollin
Edge-Minimum Saturated k-Planar Drawings (of Multigraphs)
Dominic
Bunnett
TU Berlin
Embedded stacky fans and tropical moduli spaces
Abstract.
Stacky fans are orbifold objects in discrete geometry with a rich combinatorial structure. They arise naturally as the moduli spaces of tropical curves. Explicitly, a stacky fan is built from polyhedral cones quotiented out by finite groups.We study stacky fans embedded in others and offer algorithms to compute their relative homology. We then apply this to specific loci of tropically smooth plane curves embedded in the moduli of genus 3 tropical curves. This is joint ongoing work with Michael Joswig.
Alp
Müyesser
FU Berlin
A rainbow version of Hajnal-Szemerédi Theorem
Abstract.
Let G1, … , Gn/k be a collection of graphs on the same vertex set, say [n], such that each graph has minimum degree (1-1/k+o(1))n. We show that [n] can then be tiled with k-cliques, each clique coming from a distinct graph. (Here, k is a constant and n is sufficiently large.) When all the graphs are identical, this result reduces to the celebrated Hajnal-Szemerédi Theorem. This extends a result of Joos and Kim, who considered the problem when k=2, and has applications to the study of cooperative colorings, a notion of graph coloring introduced by Aharoni, Holzman, Howard, and Sprüssel. This is joint work with Yani Pehova.
Markus
Bläser
Universität Saarbrücken
Irreversibility of tensors of minimal border rank and barriers for fast matrix multiplication
Abstract.
Determining the asymptotic algebraic complexity of matrix multiplication is a central problem in algebraic complexity theory. The best upper bounds on the so-called exponent of matrix multiplication if obtained by starting with an "efficient" tensor, taking a high power and degenerating a matrix multiplication out of it. In the recent years, several so-called barrier results have been established. A barrier result shows a lower bound on the best upper bound for the exponent of matrix multiplication that can be obtained by a certain restriction starting with a certain tensor. We prove the following barrier over the complex numbers: Starting with a tensor of minimal border rank satisfying a certain genericity condition, except for the diagonal tensor, it is impossible to prove ω = 2 using arbitrary restrictions. This is astonishing since the tensors of minimal border rank look like the most natural candidates for designing fast matrix multiplication algorithms. We prove this by showing that all of these tensors are irreversible, using a structural characterisation of these tensors. Joint work with Vladimir Lysikov.
October 2020
Karin
Schaller
FU Berlin
Mirror symmetry constructions for quasi-smooth Calabi-Yau hypersurfaces in weighted projective spaces
Abstract.
We consider a general combinatorial framework for constructing mirrors of quasi-smooth Calabi-Yau hypersurfaces defined by weighted homogeneous polynomials. Our mirror construction shows how to obtain mirrors being Calabi-Yau compactifications of non-degenerate affine hypersurfaces associated to certain Newton polytopes. This talk is based on joint work with Victor Batyrev.
Raphael
Steiner
3-Coloring Arrangements of Great-Circles: Partial Results
Rainer
Sinn
University of Leipzig
Realization spaces of polytopes
Abstract.
We study a simple model of the realization space of a polytope that lends itself to the application of the implicit function theorem. We explore how far this basic tool can carry us. Combined with quite elementary combinatorial arguments, this recovers most of the known results of “nice” realization spaces.
Cyrille
Combettes
Georgia Tech
Frank-Wolfe with New and Practical Descent Directions
Abstract.
The Frank-Wolfe algorithm has become a popular constrained optimization algorithm for it is simple and projection-free, and it has been successfully applied to a variety of real-world problems. Its main drawback however lies in its convergence rate, which can be excessively slow due to naive descent directions. In this talk, we provide a brief review and propose two methods significantly improving its performance: one for minimizing a convex objective and one for minimizing a (non)convex finite-sum objective. The first method speeds up the Frank-Wolfe algorithm by better aligning the descent direction with that of the negative gradient via a matching pursuit-style subroutine. Although the idea is reasonably natural, its advantage over the state-of-the-art Frank-Wolfe algorithms is quite considerable. The second method blends stochastic Frank-Wolfe algorithms with adaptive gradients, which set entry-wise step-sizes that automatically adjust to the geometry of the problem. Computational experiments on convex and nonconvex objectives demonstrate the advantage of our approach.
Sophie
Spirkl
University of Waterloo
k-coloring graphs with forbidden induced subgraphs
Abstract.
A k-coloring of a graph G is a function c that assigns an integer between 1 and k to every vertex of G such that c(u) is not equal to c(v) for every edge uv of G. Deciding, given a graph G, whether G has a k-coloring, is NP-hard for all k at least 3. In this talk, we will consider what happens when we restrict the input, that is: For a graph H and integer k, what is the complexity of deciding if a given graph G with no induced subgraph isomorphic to H has a k-coloring? We know the answer for many pairs H and k. Possibly the most interesting open cases are those in which H is a path; I will talk about recent results and open questions related to this.
Matthias
Schymura
BTU Cottbus-Senftberg
Computing the covering radius of a polytope with an application to Lonely Runners
Abstract.
The covering radius of a polytope is the minimal dilation factor that is needed for the dilated polytope to cover the whole space by lattice translations. This classical parameter has versatile theoretical merits in the Geometry of Numbers, Integer Programming, the Classification of Lattice Polytopes, and Cryptography. However, given your favorite polytope it's a hard task to compute its covering radius. The only prior algorithm is due to Kannan (1992), but it is hardly implementable and has a time complexity that involves a double exponentiation of the input. We develop an easily implementable and more efficient procedure based on the concept and structure of so-called last-covered points. In the talk I will explain the main ideas behind the algorithm and its analysis. I also discuss our initial motivation drawn from a variant of the Lonely Runner Conjecture, which can be formulated as the problem of bounding the covering radius of certain lattice zonotopes. This is based on joint work with Jana Csolvjecsek, Romanos Malikiosis, and Márton Naszódi.
Federico
Castillo
MPI Leipzig
The space of Minkowski Summands
Abstract.
Given a polytope P we study the set of all possible Minkowski summands. Being a Minkowski summand is equivalent to normal (or strongly combinatorial) equivalence. Different points of views provide different looking (but equivalent) parametrizations of the space of Minkowski summands as a polyhedral cone. We will go over several examples like polygons, permutohedra, cubes, and more.
Tillmann
Miltzow
Utrecht University
A practical algorithm with performance guarantees for the art-gallery problem
Abstract.
Given a closed simple polygon P, we say two points p,q see each other if the segment pq is fully contained in P. The art gallery problem seeks a minimum size set G⊂P of guards that sees P completely. The only known algorithm to solve the art gallery problem exactly is attributed to Sharir and uses algebraic methods. As the art gallery problem is ∃R-complete, it seems impossible to avoid algebraic methods without additional assumptions. To circumvent this problem, we introduce the notion of vision stability. In order to describe vision stability, consider an enhanced guard that can see "around the corner" by an angle of δ or a diminished guard whose vision is "blocked" by an angle δ by reflex vertices. A polygon P has vision stability δ if the optimal number of enhanced guards to guard P is the same as the optimal number of diminished guards to guard P. We will argue that most relevant polygons are vision-stable. We describe a one-shot algorithm that computes an optimal guard set for vision-stable polygons using polynomial time besides solving one integer program. We implemented an iterative version of the vision-stable algorithm. Its practical performance is slower, but comparable to other state-of-the-art algorithms. Our iterative algorithm is a variation of the one-shot algorithm. It delays some steps and only computes them only when deemed necessary. Given a chord c of a polygon, we denote by n(c) the number of vertices visible from c. The chord-width of a polygon is the maximum n(c) over all possible chords c. The set of vision-stable polygons admits an FPT algorithm when parametrized by the chord-width. Furthermore, the one-shot algorithm runs in FPT time when parameterized by the number of reflex vertices. Joint work by Simon Hengeveld & Tillmann Miltzow.
Mikkel
Abrahamsen
University of Copenhagen
A framework for ∃R-completeness of two-dimensional packing problems
Abstract.
We show that many natural two-dimensional packing problems are algorithmically equivalent to finding real roots of multivariate polynomials. A two-dimensional packing problem is defined by the type of pieces, containers, and motions that are allowed. The aim is to decide if a given set of pieces can be placed inside a given container. The pieces must be placed so that they are pairwise interior-disjoint, and only motions of the allowed type can be used to move them there. We establish a framework which enables us to show that for many combinations of allowed pieces, containers, and motions, the resulting problem is ∃R-complete. This means that the problem is equivalent (under polynomial time reductions) to deciding whether a given system of polynomial equations and inequalities with integer coefficients has a real solution. We consider packing problems where only translations are allowed as the motions, and problems where arbitrary rigid motions are allowed, i.e., both translations and rotations. When rotations are allowed, we show that the following combinations of allowed pieces and containers are ∃R-complete: - simple polygons, each of which has at most 8 corners, into a square, - convex objects bounded by line segments and hyperbolic curves into a square, - convex polygons into a container bounded by line segments and hyperbolic curves. Restricted to translations, we show that the following combinations are ∃R-complete: - objects bounded by segments and hyperbolic curves into a square, - convex polygons into a container bounded by segments and hyperbolic curves. Joint work by Mikkel Abrahamsen, Tillmann Miltzow, and Nadja Seiferth. The paper has been accepted for FOCS.
Hannah
Markwig
U Tübingen
Curve counting and tropical geometry
Abstract.
The talk introduces tropical geometry as a tool for enumerative geometry. In enumerative geometry, we study counts of curves satisfying certain conditions. Particularly, we consider generating series for important enumerative invariants. Some of them appear in the context of mirror symmetry and can be related to Feynman integrals. This can be used to study their properties such as quasimodularity.
Lamar
Chidiac
A lower bound of the number of independent hyperplanes in oriented paving matroids
Leonid
Monin
University of Bristol
Inversion of matrices, ML-degrees and the space of complete quadrics
Abstract.
In my talk I will discuss the following questions: (1) How many quadrics go through *k* generic points and are tangent to *m * generic hyperplanes? (2) What is the maximum likelihood degree of a general linear concentration model? (3) What is the degree of the variety *L-1* obtained as the closure of the set of inverses of matrices from a generic linear subspace *L* of Sym*n*? In fact, all three questions are equivalent! First I will briefly explain why this is the case, and then I will describe several approaches to the above questions and possible answers they lead to. This is joint work in progress with Laurent Manivel, Mateusz Michalek, Tim Seynnaeve, Martin Vodicka, Andrzej Weber, and Jaroslaw A. Wisniewski.
Sebastian
Ortmanns
Komplexität und Berechenbarkeit von robusten Schnitten in Graphen
Leonie
Selbach
Partition algorithms for weighted cactus graphs
Karin
Schaller
FU
Mirror symmetry constructions for quasi-smooth Calabi-Yau hypersurfaces in weighted projective spaces
Abstract.
We consider a general combinatorial framework for constructing mirrors of quasi-smooth Calabi-Yau hypersurfaces defined by weighted homogeneous polynomials. Our mirror construction shows how to obtain mirrors being Calabi-Yau compactifications of non-degenerate affine hypersurfaces associated to certain Newton polytopes. This talk is based on joint work with Victor Batyrev.
Felix
Schröder
Crossing-critical graphs, the bounded degree conjecture and the number 13
September 2020
Lukas
Kühne
Hebrew Universiy of Jerusalem
The Resonance Arrangement
Abstract.
The resonance arrangement is the arrangement of hyperplanes which has all nonzero 0/1-vectors in R^n as normal vectors. It is also called the all-subsets arrangement. Its chambers appear in algebraic geometry, in mathematical physics and as maximal unbalanced families in economics.In this talk, I will present a universality result of the resonance arrangement. Subsequently, I will report on partial progress on counting its number of chambers. Along the way, I will review some basics of the combinatorics of general hyperplane arrangements.
Johanna
Wiehe
The chromatic polynomial of a digraph
Maximilian
Wittmann
On deletion-perfect families of permutations
Winfried
Hochstättler
Computing Shapley values and elements from the core
August 2020
William
Trotter
Towards a Proof of the Removable Pair Conjecture
Jaeho
Shin
Seoul National University
Polytropes and Tropical Linear Spaces
Abstract.
A polytrope is a convex polytope that is expressed as the tropical convex hull of a finite number of points. It is well known that every bounded cell of a tropical linear space is a polytrope, and its converse statement has been a conjecture. We develop an elementary approach to the relationship between tropical convexity and tropical linearity, and show that the conjecture holds in dimension up to 3 and fails in every higher dimension. Thus, tropical convexity is strictly bigger than tropical linearity.
Stefan
Felsner
Not all planar graphs are in PURE-4-DIR
Martin
Balko
On ordered Ramsey numbers of tripartite 3-uniform hypergraphs
July 2020
Yue
Ren
Swansea University
Sophia
Elia
FU Berlin
and
Marie
Brandenburg
FU Berlin
Multivariate volume, Ehrhart and h*-polynomials of polytropes
Abstract.
In this talk we discuss polytropes, tropical polytopes that are also classically convex, and their related polynomials, taking time to review all necessary background material. We describe methods stemming from algebraic geometry and Ehrhart theory for computing multivariate versions of volume, Ehrhart, and h*-polynomials of lattice polytropes. Finally, we touch on the combinatorics of the coefficients of the volume polynomials.
Florian
Kohl
Aalto University
Unconditional Reflexive Polytopes
Abstract.
A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. In this talk, we investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties. In particular, we characterize unconditional reflexive polytopes in terms of perfect graphs. As a prime example, we study a type-B analogue of the Birkhoff polytope. This talk is based on joint work with McCabe Olsen and Raman Sanyal.
Benjamin Aram
Berendsohn
FU Berlin
Geometric group testing
Abstract.
Group testing is concerned with identifying t defective items in a set of m items, where each test reports whether a specific subset of items contains at least one defective. In non-adaptive group testing, the subsets to be tested are fixed in advance. By testing multiple items at once, the required number of tests can be significantly smaller than m. In fact, if t is constant, the optimal number of (non-adaptive) tests is known to be Θ(log m). We consider the problem of non-adaptive group testing in a geometric setting, where the items are points in the plane and the tests are rectangles. We present upper and lower bounds on the required number of tests under this geometric constraint. In contrast to the general, combinatorial case, the bounds in our geometric setting are polynomial in m. Joint work with László Kozma.
Tien
Tran-Ngoc
HU Berlin
GPU computing in MATLAB for a higher-order method
Amitabh
Basu
Johns Hopkins University Baltimore
Complexity of cutting plane and branch-and-bound algorithms
Abstract.
We present some results on the theoretical complexity of branch-and-bound (BB) and cutting plane (CP) algorithms for mixed-integer optimization. In the first part of the talk, we study the relative efficiency of BB and CP, when both are based on the same family of disjunctions. We focus on variables disjunctions and split disjunctions. We extend a result of Dash to the nonlinear setting which shows that for convex 0/1 problems, CP does at least as well as BB, with variable disjunctions. We sharpen this by showing that there are instances of the stable set problem where CP does exponentially better than BB. We further show that if one moves away from 0/1 polytopes, this advantage of CP over BB disappears; there are examples where BB finishes in O(1) time, but CP takes infinitely long to prove optimality, and exponentially long to get to arbitrarily close to the optimal value (for variable disjunctions). Finally, we show that if the dimension is considered a fixed constant, then the situation reverses and BB does at least as well as CP (up to a polynomial blow up). In the second part of the talk, we will discuss the conjecture that the split closure has polynomial complexity in fixed dimension, which has remained open for a while now even in 2 dimensions. We settle it affirmatively in two dimensions, and complement it with a polynomial time pure cutting plane algorithm for 2 dimensional IPs based on split cuts.
Jörg
Wolf
Partial regularity for viscous solutions to the Naiver-Stokes equations
Jürgen
Herzog
Universität Duisburg-Essen
Sortable sets and sortable simplicial complexes
Abstract.
Sortable sets of monomials is a concept which was introduced by Bernd Sturmfels. Toric algebras whose monomial generators are sortable have the nice property that their defining toric ideal has a quadratic Gröbner basis and squarefree initial ideal, and this has remarkable consequences for the toric algebra and also for the ideal generated by this sortable set of monomials. We will also introduce sortable simplicial complexes and show that the proper interval graphs are precisely the graphs whose independence complex is sortable. My lecture is based on joint papers with my coauthors Viviana Ene, Takayuki Hibi, Fahimeh Khosh-Ahang, Somayeh Moradi, Masoomeh Rahmbeigi, Marius Vladoiu, Ayesha Asloob Qureshi and Guangjun Zhu.
Torsten
Mütze
On flips in planar matchings
Michael
Anastos
FU Berlin
Sampling Colorings of Hypergraphs
Abstract.
The talk will consist of two parts, both concerning sampling colorings of (random) hypergraphs. At the first part we will study an MCMC algorithm for sampling a (near) uniform q-coloring of a simple k-uniform hypegraph with n vertices and maximum degree D. Here q > max{ C1(k) log n, C2(k) D1/(k − 1) }. For the second part we will let Wq denote the set of proper q-colorings of the random graph G(n,m), m = dn/2 and let Hq be the graph with vertex set Wq where two vertices are connected iff the corresponding proper colorings differ in a single vertex. We will show that for sufficiently large d, if q > (1+o(1)) d/log d then Hq is connected, providing an asymptotic matching upper bound to the lower bound given by Achlioptas and Coja-Oghlan. We then extend this result to random hypergraphs. This talk is based on joint work with Alan M. Frieze.
Georgi
Mitsov
HU Berlin
Elements of stability analysis for the heat equation
Stephane
Gaubert
INRIA and CMAP, Ecole polytechnique, CNRS
Understanding and monitoring the evolution of the Covid-19 epidemic from medical emergency calls: the example of the Paris area
Abstract.
We portray the evolution of the Covid-19 epidemic during the crisis of March-April 2020 in the Paris area, by analyzing the medical emergency calls received by the EMS of the four central departments of this area (Centre 15 of SAMU 75, 92, 93 and 94). Our study reveals strong dissimilarities between these departments. We show that the logarithm of each epidemic observable can be approximated by a piecewise linear function of time. This allows us to distinguish the different phases of the epidemic, and to identify the delay between sanitary measures and their influence on the load of EMS. This also leads to an algorithm, allowing one to detect epidemic resurgences. We rely on a transport PDE epidemiological model, and we use methods from Perron-Frobenius theory and tropical geometry.
Lukas
Wolff
TU Berlin
On a variational model for pattern formation in biomembranes
Laura
Escobar
Washington University in St. Louis
Wall-crossing phenomena for Newton-Okounkov bodies
Abstract.
A Newton-Okounkov body is a convex set associated to a projective variety, equipped with a valuation. These bodies generalize the theory of Newton polytopes. Work of Kaveh-Manon gives an explicit link between tropical geometry and Newton-Okounkov bodies. We use this link to describe a wall-crossing phenomenon for Newton-Okounkov bodies. This is joint work with Megumi Harada.
Manfred
Scheucher
On the Number of Order Types
David
Fabian
FU Berlin
The running time of graph bootstrap percolation for trees
Abstract.
Graph bootstrap percolation is a model introduced by Bollobás in 1968. Given a fixed, small graph H the H-bootstrap process on a graph G is the sequence (Gi) i ≥ 0 for which G0 := G and Gi is obtained from Gi-1 by adding every edge that completes a copy of H. We investigate the maximum number of steps this process needs to stabilise when H is a tree and G has n vertices. More precisely, we show that for any tree T there exists a constant cT such that the T-bootstrap process on any graph stabilises after at most cT steps.
Emilie
Pirch
HU Berlin
Introduction and implementation of an HHO method
June 2020
Sebastian
Manecke
Goethe-Universität Frankfurt
Inscribable fans, zonotopes, and reflection arrangements
Abstract.
Steiner posed the question if any 3-dimensional polytope had a realization with vertices on a sphere. Steinitz constructed the first counter example and Rivin gave a complete resolution. In dimensions 4 and up, Mnev's universality theorem renders the question for inscribable combinatorial types hopeless. However, for a given complete fan F, we can decide in polynomial time if there is an inscribed polytope with normal fan F. Linear hyperplane arrangements can be realized as normal fans via zonotopes. It turns out that inscribed zonotopes are rare and in this talk I will focus on the question of classifying the corresponding arrangements. This relates to localizatons and restrictions of reflection arrangements and Grünbaum's quest for simplicial arrangements. The talk is based on joint work with Raman Sanyal.
Arturo
Merino
On the two-dimensional knapsack problem for convex polygons
Patrick
Morris
FU Berlin
Triangle factors in pseudorandom graphs
Abstract.
We discuss a conjecture of Krivelevich, Sudakov and Szabó which states that there exists a constant c such that any n vertex d-regular graph with n ∈ 3ℕ and second eigenvalue in absolute value at most cd2/n, contains a triangle factor. This bound is best possible due a dense pseudorandom triangle free construction of Alon.
Julian
Streitberger
HU Berlin
C0 interior penalty methods for the biharmonic problem
Mara
Belotti
SISSA
Topology of rigid isotopy classes of geometric graphs
Abstract.
We study the topology of rigid isotopy classes of geometric graphs on n vertices in a d-dimensional Euclidean space. We consider in particular two different regimes: a large number of vertices for the graphs (n large) and a large dimension of the space in which the graphs live (d large). In both cases we make considerations on the number of such classes and study their Betti numbers. In particular, for the second case we register a "shift to infinity of the topology". This is joint work with A.Lerario and A.Newman.
Kaie
Kubjas
Aalto University
Log-concave density estimation and polytopes
Abstract.
In this talk, we study probability density functions that are log-concave. The log-concave maximum likelihood estimation involves tent functions, regular subdivisions and Samworth bodies, which are continuous analogues of duals of secondary polytopes. This goal of this talk is to introduce the connections between log-concave density estimation and discrete geometry. All necessary background from statistics will be given in the talk.
Ander
Lamaison
FU Berlin
Coloring graphs by translates in the circle
Abstract.
: The fractional and circular chromatic numbers are the two most studied non-integral refinements of the chromatic number of a graph. Starting from the definition of a coloring base of a graph, which originated in work related to ergodic theory, we formalize the notion of a gyrocoloring of a graph: the vertices are colored by translates of a single Borel set in the circle group, and neighbouring vertices receive disjoint translates. The corresponding gyrochromatic number of a graph always lies between the fractional chromatic number and the circular chromatic number. We investigate basic properties of gyrocolorings. In particular, we provide an example of a graph in which no optimal gyrocoloring exists, and study the separation between fractional, circular and gyrochromatic numbers. Joint work with Pablo Candela, Carlos Catalá, Robert Hancock, Adam Kabela, Dan Kraľ and Lluís Vena.
Helena
Bergold
Neighborhood polynomial in chordal graphs
Henrik
Schneider
HU Berlin
DPG for Laplace eigenvalue problem
Raman
Sanyal
Goethe University Frankfurt
Inscribable fans, type cones, and reflection groupoids
Abstract.
Steiner posed the question if any 3-dimensional polytope had a realization with vertices on a sphere. Steinitz constructed the first counter examples and Rivin gave a complete resolution. In dimensions 4 and up, Mnev's universality theorem renders the question for inscribable combinatorial types hopeless. In this talk, I will address the following refined question: Given a polytope P, is there a normally equivalent polytope with vertices on a sphere? That is, can P be deformed into an inscribed polytope while preserving its normal fan? It turns out that the answer gives a rich interplay of geometry and combinatorics, involving local reflections and type cones. This is based on joint work with Sebastian Manecke (who will continue the story in the FU discrete geometry seminar next week).
Lukas
Abel
TU Berlin
Epitaxy and Dislocations - On Variational Models and their Analysis
Guillaume
Sagnol
Screening rules for Lasso and Optimal Designs
Raphael
Steiner
Subdivisions in dense digraphs
Büşra
Sert
TU Berlin
A study on the chamber complex
Abstract.
The notion of chamber (Minkowski cone, type cone) of a polytope has had an important role in several theories, such as Minkowski decomposition, the vector partition function, etc (McMullen, Emiris et al., Henk et al., Brion et al., Strumfels, Beck and others). While the chamber of a polytope P gives us the cone of vectors that change the half-space arrangement of P without changing its normal fan, we are interested in finding all different normal fans we can obtain by moving a given set of half-spaces. In this talk, we present the chamber complex of a matrix, which gives us the collection of all chambers we can obtain from it. Moreover, we describe an algorithm to compute it. This is joint work with Zafeirakis Zafeirakopoulos.
Tamás
Mészáros
FU Berlin
Boolean dimension, components and blocks
Abstract.
We investigate the behavior of Boolean dimension with respect to components and blocks. To put our results in context, we note that for Dushnik-Miller dimension, we have that if dim(C) ≤ d for every component C of a poset P, then dim(P) ≤ max{2,d}; also if dim(B) ≤ d for every block B of a poset P, then dim(P) ≤ d + 2. By way of constrast, local dimension is well behaved with respect to components, but not for blocks: if ldim(C) ≤ d for every component C of a poset P, then ldim(P) ≤ d+2; however, for every d ≥ 4, there exists a poset P with ldim(P) = d and dim(B) ≤ 3 for every block B of P. In this paper we show that Boolean dimension behaves like Dushnik-Miller dimension with respect to both components and blocks: if bdim(C) ≤ d for every component C of P, then bdim(P) ≤ 2+d+4·2d; also if bdim(B) ≤ d for every block B of P, then bdim(P) ≤ 19 + d + 18·2d. This is joint work with Piotr Micek and William T. Trotter.
Carsten
Carstensen
HU Berlin
Introduction to dG4plates
Tien
Tran-Ngoc
HU Berlin
Unstabilized Hybrid High-Order method for a class of degenerate convex minimization problems
Shah
Faisal
HU Berlin
What is an ergodic decomposition of invariant measures?
Abstract.
Ergodic systems, being indecomposable, are from the main objects of study in dynamical systems. If a system is not ergodic, it is natural to ask the following question: Is it possible to split it into ergodic systems in such a way that the study of the former reduces to the study of latter ones? In this talk, we will answer this question for measurable maps defined on complete separable metric spaces with Borel probability measure. No background in ergodic theory is assumed! The contents of the talk are taken from our preprint Ergodic Decomposition, to appear in Indagationes Mathematicae. For technical reasons, the talk could not be recorded. However, its slides can be found here.
Sven
Jäger
Non-Clairvoyant Precedence Constrained Scheduling
Jonathan
Rollin
Sparse Ordered Ramsey Graphs
José
Samper
MPI MiS Leipzig
How to complete matroids.
Abstract.
The goal of this talk is to present the big picture of one of my favorite problems: matroids enjoy several properties that are ideal to be studied with techniques from polyhedral geometry and combinatorial topology, but applying those techniques typically take us out of the realm of matroid theory. I am interested in the minimal class of additional objects needed to apply the toolkits freely. Prominent examples of this phenomenon include the universal valuation theorem of Derksen and Fink, or the matroid flip technique of Adiprasito, Huh and Katz. I will discuss many flavors of this phenomenon, including some new ones I have stumbled upon recently and try to compare them. Along the way I will present many open questions that aim to understand them better. The talk will include some parts of my work with Federico Castillo, Jeremy Martin and Alex Heaton.
Philipp
Bringmann
HU Berlin
Towards an efficient implementation of newest-vertex bisection with
separate marking in MATLAB
Taylor
Brysiewicz
Texas A&M University
The degree of Stiefel manifolds
Abstract.
The Stiefel manifold is the set of orthonormal bases for k-planes in an n-dimensional space. We compute its degree as an algebraic variety in the space of k-by-n matrices using techniques from classical algebraic geometry, representation theory, and combinatorics. We give a combinatorial interpretation of this degree in terms of non-intersecting lattice paths. (This is joint work with Fulvio Gesmundo)
May 2020
Felix
Schröder
A better approximation for the longest non-crossing spanning tree of a point set
Lei
Xue
University of Washington
A Proof of Grünbaum's Lower Bound Conjecture for general polytopes
Abstract.
In 1967, Grünbaum conjectured that any $d$-dimensional polytope with $d+s\leq 2d$ vertices has at least \[\phi_k(d+s,d) = {d+1 \choose k+1 }+{d \choose k+1 }-{d+1-s \choose k+1 } \] $k$-faces. In the talk we will discuss the proof of this conjecture and also also characterize the cases in which equality holds.
Johannes
Storn
Universität Bielefeld
On the Sobolev and Lp stability of the L² projection
Thorsten
Theobald
Goethe University Frankfurt
Conic stabilility of polynomials and spectrahedra
Abstract.
In the geometry of polynomials, the notion of stability is of prominent importance. The purpose of the talk is to discuss its recent generalization to conic stability. Given a convex cone K in real n-space, a multivariate polynomial f in C[z] is called K-stable if it does not have a root whose vector of the imaginary parts is contained in the interior of $K$. If $K$ is the non-negative orthant, then $K$-stability specializes to the usual notion of stability of polynomials. In particular, we focus on $K$-stability with respect to the positive semidefinite cone and develop stability criteria building upon the connection to the containment problem for spectrahedra, to positive maps and to determinantal representations. These results are based on joint work with Papri Dey and Stephan Gardoll.
Sarah
Morell
Minimum-cost integer circulations in given homology classes
Stefan
Felsner
Bundeling crossings of strings
Jörn
Wichmann
Universität Bielefeld
The parabolic p-Laplacian with fractional differentiability
Abstract.
We study the parabolic p-Laplacian system in a bounded domain. We deduce optimal convergence rates for the space-time discretization based on an implicit Euler scheme in time. Our estimates are expressed in terms of Nikolskii spaces and therefore cover situations when the (gradient of) the solution has only fractional derivatives in space and time. The main novelty is that, different to all previous results, we do not assume any coupling condition between the space and time resolution h and τ. The theoretical error analysis is complemented by numerical experiments. This is joint work with Dominic Breit, Lars Diening, and Johannes Storn.
Luis Carlos
Garcia Lirola
Kent State University
Volume product and metric spaces
Abstract.
Given a finite metric space M, the set of Lipschitz functions on M with Lipschitz constant at most 1 can be identified with a convex polytope in R^n. In this talk, we will show that there is a strong connection between the geometric properties of this polytope (as the vertices and the volume product) and the properties of the metric space M. We will also relate this study with a famous open problem in Convex Geometry, the Mahler conjecture, on the product of the volume of a convex body and its polar. This is a joint work with M. Alexander, M. Fradelizi, and A. Zvavitch.
Sara
Lamboglia
Goethe-Universität Frankfurt
Tropical convex hulls of polyhedral sets
Abstract.
In this talk I will present some recent results on the interplay between tropical and classical convexity. In particular, I will focus on the tropical convex hull of convex sets and fans and their explicit computation. This will lead to a lower bound on the degree of tropical fan curves and to a description of tropically convex cones. This is joint ongoing work with Cvetelina Hill and Faye Pasley Simon.
Jens
Schmidt
Dynamics of Cycles in Polyhedra I: The Isolation Lemma
Sophie
Puttkammer
HU Berlin
Notes on Morley in 3D
Georg
Loho
London School of Economics
Oriented Matroids from Triangulations of Products of Simplices
Abstract.
Classically, there is a rich theory in algebraic combinatorics surrounding the various objects associated with a generic real matrix. Examples include regular triangulations of the product of two simplices, coherent matching fields, and realizable oriented matroids. In this talk, we will extend the theory by skipping the matrix and starting with an arbitrary triangulation of the product of two simplices instead. In particular, we show that every polyhedral matching field induces oriented matroids. The oriented matroid is composed of compatible chirotopes on the cells in a matroid subdivision of the hypersimplex. Furthermore, we give a corresponding topological construction using Viro’s patchworking. This talk will also sketch the relationship between Baker-Bowler’s matroids over hyperfields and our work. This is joint work with Marcel Celaya and Chi-Ho Yuen.
Luis
Ferroni Rivetti
Uni. Bologna
The Ehrhart Polynomial of some Matroid Polytopes
Abstract.
In this seminar we will talk about the Ehrhart Theory of the polytopes of two interesting families of matroids: uniform matroids and minimal matroids. They provide two important examples of infinite families of matroids which support a Conjecture posed by De Loera et al, regarding the positivity of the coefficients of the Ehrhart polynomials of matroid polytopes. We will also discuss how the Ehrhart positivity of minimal matroids is related with the operation of circuit-hyperplane relaxation of a matroid, and we will make brief analysis of the Ehrhart polynomial as a matroid invariant.
André
Schulz
Representing Graphs by Polygons with Side Contacts in 3D
Carsten
Carstensen
HU Berlin
Abstract nonconforming schemes
Martin
Winter
TU Chemnitz
Edge-Transitive Polytopes
Abstract.
Despite the long history of the study of symmetric polytopes, aside from two extreme cases, the general transitivity properties of convex polytopes are still badly understood. For example, it has long been known that there are five flag-transitive (aka. regular) polyhedra (called, Platonic solids), six flag-transitive 4-polytopes and exactly three flag-transitive d-polytopes for any d>=5. The symmetry requirement of flag-transitivity is therefore quite restrictive for convex polytopes. On the other end of the spectrum, the class of vertex-transitive (aka. isogonal) polytopes is as rich as the category of finite groups. There seems to be little known about intermediate symmetries, like transitivity on edges or k-faces for any k>=2, and their interactions.In this talk we discuss the next most accessible transitivity, namely, edge-transitivity of convex polytopes. That is, we ask for a classification of convex polytopes in which all edges are identical under the symmetries of the polytope. Despite this restriction feeling more similar to vertex-transitivity than to flag-transitivity, we will see that the contrary seems to be the case: the class of edge-transitive polytopes appears to be quite restricted. We give, what we believe to be, a complete list of all edge-transitive polytopes, as well as a full classification for certain interesting sub-classes. Thereby, we show how edge-transitive polytopes can be studied with the tools of spectral graph theory.
April 2020
Max
Hlavacek
UC Berkeley
Subdivisions of Shellable Complexes
Abstract.
Combinatorialists often ask whether certain polynomials are real rooted; today we focus on h-polynomials, which encode the number of faces of cell complexes. When proving that a polynomial is real-rooted, we often rely on the theory of interlacing polynomials and their recursive nature. In this paper, we relate the recursive nature of interlacing polynomials to the recursive structure, typically termed 'shellability', of cell complexes. In this talk, we describe how these ideas can be used to positively answer a question posed first by Brenti and Welker, and then again by Mohammadi and Welker, on whether the h-polynomial of the barycentric subdivision of a cubical polytope is real rooted. We then briefly describe other subdivisions of shellable complexes in which these techniques apply.
Raphael
Steiner
Strongly Pfaffian Graphs
Marek
Kaluba
Random polytopes in Machine Learning
Abstract.
In this talk I will describe very recent work on analyzing different models of auto-encoder neural networks using random polytopes. In general, a class of neural networks known as auto-encoders can be understood as a low- dimensional (piecewise linear) embedding ℝᴺ → ℝⁿ (with N >> n), where points in the domain are e.g. images (in their pixel form) and each dimension the range extracts a single "feature".Although understanding the properties of these embedings is crucial for explainability (e.g. interpreting the decisions taken by the network), the research in this direction has been undertaken only from the statistical point of view. We introduce `random polytope descriptors` which try to approximate datasets in the feature space by possibly tight convex bodies. This is the first attempt at understanding the very coarse geometry of embeddings provided by neural networks. Using such descriptors one can answer e.g. if the natural clusters of of images has been preserved or distorted by the neural network, and if the embedding actually separates them (the question of entanglement).We use random polytope descriptors to examine the behaviour of auto-encoder networks in both vanilla and variational form, and provide first evidence for susseptibility of the latter to out-of-distribution attacks.This is joint work with L.Ruffs (TUB, Department of Electrical Engineering and Computer Science) and M.Joswig (TUB, Chair of Discrete Mathematics/Geometry; MPI for Mathematics in the Sciences, Leipzig).
Francisco
Criado
Borsuk’s problem
Abstract.
In 1932, Borsuk proved that every 2-dimensional convex body can be divided in three parts with strictly smaller diameter than the original. He also asked if the same would hold for any dimension d: is it true that every d-dimensional convex body can be divided in (d+1) pieces of strictly smaller diameter?The answer to this question is positive in 3 dimensions (Perkal 1947), but in 1993, Kahn and Kalai constructed polytopal counterexamples in dimension 1325. Nowadays, we know that the answer is "no" for all dimensions d>=64 (Bondarenko and Jenrich 2013). It remains open for every dimension between 4 and 63In this talk we introduce the problem, and we show the proofs for dimension 2 and 3. plus computational ideas on how to extend these solutions to dimension 4.
Rui
Ma
Universität Duisburg Essen
Adaptive least-squares finite element methods for non-selfadjoint indefinite second-order linear elliptic problems
Abstract.
In this talk we establish the convergence of adaptive least-squares finite element methods for second-order linear non-selfadjoint indefinite elliptic problems in three dimensions. The error is measured in the L² norm of the flux variable and then allows for an adaptive algorithm with collective marking. The axioms of adaptivity apply to this setting and guarantee the rate optimality for sufficiently small initial mesh-sizes and bulk parameter.
Karl
Däubel
The Maximum Leaf Spanning Tree Problem on Grid Graphs
Mikkel
Abrahamsen
University of Copenhagen
Geometric Multicut: Shortest Fences for Separating Groups of Objects in the Plane
Abstract.
We study the following separation problem: Given a collection of colored objects in the plane, compute a shortest "fence" F, i.e., a union of curves of minimum total length, that separates every two objects of different colors. Two objects are separated if F contains a simple closed curve that has one object in the interior and the other in the exterior. We refer to the problem as GEOMETRIC k-CUT, where k is the number of different colors, as it can be seen as a geometric analogue to the well-studied multicut problem on graphs. We first give an O(n^4 log^3 n)-time algorithm that computes an optimal fence for the case where the input consists of polygons of two colors and n corners in total. We then show that the problem is NP-hard for the case of three colors. Finally, we give a 4/3*1.2965-approximation algorithm. Joint work with Panos Giannopoulos, Maarten Löffler, and Günter Rote. Presented at ICALP 2019.
Jonathan
Spreer
Univ. Sydney
Linking topology and combinatorics: Width-type parameters of 3-manifolds
Abstract.
Many fundamental topological problems about 3-manifolds are algorithmically solvable in theory, but continue to withstand practical computations. In recent years some of these problems have been shown to allow efficient solutions, as long as the input 3-manifold comes with a sufficiently "thin" presentation. More specifically, a 3-manifold given as a triangulation is considered thin, if the treewidth of its dual graph is small. I will show how this combinatorial parameter, defined on a triangulation, can be linked back to purely topological properties of the underlying manifold. From this connection it can then be followed that, for some 3-manifolds, we cannot hope for a thin triangulation.
Manfred
Scheucher
7-gons in 3-space meet cadical
Jonathan
Spreer
University of Sydney
Linking topology and combinatorics: Width-type parameters of 3-manifolds
Abstract.
Many fundamental topological problems about 3-manifolds are algorithmically solvable in theory, but continue to withstand practical computations. In recent years some of these problems have been shown to allow efficient solutions, as long as the input 3-manifold comes with a sufficiently "thin" presentation.More specifically, a 3-manifold given as a triangulation is considered thin, if the treewidth of its dual graph is small. I will show how this combinatorial parameter, defined on a triangulation, can be linked back to purely topological properties of the underlying manifold. From this connection it can then be followed that, for some 3-manifolds, we cannot hope for a thin triangulation.This is joint work with Kristóf Huszár and Uli Wagner
Khai Van
Tran
Derandomizing Unconstrained Submodular Function Maximization
Alina
Stancu
Concordia University Montréal
Examples and usage of discrete curvature in convex geometry problems
Abstract.
Discretization of curvature has found numerous applications particularly in the modeling of moving smooth boundaries separating two different materials and image processing. Nonetheless, there are usually several ways to address discretization of curvature, even in the case of a 1-dimenional object (a curve) depending in essence of the role of the curvature in the physical process or the problem considered. I will present two models of discretization of curvature that I was personally involved with in relation to two geometric problems where the objects considered were also convex.
Fleurienne
Bertrand
HU Berlin
First order least-squares formulations for eigenvalue problems
Abstract.
In this talk we discuss spectral properties of operators associated with the least-squares finite element approximation of elliptic partial differential equations. The convergence of the discrete eigenvalues and eigenfunctions towards the corresponding continuous eigenmodes is studied and analyzed with the help of appropriate L² error estimates. A priori and a posteriori estimates are proved. This is joint work with Daniele Boffi.
Felix
Schröder
Enumeration of topological drawings of K_n
Hannah
Sjöberg
FU Berlin
Do alcoved polytopes have unimodal h*-vectors?
Abstract.
Unimodal sequences are widely studied in combinatorics. They can for example show up as a result of some algebraic properties of the underlying object. The h-vectors of simplicial complexes and the Ehrhart h*-vectors of lattice polytopes are examples for sequences that are often studied for unimodality. We will look at a certain class of lattice polytopes, alcoved polytopes and describe some conditions under which their h*-vectors are unimodal.
Ayush Kumar
Tewari
Forbidden patterns in tropical planar curves and Panoptagons
Dominic
Bunnett
Geometric discriminants and moduli
Abstract.
A discriminant is a subset of a function space consisting of singular functions. We shall consider discriminants of finite dimensional vector spaces of polynomials defining hypersurfaces in a toric variety. In the best case scenario (for plane curves for example), the discriminant is an irreducible hypersurface, however this is not the case for almost any other toric variety. We discuss positive results on the geometry of discriminants of weighted projective spaces and how to compute them via the A-discriminants of Gelfand, Kapranov and Zelevinsky.
Christoph
Hertrich
Representation Benefits of Deep Feedforward Networks
Matt
Beck
FU Berlin / San Francisco State Uni.
The Arithmetic of Coxeter Permutahedra
Abstract.
Ehrhart theory measures a polytope P discretely by counting the number of lattice points inside its dilates P, 2P, 3P, ... We compute the Ehrhart quasipolynomials of the standard Coxeter permutahedra for the classical Coxeter groups. A central tool is a description of the Ehrhart theory of a rational translate of an integer zonotope.
Stefan
Felsner
Pseudolinear Drawings
Robert
Loewe
Minkowski decompositions of generalized associahedra
Abstract.
We give an explicit subword complex description of the rays of the type cone of the g-vector fan of an finite type cluster algebra with acyclic initial seed. In particular, this yields a non-recursive description of the Newton polytopes of the F-polynomials as conjectured by Brodsky and Stump. We finally show that for finite type cluster algebras, the cluster complex is isomorphic to the totally positive part of the tropicalization of the cluster variety as conjectured by Speyer and Williams.
Rico
Raber
On the Robustness of Potential-Based Flow Networks
Piotr
Micek
Adjacency Labelling for Planar Graphs (and Beyond)
Lars
Kastner
Hyperplane arrangements in polymake
Abstract.
I will report on the implementation of hyperplane arrangements in polymake. Hyperplane arrangements appear in a wide variety of applications in tropical and algebraic geometry. An important construction is the induced subdivision. We will discuss the new HyperplaneArrangement object, its properties and our simple algorithm for constructing the cell subdivision.This is joint work with Marta Panizzut.
March 2020
Karl
Däubel
Some Aspects of Graph Sparsification in Theory and Practice
Paul
Vater
Patchworking real tropical hypersurfaces
Abstract.
We take a look at the tropical version of Viro's combinatorial patchworking method, as well as a recent implementation in polymake. In particular we investigate an efficient way of computing the homology with Z_2 coefficients of such a patchworked hypersurface.
Andrew
Newman
A two-step random polytope model
Abstract.
We consider a model for generating non-simple, non-simplicial random polytopes. The first step of the random process generates a random polytope P via random hyperplanes tangent to the d-dimensional sphere; the second step is given by the convex hull of a binomial sample of the vertices of P. In this talk we will discuss some ongoing work establishing results about the expected complexity of such polytopes.
Raphael
Steiner
Dichromatic number and forbidden subdivisions
Oguzhan
Yuruk
Understanding the Parameter Regions of Multistationarity in Dual
Phosporylation Cycle via SONC
Abstract.
Parameterized ordinary differential equation systems are crucial for modeling in biochemical reaction networks under the assumption of mass-action kinetics. Existence of multiple positive solutions in systems arising from biochemical reaction network are crucial since it is linked to cellular decision making and memory related on/off responses. Recent developments points out that the multistationarity, along with some other qualitative properties of the solutions, is related to various questions concerning the signs of multivariate polynomials in positive orthant. In this work, we provide further insight to the set of kinetic parameters that enable or preclude multistationarity of dual phosphorylation cycle by utilizing circuit polynomials to find symbolic certificates of nonnegativity. This is a joint work with Elisenda Feliu, Nidhi Kaihnsa and Timo de Wolff.
Gonzalo
Muñoz
Universidad de O'Higgins
A Sample Size-Efficient Polyhedral Encoding and Algorithm for Deep Neural Network Training Problems
Abstract.
Deep Learning has received much attention lately, however, results regarding the computational complexity of training deep neural networks have only recently been obtained. Moreover, all current training algorithms for DNNs with optimality guarantees possess a poor dependency on the sample size, which is typically the largest parameter. In this work, we show that the training problems of large classes of deep neural networks with various architectures admit a polyhedral representation whose size is linear in the sample size. This provides the first theoretical training algorithm with provable worst-case optimality guarantees whose running time is polynomial in the size of the sample.
Khai Van
Tran
Symmetric Rendezvous-on-the-Line with Unkown Initial Distance
Nicolas
Schneider
Characterizing equatable graphs – node balancing by edge increments and decrements
February 2020
Ralph
Morrison
Williams College
Tropically planar graphs: geometry and combinatorics
Abstract.
Tropically planar graphs are graphs that arise as the skeletons of smooth tropical plane curves, which are dual to regular unimodular triangulations of lattice polygons. In this talk we study what geometric properties such graphs have in the moduli space of metric graphs, as well as the combinatorial properties these graphs satisfy. This talk will include older work with Brodsky, Joswig, and Sturmfels; newer work with Coles, Dutta, Jiang, and Scharf; and ongoing work with Tewari.
Thomas
Yahl
Texas A&M Univ.
Solving decomposable sparse systems
Abstract.
Solutions to polynomial systems lying in a family can be identified with fibres of a branched cover. A branched cover decomposables if it factors on an open set through nontrivial branched covers. This structure can be exploited to numerically compute solutions via homotopy continuation methods. This leads to an algorithm for numerically solving systems lying in a family by completely factoring the associated branched cover. We study this in the case of sparse polynomial systems, where these decompositions can be explicitly described and their fibres can be computed as solutions to simpler polynomial systems.
Ágnes
Cseh
The complexity of cake cutting with unequal shares
Pamela
Harris
Williams College
Recent results on Kostant’s partition function
Abstract.
In this talk we introduce Kostant’s partition function which counts the number of ways to represent a particular weight (vector) as a nonnegative integral sum of positive roots of a Lie algebra. We provide two fundamental uses for this function. The first is associated with the computation of weight multiplicities in finite-dimensional irreducible representations of classical Lie algebras, and the second is in the computation of volumes of flow polytopes. We provide some recent results in the representation theory setting, and state a direction of ongoing research related to the computation of the volume of a flow polytope associated to a Caracol diagram.
Jonathan
Wolff
Zur Kombinatorik von Stick-Graphen
Felix
Schröder
TU Berlin
What is the dimension of a partial order?
Abstract.
Partial orders are among the most basic combinatorial structures, related to foundational mathematics such as properties of set systems as well as applied mathematics such as scheduling. We will look at multiple ways to visualize these posets, yielding a combinatorial definition of their dimension. We will then investigate the dimension of some special posets.
Andres
Vindas Melendez
Univ. of Kentucky
Equivariant Volumes of the Permutahedron
Abstract.
Motivated by the generalization of Ehrhart theory with group actions, this project is the prequel to the equivariant Ehrhart theory of the permutahedron (which Mariel spoke about in December). The fixed polytopes of the permutahedron are the polytopes that are fixed by acting on the permutahedron by a permutation. We prove some general results about the fixed polytopes/slices. In particular, we compute their dimension, show that they are combinatorially equivalent to permutahedra, provide hyperplane and vertex descriptions, and prove that they are zonotopes. Lastly, we obtain a formula for the volume of these fixed slices, which is a generalization of Richard Stanley's result of the volume for the standard permutahedron. This is joint work with Federico Ardila (San Francisco State Unviersity) and Anna Schindler (North Seattle College).
Manfred
Scheucher
Fast Computation of Chirotopes
Antonio
Macchia
FU Berlin
Binomial edge ideals of bipartite graphs
Abstract.
Binomial edge ideals are ideals generated by binomials corresponding to the edges of a graph, naturally generalizing the ideals of 2-minors of a generic matrix with two rows. They also arise naturally in the context of conditional independence ideals in Algebraic Statistics.We give a combinatorial classification of Cohen-Macaulay binomial edge ideals of bipartite graphs providing an explicit construction in graph-theoretical terms. In the proof we use the dual graph of an ideal, showing in our setting the converse of Hartshorne's Connectedness theorem.As a consequence, we prove for these ideals a Hirsch-type conjecture of Benedetti-Varbaro.This is a joint work with Davide Bolognini and Francesco Strazzanti. Organizer: DM/G
Joscha
Matysiak
HU Berlin
Patrick
Morris
FU Berlin
Vertex Ramsey properties of randomly perturbed graphs
Abstract.
Given graphs F,H and G, we say that G is (F,H)v -Ramsey if every red/blue vertex colouring of G contains a red copy of F or a blue copy of H. Results of Łuczak, Ruciński and Voigt and, subsequently, Kreuter determine the threshold for the property that the random graph G(n,p) is (F,H)v -Ramsey. In this talk we consider the sister problem in the setting of randomly perturbed graphs. In particular, we determine how many random edges one needs to add to a dense graph to ensure that with high probability the resulting graph is (F,H)v -Ramsey for all pairs (F,H) that involve at least one clique. This represents joint work with Shagnik Das and Andrew Treglown.
Adrian
Langer
U Warsaw
On smooth projective $\mathscr{D}$-affine varieties
Abstract.
A $\mathscr{D}$-affine variety is a variety on which left $\mathscr{D}$-modules behave as on an affine variety. Unlike in the case of $\mathscr{O}$-modules there exist smooth projective varieties that are $\mathscr{D}$-affine (e.g., rational homogeneous varieties in characteristic zero). I will survey known results on smooth projective $\mathscr{D}$-affine varieties. In particular, I will classify $\mathscr{D}$-affine smooth projective surfaces. In positive characteristic, a basic tool that I use is a new generalization of Miyaoka's generic semipositivity theorem.
José
Samper
Max Planck Institut Leipzig
Dual matroid polytopes and the independence complex of a matroid
Abstract.
A shelling order of a simplicial/polytopal complex is an ordering of the top dimensional faces that allows us to understand various properties of the underlying complex (when it exists). Empirically, some shelling orders are better than others in the sense that they are easier to analyze or come equipped with . This is especially notable for complexes that admit many shelling orders, like polytopes and and matroid independence complexes. We propose a strange connection, linking shelling orders of dual matroid polytopes to shelling orders of independence complexes. In particular, we show that several classical theorems about shellability of matroids have geometric interpretations. We use this to address to propose a new strategy for a 1977 conjecture of R. Stanley about face numbers of independence complexes: that the h-vector is a pure O-sequence. The talk is based on joint work with Alex Heaton.
Lionel
Pournin
Université Paris 13
Recent results on the diameter of lattice polytopes
Abstract.
Several new results about the largest possible diameter of a lattice polytope contained in the hypercube [0,k]^d, a quantity related to the complexity of the simplex algorithm, will be presented. Upper bounds on this quantity have been known for a couple of decades and have been improved recently. In this lecture, conjecturally sharp lower bounds on this quantity will be presented for all d and k, as well as exact asymptotic estimates when d is fixed and k grows large. These lower bounds are obtained by computing the largest diameter a lattice zonotope contained in the hypercube [0,k]^d can have, answering a question by Günter Rote. This talk is based on joint work with Antoine Deza and Noriyoshi Sukegawa.
Petra
Schwer
OVGU Magdeburg
Building bridges between Geometry and Algebra
Katharina
Jochemko
KTH Stockholm
Generalized permutahedra: Minkowski linear functionals and Ehrhart positivity
Abstract.
Generalized permutahedra form a combinatorially rich class of polytopes that naturally appear in many areas of mathematics such as combinatorics, geometry, optimization and statistics. They comprise many important classes of polytopes, for example, matroid polytopes. We study functions on generalized permutahedra that behave linearly with respect to dilation and taking Minkowski sums. We give a complete classification of all positive, translation-invariant Minkowski linear functionals on permutahedra that are invariant under permutations of the coordinates: they form a simplicial cone and we explicitly describe the generators. We apply our results to prove that the linear coefficients of Ehrhart polynomials of generalized permutahedra are nonnegative, verifying conjectures of De Loera-Haws-Koeppe (2009) and Castillo-Liu (2018) in this case. This is joint work with Mohan Ravichandran.
Amy
Wiebe
FU Berlin
Combining Realization Space Models of Polytopes
Abstract.
In this talk I will present a model for the realization space of a polytope which represents a polytope by its slack matrix. This model provides a natural algebraic relaxation for the realization space, and comes with a defining ideal which can be used as a computational engine to answer questions about the realization space. We will see how this model is related to more classical realization space models (representing realizations by Gale diagrams or points of the Grassmannian). In particular, we will see these relationships can be used to improve computational efficiency of the slack model.
Pranshu
Gupta
TU Hamburg
Pattern gadgets and highly Ramsey-simple graphs
Abstract.
We say a graph G is q-Ramsey for another graph H if in every q-colouring of the edges of G there is a monochromatic copy of H. Additionally, if no proper subgraph of G is q-Ramsey for H, we say that G is minimal q-Ramsey for H. Let Mq(H) denote the set of all minimal q-Ramsey graphs for H. In 1976, Burr, Erdős, Lovász initiated the study of properties of this set. In particular, the parameter sq(H)= min { δ(G) : G ∈ Mq(H) } has been of considerable interest. It is not difficult to show that sq(H) ≥ qδ(H) - (q-1). We call a graph H q-Ramsey simple if this lower bound is the correct value of sq(H). In 2010, Szabó et al. showed that many classes of bipartite graphs are 2-Ramsey simple. We define a graph H to be highly q-Ramsey simple if it is q-Ramsey simple and for every positive integer k there exists a minimal Ramsey graph for H with at least k vertices of degree sq(H). We develop a new tool, called a pattern gadget, which helps us show that cycles of length at least four are highly q-Ramsey simple for all q ≥ 2. We also discuss similar results for other graphs.This is a joint work with Simona Boyadzhiyska and Dennis Clemens.
Yanitsa
Pehova
University of Warwick
Characterisation of quasirandom permutations by a pattern sum
Abstract.
We say that a sequence {π_i} of permutations is quasirandom if, for each k>1 and each σ∈S_k, the probability that a uniformly chosen k-set of entries of π_i induces σ tends to 1/k! as i tends to infinity. It is known that a much weaker condition already forces π_i to be quasirandom; namely, if the above property holds for all σ∈S4. We further weaken this condition by exhibiting sets S⊆S4, such that if randomly chosen four entries of π_i induce an element of S with probability tending to |S|/24, then {π_i} is quasirandom. Moreover, we are able to completely characterise the sets S with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight. This is joint work with Timothy Chan, Daniel Král', Jon Noel, Maryam Sharifzadeh and Jan Volec.
Sándor
Kisfaludi-Bak
Max Planck Institut Saarbrücken
Separators and exact algorithms for geometric intersection graphs
Abstract.
Given n ball-like objects in some metric space, one can define a geometric intersection graph where the vertices correspond to objects, and edges are added for pairs of objects with a non-empty intersection. A separator of a graph is a small or otherwise "nice" vertex set whose removal disconnects the graph into two roughly equal parts. In this talk, we will see some separator theorems for intersection graphs in Euclidean and hyperbolic space. One can use these separators to design simple divide and conquer algorithms for several classical NP-hard problems. It turns out that well-designed separators often lead to subexponential algorithms with optimal running times (up to constant factors in the exponent) under the Exponential Time Hypothesis (ETH).
January 2020
Paul
Hager
TU Berlin
What is gaussian multiplicative chaos?
Abstract.
We will talk about the random measure which was originally introduced by Kahane in 1985 with the motivation of giving a rigorous construction to the Kolmogorov-Obukhov model of fully developed turbulences. In particular we are going to discuss the applications, properties and different approaches to its construction.
Philipp
Bringmann
HU Berlin
Adaptive least-squares finite element method with optimal convergence rates
Elise
Walker
Texas A&M Univ.
Toric degenerations and homotopy methods from finite Khovanskii bases
Abstract.
Homotopies are useful numerical methods for solving systems of polynomial equations. Embedded toric degenerations are one source for homotopy algorithms. In particular, if a projective variety has a toric degeneration, then linear sections of that variety can be optimally computed using the polyhedral homotopy. Any variety whose coordinate ring has a finite Khovanskii basis is known to have a toric degeneration. We provide embeddings for this Khovanskii toric degeneration to compute general linear sections of the variety. This is joint work with Michael Burr and Frank Sottile.
Michał
Seweryn
Jagellonian University, Krakow
Erdős-Hajnal properties for powers of sparse graphs
Abstract.
The notion of nowhere dense classes of graphs attracted much attention in recent years and found many applications in structural graph theory, algorithmics and logic. The powers of nowhere dense graphs do not need to be sparse, for instance the second power of star graphs are complete graphs. However, it is believed that powers of sparse graphs inherit somewhat simple structure. In this spirit, we show that for a fixed nowhere dense class of graphs and a positive integer d, in any n-vertex graph G in the class, there are disjoint vertex subsets A and B, each of size almost linear in n, such that in the dth power of G, either there is no edge between A and B, or there are all possible edges between A and B. This is joint work with Marcin Briański, Piotr Micek and Michał Pilipczuk.
Felix
Schröder
3-face colorings and Tutte's 3-flow conjecture
Matias
Villagra
Pontifical Catholic University of Chile
On a canonical symmetry breaking technique for polytopes
Abstract.
Given a group of symmetries of a polytope, a Fundamental Domain is a set of R^n that aims to select a unique representative of symmetric vectors, i.e. such that each point in the set is a unique representative under its G-orbit, effectively eliminating all isomorphic points of the polytope. The canonical Fundamental Domain found in the literature, which can be constructed for any permutation group, is NP-hard to separate even for structurally simple groups whose elements are disjoint involutions (Babai & Luks 1983).We consider a recent set of inequalities that has been implemented in CPLEX as a symmetry breaking technique for arbitrary finite permutation groups (Salvagnin 2018). We show a strong connection of this set with the set of lexicographically maximal vectors (LEX), and show that it defines a Fundamental Domain with quadratic many facets on the dimension, yielding a polynomial time separation algorithm. Moreover, we study when LEX defines a closed set, which suggests a stronger way of breaking symmetries.This is joint work with José Verschae and Léonard von Niederhäusern.
Alexander
Heinlein
Universität zu Köln
Overlapping Schwarz preconditioning techniques for nonlinear problems
Georgi
Mitsov
HU Berlin
Well-posedness of generalized dPG methods with locally weighted test-search norms for the heat equation
Charlotte
Knierim
ETH Zürich
Long Cycles, Heavy Cycles and Cycle Decompositions in Directed Graphs
Abstract.
In this talk I show a connection between heavy cycles and cycle decompositions. We prove that every directed Eulerian graph can be decomposed into at most O(n log Δ) disjoint cycles, thus making progress towards the conjecture by Bollobás and Scott, and matching the best known upper bound from the undirected case. This also implies the existence of long cycles differing to the Erdős-Gallai bound for undirected graphs in only a log factor Our approach is based on finding heavy cycles in certain edge-weightings of directed graphs. As a further consequence of our techniques, we prove that for every edge-weighted digraph in which every vertex has out-weight at least 1, there exists a cycle with weight at least Ω(log log n/log n), thus resolving a question by Bollobás and Scott.This is joint work with Maxime Larcher, Anders Martinsson and Andreas Noever.
Ezra
Waxman
TU Dresden
Random Models for the Distribution of Primes with a Prescribed Primitive Root
Abstract.
The Hardy-Littlewood Conjecture (also known as the k-tuple conjecture) is a vast generalization of the twin prime conjecture. In particular, it gives an asymptotic count for the number of integer pairs (n,n+d) such that both n and n+d are prime. In this talk, I will discuss preliminary results concerning the application of these heuristic models to the distribution of primes with a prescribed primitive root. Specifically, for integers \\(g\\) \geq 2\\) and \\(d \\in 2\mathbb{N}\\), I will present a conjecture for the number of prime pairs \\(p,p+d\\) such that \\(g\\) is a primitive root modulo both \\(p\\) and \\(p+d\\). Time permitting, I will also introduce preliminary findings concerning the distribution of primes with a prescribed primitive root, across short intervals. Joint work with Magdaléna Tinková & Mikuláš Zindulka.
Antonia
Adamik
On Equilibria in Atomic Splittable Flow Over Time Games
Michał
Włodarczyk
Eindhoven
Losing treewidth by separating subsets: on approximation of vertex/edge deletion problems
Abstract.
Consider the problem of deleting the smallest set S of vertices (resp. edges) from a given graph G such that the induced subgraph (resp. subgraph) G\S belongs to some class H. I will cover the case where graphs in H have treewidth bounded by t, and give a general framework to obtain approximation algorithms basing on two ingredients:1) approximation algorithms for the k-Subset Separator problem,2) FPT algorithms parameterized by the solution size. For the vertex deletion setting, this new framework combined with the current best result for k-Subset Vertex Separator, improves approximation ratios for basic problems such as k-Treewidth Vertex Deletion and Planar F Vertex Deletion. Our algorithms are simpler than previous works and give the first deterministic and uniform approximation algorithms under the natural parameterization.I will talk about what it means for several important graph classes and how the bounded treewidth property is exploited. I will present a sketch of the proof for the H Vertex Deletion algorithm and explain the differences between deleting vertices or edges.
Gweneth Anne
McKinley
MIT, Cambridge, USA
Super-logarithmic cliques in dense inhomogeneous random graphs
Abstract.
In the theory of dense graph limits, a graphon is a symmetric measurable function W from [0,1]^2 to [0,1]. Each graphon gives rise naturally to a random graph distribution, denoted G(n,W), that can be viewed as a generalization of the Erdös-Rényi random graph. Recently, Dolezal, Hladky, and Mathe gave an asymptotic formula of order log n for the size of the largest clique in G(n,W) when W is bounded away from 0 and 1. We show that if W is allowed to approach 1 at a finite number of points, and displays a moderate rate of growth near these points, then the clique number of G(n,W) will be of order √n almost surely. We also give a family of examples with clique number of order n^c for any c in (0,1), and some conditions under which the clique number of G(n,W) will be o(√n) or ω(√n). This talk assumes no previous knowledge of graphons.
Haim
Kaplan
Tel Aviv University
Privately Learning Thresholds
Abstract.
We study the problem of computing a point in the convex hull, and the related problem of computing a separating hyperplane, under the constraint of differential privacy. Intuitively, differential privacy means that our output should be robust to small changes in the input (for example to adding or deleting a point). We study the minimum size of the input needed to achieve such a private computation (sample complexity) and its time complexity. Even in one dimension the problem is non-trivial and we will first focus on this case. Several interesting open problems will be presented as well. No previous background on differential privacy will be assumed.
Jean-Philippe
Labbé
AG Diskrete Geometrie
Combinatorial Foundations for Geometric Realizations of Subword Complexes of Coxeter Groups
Abstract.
Let d>=1 and C be a simplicial complex homeomorphic to a (d-1)-dimensional sphere. "Is there a d-dimensional simplicial convex polytope P whose face lattice is isomorphic to that of C?" This is a classical problem in discrete geometry going back to Steinitz who proved that every 2-dimensional simplicial sphere is the boundary of a polytope. The determination of the polytopality of subword complexes is a resisting instance of Steinitz's problem. Indeed, since their introduction more than 15 years ago, subword complexes built up a wide portfolio of relations and applications to many other areas of research (Schubert varieties, cluster algebras, associahedra, tropical Grassmannians, to name a few) and a lot of efforts has been put into realizing them as polytopes, with relatively little success. In this talk, I will present some reasons why this problem resisted so far, and present a glimpse of a novel approach to study the problem grouping together Schur functions, combinatorics of words, and oriented matroids.
Max
Richter
Combinatorial properties of topological drawings of complete graphs
Andres R.
Vindas Melendez
University of Kentucky
The Equivariant Ehrhart Theory of the Permutahedron
Abstract.
In 2010, Stapledon described a generalization of Ehrhart theory with group actions. In 2018, Ardila, Schindler, and I made progress towards answering one of Stapledon's open problems that asked to determine the equivariant Ehrhart theory of the permutahedron. We proved some general results about the fixed polytopes of the permutahedron, which are the polytopes that are fixed by acting on the permutahedron by a permutation. In particular, we computed their dimension, showed that they are combinatorially equivalent to permutahedra, provided hyperplane and vertex descriptions, and proved that they are zonotopes. Lastly, we obtained a formula for the volume of these fixed polytopes, which is a generalization of Richard Stanley's result of the volume for the standard permutahedron. Building off of the work of the aforementioned, we determine the equivariant Ehrhart theory of the permutahedron, thereby resolving the open problem. This project presents combinatorial formulas for the Ehrhart quasipolynomials and Ehrhart Series of the fixed polytopes of the permutahedron, along with other results regarding interpretations of the equivariant analogue of the Ehrhart series. This is joint work with Federico Ardila (San Francisco State University) and Mariel Supina (UC Berkeley).
Joscha
Matysiak
HU Berlin
This presentation is rescheduled on February 12th.
Frederik
Garbe
Czech Academy of Sciences
Limits of Sequences of Latin Squares
Abstract.
We introduce a limit theory for sequences of Latin squares paralleling the ones for dense graphs and permutations. The limit objects are certain distribution valued two variable functions, which we call Latinons, and left-convergence is defined via densities of kxl subpatterns of Latin Squares. The main result is a compactness theorem stating that every sequence of Latin squares of growing orders has a Latinon as an accumulation point. Furthermore, our space of Latinons is minimal, as we show that every Latinon can be approximated by Latin squares. This relies on a result of Keevash about combinatorial designs. We also introduce an analogue of the cut-distance and prove counterparts to the counting lemma, sampling lemma and inverse counting lemma.This is joint work with R. Hancock, J. Hladky, and M. Sharifzadeh.
Antareep
Mandal
HU Berlin
Sup-norm bound for the average over an orthonormal basis of Siegel cusp forms
Abstract.
This talk updates our progress towards obtaining a sup-norm bound for the average over an orthonormal basis of Siegel cusp forms by relating it to the discrete spectrum of the heat kernel of the Siegel-Maass Laplacian. We begin by generalizing the relation between the cusp forms and the Maass forms in higher dimensions. Then we obtain the spherical functions for the Laplace-Beltrami operator in Siegel upper half-space by reduction to the complex case and make a weight correction to make them correspond to the Siegel-Maass Laplacian. Then we use these to construct the heat kernel, whose analysis leads to the weight dependence of the sup-norm bound.
Gorav
Jindal
Aalto University, Helsinki
On the Complexity of Symmetric Polynomials
Abstract.
The fundamental theorem of symmetric polynomials states that for a symmetric polynomial f_Sym in C[x1,x2,...,xn], there exists a unique "witness" f in C[y1,y2,...,yn] such that f_Sym=f(e1,e2,...,en), where the e_i's are the elementary symmetric polynomials. In this work, we study the arithmetic complexity L(f) of the witness f as a function of the arithmetic complexity L(f_Sym) of f_Sym. We show that the arithmetic complexity L(f) of f is bounded by poly(L(f_Sym),deg(f),n). Prior to this work, only exponential upper bounds were known for L(f). The main ingredient in our result is an algebraic analogue of Newton’s iteration on power series.As a corollary of this result, we show that if VP is not equal to VNP then there are symmetric polynomial families which have super-polynomial arithmetic complexity. This is joint work with Markus Bläser.
Alan
Arroyo
Institute of Science and Technology Austria
Obstructions in graph drawings
Abstract.
Many nice results in graph theory involve the characterization of a graph class in terms of a set of forbidden obstructions. In this talk I will consider the problem of understanding whether a class of graph drawings can be characterized by a set of forbidden obstructions. I will do an overview on interesting recent results that fall under this theme, but I will emphasize more on those related to geometric arrangements.
Holger
Dell
IT University of Copenhagen
Counting and sampling small subgraphs
Abstract.
In this talk, we discuss various algorithmic techniques used for counting and sampling subgraphs in a large input graph. The focus of the talk is on the mathematical foundations. We start with the beautiful technique of Color-Coding (Alon, Yuster, Zwick 1995), and we discuss various generalizations based on group algebras (Koutis 2008) and on the exterior algebra (Brand, D, Husfeldt 2018). These techniques are most useful for sampling, which is equivalent to approximate counting. On the other hand, the fastest known algorithm to exactly count subgraphs that are isomorphic to a graph H (Curticapean, D, Marx 2017) is based on the foundations of Lovász' theory of graph limits.
Jörg
Wolf
On the Serrin condition for one velocity component of solutions to the Navier-Stokes equations
Ander
Lamaison
FU Berlin
What is the behavior of random graphs?
Abstract.
We construct a graph randomly by taking $n$ vertices and, for every pair of them, we draw an edge between them with probability $p$. As it turns out, if we slowly increase $p$, the properties of the graph that we obtain change drastically at certain values of $p$. In this talk we introduce the concept of threshold probability, and discuss what properties we expect to see in the random graph for different ranges of $p$.
Bhargav
Narayanan
Rutgers U
Thresholds
Johannes
Hofscheier
Nottingham
Generalised flatness constants, spanning lattice polytopes, and the Gromov width
Abstract.
In this talk, I will present joint work with Averkov and Nill where we motivate some new directions of research regarding the lattice width of convex bodies. Based on the conclusion that convex bodies of large width contain a unimodular copy of a standard simplex, we will discuss relations to recent results on spanning lattice polytopes and how this can be viewed as the beginning of the study of generalised flatness constants. We will conclude the talk by outlining connections between the lattice width of a Delzant polytope and the Gromov width of its associated toric symplectic manifold. No prior knowledge to symplectic geometry is assumed.
Stefan
Felsner
Remarks on Plattenbauten
Simon
Telen
KU Leuven
Numerical Root Finding via Cox Rings
Abstract.
In this talk, we consider the problem of solving a system of (sparse) Laurent polynomial equations defining finitely many nonsingular points on a compact toric variety. The Cox ring of this toric variety is a generalization of the homogeneous coordinate ring of projective space. We work with multiplication maps in graded pieces of this ring to generalize the eigenvalue, eigenvector theorem for root finding in affine space. We use numerical linear algebra to compute the corresponding matrices, and from these matrices a set of homogeneous coordinates of the solutions. Several numerical experiments show the effectiveness of the resulting method, especially for solving (nearly) degenerate, high degree systems in small numbers of variables.
Philipp
Bringmann
HU Berlin
Convergence proofs for adaptive LSFEMs and implementation of the octAFEM3D software package
Bhargav
Narayanan
Rutgers University
Disproportionate Division
Abstract.
A finite number of agents, each with their own measure of utility, would like to cut up a piece of cake amongst themselves. How efficiently can they do this? Almost everything one would like to know about this problem is known in the case where the agents all want a fair share of the cake. The more general problem where the agents have different claims to the cake is less well understood; in this talk, I shall present a new, efficient division procedure for the general problem that yields nearly optimal bounds. We’ll take the scenic route to the problem, first through some algebraic topology, and then end up at some combinatorics.
Dragos
Fratila
U Strasbourg
The Jordan-Chevalley decomposition for semistable $G$-bundles on elliptic curves
Abstract.
For projective curves of arithmetic genus one, there are three possibilities: nodal curve, cuspidal curve, elliptic curve. For $G$ a reductive group one can consider semistable $G$-bundles on such a curve and their moduli stack. In the nodal case one recovers the adjoint quotient $G/G$ and in the cuspidal case the Lie algebra version of the adjoint quotient. The elliptic curve case could be considered as a further "exponentiation" of the group $G$. We will review the Jordan-Chevalley decomposition in the Lie algebra and Lie group case and then proceed to explain how one can formulate such a decomposition for $G$-bundles over an elliptic curve. Then we'll see that the Jordan-Chevalley decomposition can be also expressed in terms of the existence of a certain stratification of the moduli stack of semistable $G$-bundles. In the nodal or cuspidal case the strata are described in terms of nilpotent cones of certain Levi or pseudo-Levi subgroups. I will explain a similar description in the elliptic case. Finally, I hope to explain how such a stratification can be used to study certain automorphic sheaves on $\mathrm{Bun}_G(E)$ for an elliptic curve in analogy to the study of character sheaves for groups or Lie algebras. This is joint work with Sam Gunningham and Penghui Li.
Shahrzad
Haddadan
MPI Saarbrücken
Algorithms for top-k Lists and Social Networks
Abstract.
Today’s massive and dynamic data sets have motivated many computer scientists and mathematicians to study classical problems in combinatorics and graph theory in various settings. In certain settings the algorithms’ access to the data is restricted while in others the resources are limited. In this talk we explore such settings in three directions: ranking of objects, property testing in social networks and finding communities in dynamic graphs.In the first part, we discuss algorithms on permutations as well as prefixes of permutations also known as top-k lists. The study of later particularly arises in big data scenarios when analysis of the entire permutation is costly or not interesting. We start by discussing random walks on the set of full rankings or permutations of n objects, a set whose size is n!. Since 1990s to today, various versions of this problem have been studied, each for different motivation.The second part of the talk is devoted to property testing in social networks: given a graph, an algorithm seeks to approximate several parameters of a graph just by accessing the graph by means of oracles and while querying these oracles as few times as possible. We introduce a new oracle access model which is applicable to social networks, and assess the complexity of estimating parameters such as number of users (vertices) in this model.In the third part of the talk, we introduce a new dynamic graph model which is based on triadic closure: a friendship is more likely to be formed between a pair of users with a larger number of mutual friends. We find upper bounds for the rate of graph evolution in this model and using such bounds we devise algorithms discovering communities. I will finish this talk by proposing new directions and presenting related open problems.
Andrei
Asinowski
Universität Klagenfurt
Vectorial kernel method and patterns in lattice paths
Abstract.
A directed lattice path is a polygonal line which starts at the origin and consists of several vectors of the form (1, y) (where y belongs to a fixed set of integers) appended to each other. Enumeration of different kinds of lattice paths (walks/bridges/meanders/excursions) was accomplished by Banderier and Flajolet in 2002. We refine and generalize their results by studying lattice paths that avoid a fixed pattern (or several patterns). To this end, we develop a "vectorial kernel method" – a unified framework for enumeration of words generated by a counter automaton. Another improtant tool is the "autocorrelation polynomial" that encodes self-overlappings of a pattern, and its generalization: the "mutual correlation matrix" for several patterns. (This talk is based on joint works with Cyril Banderier, Axel Bacher, Bernhard Gittenberger and Valerie Rointer.)
Eric
Fusy
École Polytechnique Paris
Bijections between families of walks using oriented planar maps
Abstract.
When counting walks (with a given step-set), an equi-enumeration phenomenom is often observed between a stronger constraint on the domain and a stronger constraint on the position of the endpoint (a classical one-dimensional example is the fact that positive walks of length 2n are in bijection with walks of length 2n ending at 0, both being counted by the central binomial coefficient). I will show examples of such relations for 2d walks where the equi-enumeration can be bijectively explained using planar maps endowed with certain orientations (Schnyder woods, bipolar orientations).
Oliver
Sander
TU Dresden
Geometric Finite Element
Joseph
Doolittle
FU Berlin
Non-Inscribable 4-polytopes
Abstract.
Following work of Chen and Padrol, and many others, we studied the question of which polytopes can be inscribed, realized with all vertices on a sphere. We focused on dual-to-cyclic polytopes, and found a nearly complete characterization of when these polytopes are indescribable. The obstruction to inscribability used comes from classical plane geometry, Miquel's 5 circle theorem. We further found the first example of an f-vector for which every polytope with that f-vector cannot be inscribed. In this talk, we will explain the methods and insights leading to these results, which include algebraic geometry, linear algebra, combinatorics, and plane geometry. This is joint work with Jean-Philippe Labbé, Carsten Lange, Rainer Sinn, Jonathan Spreer, and Günter M. Ziegler.
Giulia
Codenotti
FU Berlin
Unimodular covers of 3-dimensional parallelepipeds and Cayley sums
Sophie
Puttkammer
HU Berlin
Remarks on optimal convergence rates for guaranteed lower eigenvalue bounds with a modified nonconforming method
Lukas
Kühne
Hebrew University of Jerusalem
Matroid representations by c-arrangements are undecidable
Abstract.
A matroid is a combinatorial object based on an abstraction of linear independence in vector spaces and forests in graphs. It is a classical question to determine whether a given matroid is representable as a vector configuration over a field. Such a matroid is called linear.This talk is about a generalization of that question from vector configurations to c-arrangements.A c-arrangement for a fixed c is an arrangement of dimension c subspaces such that the dimensions of their sums are multiples of c. Matroids representable as c-arrangements are called multilinear matroids.We prove that it is algorithmically undecidable whether there exists a c such that a given matroid has a c-arrangement representation. In the proof, we introduce a non-commutative von Staudt construction to encode an instance of the uniform word problem for finite groups in matroids of rank three.The talk is based on joint work with Geva Yashfe.
Frank
Sottile
Texas University
Irrational toric varieties and the secondary polytope
Abstract.
The secondary fan of a point configuration A in R^n encodes all regular subdivisions of A. These subdivisions also record all limiting objects obtained by weight degenerations of the irrational toric variety X_A parameterized by A. The secondary fan is the normal fan of the secondary polytope. We explain a functorial construction of R^n-equivariant cell complexes from fans that, when applied to the secondary fan, realizes the secondary polytope as the moduli space of translations and degenerations of X_A. This extends the work of Kapranov, Sturmfels and Zelevinsky (who established this for complex toric varieties when A is integral) to all real configurations A.
December 2019
Raphael
Steiner
Another Brick in the Wall
Henrik
Schneider
HU Berlin
DPG for the Laplace eigenvalue problem
Olaf
Parczyk
LSE
The size-Ramsey number of tight 3-uniform paths
Abstract.
Given a hypergraph H, the size-Ramsey number is the smallest integer m such that there exists a graph G with m edges with the property that in any colouring of the edges of G with two colours there is a monochromatic copy of H. Extending on results for graphs we prove that the size Ramsey number of the 3-uniform tight path on n vertices is linear in n. This is joint work with Jie Han, Yoshiharu Kohayakawa, and Guilherme Mota.
Zhen
Liu
Modeling and Optimization for the Snapshot Imaging Polarimeter
Lasse
Pyka
On the existence of minimizers for a variational model of martensitic microstructures
Alexander
Heaton
Max Planck Institut Leipzig
Applications of algebra to the geometry of materials
Abstract.
The upcoming Thematic Einstein Semester entitled "Geometric and Topological Structure of Materials" will focus on recent advancements in computational materials science. In this talk, we will present two interesting areas where algebra can play a role in future research. First, in the design and analysis of novel materials, questions of rigidity of frameworks arise. For generic configurations, matroid-theoretic and combinatorial techniques can be used to study rigidity. But the frameworks arising in applications are rarely generic, therefore deciding local rigidity presents a difficult computational problem which connects to the numerical algebraic geometry of varieties and semi-algebraic sets. Second, both point configurations and also polycrystalline materials give rise to cellular decompositions of 3-dimensional space. We will discuss how a Fourier-type analysis of these cells can be used to understand their approximate symmetry. The representation theory of SO(3) and its finite subgroups can be used to extract Fourier coefficients from the surface normal density of a given cell.
Janusz
Ginster
Many-particle limits in molecular solvation
Paweł
Dłotko
Swansea University
Topology in Action
Abstract.
In this talk we will focus on a number of practical problems, originating from the theory of dynamical systems and materials science, all the way to medicine and data science. In all of them we will identify certain shapes that carry important information required to solve those problems. We will introduce standard and new tools of Topological Data Analysis and see how to apply them to the discussed scenarios.
Mariel
Supina
FU Berlin
/
UC Berkeley
Equivariant Ehrhart Theory of the Permutahedron
Abstract.
Ehrhart theory is a topic in geometric combinatorics which involves counting the lattice points inside of lattice polytopes. Stapledon (2010) introduced equivariant Ehrhart theory, which combines discrete geometry, combinatorics, and representation theory to give a generalization of Ehrhart theory that accounts for the symmetries of polytopes. In this talk, I will discuss joint work with Ardila and Vindas-Meléndez (2019) on answering one of Stapledon's open questions: determining the equivariant Ehrhart theory for the permutahedron, and verifying his Effectiveness Conjecture in this special case.
Erik
Tadewaldt
Superpermutations And Super-Patterns
Simona
Boyadzhiyska
FU Berlin
On Ramsey minimal graphs for cliques
Abstract.
A graph G is Ramsey for another graph H if any two-coloring of the edges of G contains a monochromatic copy of H. We call G Ramsey minimal for H if G is Ramsey for H but no proper subgraph of G is. In this talk, we will present a result of Rödl and Siggers, showing that, for any k≥3 and n sufficiently large, there exist many graphs on at most n vertices that are Ramsey minimal for the k-clique. More precisely, we will show that, for any k≥3, there exist constants c(k)>1 and n0(k) such that, for all n≥n0(k), there are at least cn^2 Ramsey minimal graphs for Kk on at most n vertices.This talk is based on the paper "On Ramsey minimal graphs" by V. Rödl and M. Siggers.
Benedikt
Gräßle
HU Berlin
Numerical Analysis with HHO methods
Anil
Aryasomayajula
IISER Tirupati, India
On the holomorphic version of a conjecture by Sarnak
Abstract.
In 1995, Iwaniec and Sarnak computed estimates of Hecke Eigen Maass forms associated to co-compact arithmetic subgroups of $\mathrm{SL}(2,\mathbb{R})$, and Sarnak went on to make a conjecture on the growth of Hecke Eigen Maass forms. Adapting the arguments of Iwaniec and Sarnak to the setting of Hecke Eigen cusp forms, we discuss a holomorphic version of the conjecture of Sarnak.
Philipp
Warode
Complexity and Parametric Computation of Equilibria in Atomic Splittable Congestion Games via Weighted Block Laplacians
Marcel
Celaya
Georgia Institute of Technology
The linear span of lattice points in a half-open lattice parallelepiped
Abstract.
The problem of deciding if there exists a lattice point inside a half-open parallelepiped on a given rational hyperplane is known to be NP-complete. In contrast, the question of deciding if all lattice points in a half-open parallelepiped lie on a given hyperplane has a much nicer answer. In this talk I will explain this answer in detail, and outline some applications.
Svante
Linusson
Königlich Technische Hochschule Stockholm
Limit shape of shifted staircase SYT
Abstract.
A shifted tableau of staircase shape has row lengths n,n-1,...,2,1 adjusted on the right side and numbers increasing along rows and columns. Let the number in a box represent the height of a point above that box, then we have proved that the points for a uniformly chosen random shifted staircase SYT in the limit converge to a certain surface in three dimensions. I will present this result and also how this implies, via properties of the Edelman–Greene bijection, results about random 132-avoiding sorting networks, including limit shapes for trajectories and intermediate permutations. (Based on joint work with Samu Potka and Robin Sulzgruber.)
Josué
Tonelli-Cueto
TU Berlin
What is best voting system?
Abstract.
Democracy might look like a straightforward process: whatever is the option preferred by the group should be the chosen option. But what does it mean to be the preferred option by the group? To answer this question, several voting systems (Condorcet's method, plurality voting, Borda count...), each with its advantages and disadvantages, have been proposed. As it has become clear in current times, how we vote influences radically the outcome of the election process. Can't we just choose the objectively best possible voting system? In this talk, we will see what mathematics has to say about this.
Irem
Portakal
Univ. Magdeburg
Kazhdan-Lusztig varieties with a view towards T-varieties
Abstract.
Lee and Masuda investigate the closure of the usual torus orbit in Schubert variety X_w in full flag variety. The maximal cones of the associated normal fan of this toric variety are encoded by permutations smaller than w (in Bruhat sense). It turns out that these cones are edge cones of certain directed graphs. On the other hand, Kazhdan and Lusztig prove a relation between Kazhdan-Lusztig varieties and affine neighborhoods of torus fixed points in the Schubert variety X_w. In this talk, we aim to bring these two results together in order to investigate the usual torus action on Kazhdan-Lusztig varieties using directed graphs. This is an ongoing-project with Donten-Bury and Escobar-Vega.
Céline
Torres
Universität Zürich
Stability of the Helmholtz equation with highly oscillatory coefficients
Abstract.
Existence and uniqueness of the heterogeneous Helmholtz problem on bounded domains can be shown using a unique continuation principle in Fredholm's alternative. This results in an energy estimate of the problem, with a stability constant that is not directly explicit in the coefficients or the wave number. We show, that for highly oscillatory coefficients, the solution can exhibit a localised wave. As a result the stability grows exponentially in the wave number. We discuss how this constant enters in the condition for quasi-optimality, when using a hp-Finite Element Method to discretise the problem and the difficulties that arise therein.
David
Fabian
FU Berlin
On infinite α-strong Sidon sets
Abstract.
Given a real parameter 0≤α<1, an α-strong Sidon set is a subset S⊂N of the positive integers such that for any w,x,y,z∊S with {w,x}≠{y,z} one has |(w+x)-(y+z)|≥ max{wα,xα,yα,zα}. In this talk we focus on the setting when S is infinite and study the asymptotics of the counting function S(n):=|S∩[n]|. By generalizing a construction of Cilleruelo we show that there exists an α-strong Sidon set S with counting function S(n) = n\sqrt{(1+α)^2+1}-(1+α)+o(1). Finally, we take a look at how α-strong Sidon sets can be used to bound the density of the largest Sidon set contained in a random infinite subset of the positive integers. We also describe how these results can be extended to Bh-sets. This is joint work with Juanjo Rué and Christoph Spiegel.
Martin
Gallauer
U Oxford
Simplicity of Tannakian categories, and applications
Abstract.
In this talk, I want to discuss two classification problems in algebraic geometry: (1) Given a variety, classify its constructible sheaves (in the derived sense). (2) Given a field, classify its motives (in the derived sense). One aspect which connects the two problems is the appearance of Tannakian categories (in the derived sense). I will draw attention to the fact that these are "simple", and explain how this allows us to make progress on the two classification problems.
Guillaume
Sagnol
Second-Order Stochastic Dominance and Applications in Scheduling
Viktor
Tkachenko
Inst. of Mathematics, Nat. Academy of Sciences of Ukraine
Forced frequency locking of periodic solutions to impulsive differential equations
November 2019
Janusz
Ginster
HU Berlin
What is the role of
convexity in variational problems?
Abstract.
A classical problem in the Calculus of Variations is to minimize an integral functional in a certain class of functions. In this talk we present several simple examples which illustrate why and how non-convexity of the integrand can result in the non-existence of minimizers or the formation of small-scale oscillations.
Barbara
Zwicknagl
HU Berlin
Mathematical variations on pattern formation in smart materials
Zuzka
Patáková
IUUK
Around Radon's theorem
Abstract.
Radon’s theorem is one of the cornerstones of convex geometry. It implies many of the key results in the area such as Helly’s theorem and, as recently shown by Andreas Holmsen and Dong-Gyu Lee, also its more robust version, fractional Helly’s theorem together with a colorful strengthening of Helly’s theorem. Consequently, this yields an existence of weak epsilon nets and a (p,q)-theorem. We show that we can obtain these results even without assuming convexity, replacing it with very weak topological conditions. Moreover, using a recent result we have obtained together with Gil Kalai, we show that, under the weak topological conditions, the fractional Helly number for open sets in the plane or on a surface is reasonably small. Special case of this result settles a conjecture of Andreas Holmsen, Minki Kim, and Seunghun Lee about an existence of a (p,q)-theorem for open subsets of a surface.
Ariel
Liebman
Monash University
Optimisation, Machine Learning and AI for Rapid Grid Decarbonisation
Abstract.
The national and transcontinental electricity grids of today are based on devices such as coal furnaces, steam turbines, copper and steel wires, electric transformers, and electromechanical power switches that have remained unchanged for 100 years. However imperceptibly, the components and operational management of this great machine, the grid, has began to change irreversibly. This is fortunate, as climate science tells us we must reduce CO<sub>2</sub> emissions from the energy sector to zero by 2050 and to 50% of current levels by 2030 if we are to prevent dangerous climate changes in future world that is over 1.5 degree hotter that today. Now utility scale wind and solar PV farms as large as coal, gas and nuclear generators are being deployed more cheaply than it is possible to build and operate generators using older technologies. In some cases, even these new technologies can be cheaper that even merely the operating costs of older technologies. In addition, low cost rooftop solar PV has also enabled consumers to become self-suppliers and also contributors to the supply of energy for their neighbours. Moreover, the “dumb” grid of the past, is becoming “smarter”. This is enabled through a combination of ubiquitous low-cost telecommunication and programmable devices at the edge of the grid such as smart meters, smart PV inverters, smart air conditioners and home energy management systems. The final component is the electrification of the private transport system that will finally eliminate the need for fossil fuels. The implications of this are that it is now necessary to rapidly replan and reinvest in the energy system at rates and in ways that are unprecedented in industrial civilisations history. While the majority of hardware technology already exist, the missing piece of the puzzle are new computers science technologies, and particularly Optimisation, Machine Learning, Forecasting and Data analytics methods needed to plan and operate this rapidly transforming system. In this talk I will describe a range of ways existing computer science tools in the Optimisation, AI, ML and other areas we and others are enhancing in order to better operate and plan the existing power system. I will focus on identifying emerging research opportunities in areas that are needed to complete the transformation to a cheaper, smarter and zero carbon energy system.
Stefan
Sauter
Universität Zürich
𝒟ℋ2 Matrices and their Application to Scattering Problems
Abstract.
The sparse approximation of high-frequency Helmholtz-type integral operators has many important physical applications such as problems in wave propagation and wave scattering. The discrete system matrices are huge and densely populated; hence their sparse approximation is of outstanding importance. In our talk we will generalize the directional ℋ2 -matrix techniques from the "pure" Helmholtz operator Lu=−Δu+z2u with z=−ik;k real, to general complex frequencies z with Re(z)>0. In this case, the fundamental solution decreases exponentially for large arguments. We will develop a new admissibility condition which contain Re(z) and Im(z) in an explicit way and introduce the approximation of the integral kernel function on admissible blocks in terms of frequency-dependent directional expansion functions. We present an error analysis which is explicit with respect to the expansion order and with respect to the real and imaginary part of z. This allows us to choose the variable expansion order in a quasi-optimal way depending on Re(z) but independent of, possibly large, Im(z). The complexity analysis is explicit with respect to Re(z) and Im(z) and shows how higher values of Re(z) reduce the complexity. In certain cases, it even turns out that the discrete matrix can be replaced by its nearfield part. Numerical experiments illustrate the sharpness of the derived estimates and the efficiency of our sparse approximation. This talk comprises joint work with S. Börm, Christian-Albrechts-Universität Kiel, Germany and M. Lopez-Fernandez, Sapienza Universita di Roma, Italy.
Günter
Rote
Enumerating and Counting Pseudoline Arrangements
Shagnik
Das
FU Berlin
Ramsey games near the critical threshold
Abstract.
A well-known result of Rödl and Ruciński states that for any graph H there exists a constant C such that if p ≥ C n- 1/m_2(H), then the random graph Gn,p is a.a.s. H-Ramsey, that is, any 2-colouring of its edges contains a monochromatic copy of H. Aside from a few simple exceptions, the corresponding 0-statement also holds; that is, there exists c>0 such that if p≤ cn-1/m_2(H) the random graph Gn,p is a.a.s. not H-Ramsey. We show that near this threshold, even when Gn,p is not H-Ramsey, it is often extremely close to being H-Ramsey. More precisely, we prove that for any constant c > 0 and any strictly 2-balanced graph H, if p ≥ c n-1/m_2(H), then the random graph Gn,p a.a.s. has the property that every 2-edge-colouring without monochromatic copies of H cannot be extended to an H-free colouring after Ω(1) extra random edges are added. This generalises a result by Friedgut, Kohayakawa, Rödl, Ruciński and Tetali, who in 2002 proved the same statement for triangles, and addresses a question raised by those authors. We also provide a wide variety of examples to show that these theorems need not hold when H is not strictly 2-balanced, and extend a result of theirs in the 3-coloured setting. This is joint work with David Conlon, Joonkyung Lee and our very own Tamás Mészáros.
Thorsten
Heidersdorf
U Bonn
Tensor categories in positive characteristic
Abstract.
Some of the most important tensor categories over a field come from representations of algebraic groups. A celebrated theorem of Deligne asserts that every tensor category of subexponential growth is the representation category of an algebraic supergroup scheme. The theorem is no longer true in characteristic $p > 0$. Counterexamples arise from representations of the cyclic group $\mathbb{Z}/p\mathbb{Z}$. Ostrik proposed a conjectural extension, but recently Benson-Etingof constructed an infinite chain of counterexamples in characteristic 2. These categories are closely related to representations of algebraic groups and the question if monoidal categories admit abelian envelopes. I will give an overview about these results and discuss some recent developments.
Manfred
Scheucher
Techniche Universität Berlin
Topological Drawings meet SAT Solvers and Classical Theorems of Convex Geometry
Abstract.
In a simple topological drawing of the complete graph $K_n$, vertices are mapped to points in the plane, edges are mapped to simple curves connecting the corresponding end points, and each pair of edges intersects at most once, either in a common vertex or in a proper crossing. We discuss an axiomatization of simple drawings and for various sub-classes and present a SAT model. With the aid of modern SAT solvers, we investigate some famous and important classical theorems from Convex Geometry (such as Caratheodory’s, Helly's, Kirchberger's Theorem, and the Erdös-Szekeres Theorem) in the context of simple drawings. This is joint work with Helena Bergold, Stefan Felsner, Felix Schröder, and Raphael Steiner. Research is in progress.
Georg
Loho
London School of Economics and Political Science
Signed tropical convexity
Abstract.
Convexity for the max-plus algebra has been studied from different directions including discrete geometry, scheduling, computational complexity. As there is no inverse for the max-operation, this used to rely on an implicit non-negativity assumption. We remove this restriction by introducing `signed tropical convexity'. This allows to exhibit new phenomena at the interplay between computational complexity and geometry. We obtain several structural theorems including a new Farkas lemma and a Minkowski-Weyl theorem for polytopes over the signed tropical numbers. Our notion has several natural formulations in terms of balance relations, polytopes over Puiseux series and hyperoperations.
Yana
Scotar
Donetsk National University, Vinnytsa, Ukraine
Canonical systems and their Weyl functions
Camille
Combe
Univ. Strasbourg
Cubic realizations of Tamari interval lattices
Abstract.
We introduce cubic coordinates, which are integer words encoding intervals in the Tamari lattices. Cubic coordinates are in bijection with interval-posets, themselves known to be in bijection with Tamari intervals. We will see that in each degree the set of cubic coordinates forms a lattice, isomorphic to the lattice of Tamari intervals. Geometric realizations are naturally obtained by placing cubic coordinates in space, highlighting some of their properties. We will present the cellular structure of these realizations. Finally, we will see a proof generalizing the result of Björner and Wachs about the EL-shellability of Tamari posets for Tamari interval posets.
Ma
Rui
HU Berlin
Convergence of the adaptive nonconforming element method for an obstacle problem
Tamás
Mészáros
FU Berlin
Greedy maximal independent sets via local limits
Abstract.
The greedy algorithm for finding a maximal independent set in a graph G on vertices V = {v1, ..., vn} can be described as follows. Let σ be a permutation of [n] chosen uniformly at random. Starting from an empty set R, at step i add vσ(i) to the set R if and only if R∪{vσ(i)} is an independent set in G. This very natural algorithm has been studied extensively in various settings in combinatorics, probability, computer science - and even in chemistry. In this talk we present a natural and general framework for calculating the asymptotics of the proportion of the yielded independent set for sequences of (possibly random) graphs, involving a useful notion of local convergence. We use this framework both to give short and simple proofs for results on previously studied families of graphs, such as paths and binomial random graphs, and to study new ones, such as random trees. This is joint work with Michael Krivelevich, Peleg Michaeli and Clara Shikhelman.
Marco
Flores
HU Berlin
Cohomological dimension of projective schemes in pro-p towers
Abstract.
If $X$ is a variety of dimension $d$ over an algebraically closed field, its étale cohomology groups with coefficients in any constructible sheaf vanish in degree greater than $2d$. Moreover, if $X$ is affine they already vanish in degree greater than $d$ by Artin's vanishing theorem. The last is not true for projective varieties. However, Scholze showed that if $X$ is a complex projective variety of dimension $d$ and $p$ is a prime number, there is a specific tower of $p$-power degree covers of $X$ such that the direct limit of étale cohomology groups with $ extbackslash mathbb{F}_p$-coefficients does vanish in degree greater than $d$. In this talk we present a new proof of this result, by Hélène Esnault, which moreover works over any algebraically closed field of characteristic different from $p$.
Alexandra
Quitmann
Percolation and its convergence to Stochastic Loewner Evolution
Luc
Pronzato
Design of Computer Experiments based on Bayesian Quadrature
Ansgar
Freyer
Technische Universität Berlin
Discrete analogs of an inequality by Meyer
Abstract.
In 1988, Mathieu Meyer presented a lower bound on the volume of a convex body in terms of the volumes of its sections with the coordinate hyperplanes. Our aim is to "discretize" this inequality by replacing the volume by the lattice point enumerator. It turns out that such a discrete analog cannot exist for general convex bodies. Moreover, it will not imply its continuous counterpart. We prove inequalities for the case of o-symmetric bodies (using arithmetic progressions) and unconditional bodies (by proving a universal bound for the lattice point enumerator of unconditional lattice polytopes). Moreover, the talk will touch on a reverse inequality and other related problems.
Matthias
Reitzner
Universität Osnabrück
Stars of Empty Simplices
Abstract.
Consider an n-element point set in general position in d-dimensional space. For a k-element subset the degree is the number of empty d-simplices with this k-set as base. We investigate the maximal degree of a random point set consisting of n independently and uniformly chosen points from a compact set.
Leonid
Pestov
Immanuel Kant Baltic Federal University, Kaliningrad, Russia
On lens rigidity of 2-dimensional Riemannian manifolds
Georg
Lehner
FU Berlin
What is a topological surface?
Abstract.
Everyone knows what a two dimensional surface is. If we allow surfaces to bend or stretch, but not rip and tear, how many fundamentally different surfaces are there? The answer is known since the 1860's but it hasn't lost its simple elegance since: Everything can be obtained by cutting a number of holes into a sphere and glueing into some of them either handles or Möbius strips. We present the simple ZIP proof due to Conway and we will draw lots and lots of pictures. No formulas needed!
Takayuki
Hibi
Osaka Univ.
Gorenstein polytopes
Abstract.
A Gorenstein polytope is a lattice polytope one of whose dilated polytopes is a reflexive polytope. In my talk, after reviewing Gorenstein polytopes from a viewpoint of enumeration of lattice points, the conjecture that every (0, 1)-polytope is a face of a Gorenstein (0, 1)-polytope will be discussed. No special knowledge will be required to understand my talk.
Matthias
Mnich
Degree-Bounded Polymatroids, with Applications to the Many-Visits TSP
Torsten
Mütze
The central levels problem and symmetric chains in the hypercube
Ornela
Mulita
SISSA
Smoothed Adaptive Finite Element Methods
Abstract.
We propose a new algorithm for Adaptive Finite Element Methods based on Smoothing iterations (S-AFEM). The algorithm is inspired by the ascending phase of the V-cycle multigrid method: we replace accurate algebraic solutions in intermediate cycles of the classical AFEM with the application of a prolongation step, followed by a fixed number of few smoothing steps. The method reduces the overall computational cost of AFEM by providing a fast procedure for the construction of a quasi-optimal mesh sequence with large algebraic error in the intermediate cycles. Indeed, even though the intermediate solutions are far from the exact algebraic solutions, we show that their a-posteriori error estimation produces a refinement pattern that is substantially equivalent to the one that would be generated by classical AFEM, at a considerable fraction of the computational cost. In this talk, we will quantify rigorously how the error propagates throughout the algorithm, and then we will provide a connection with classical a posteriori error analysis. Finally, we will present a series of numerical experiments that highlights the efficiency and the computational speedup of S-AFEM.
Christoph
Spiegel
UPC Barcelona
A step beyond Freiman's theorem for set addition modulo a prime
Abstract.
Freiman's 2.4-Theorem states that any subset A of Zp satisfying |2A| ≤ 2.4 |A| - 3 and |A| < p/35 can be covered by an arithmetic progression of length at most |2A| - |A| + 1. A more general result of Green and Ruzsa implies that this covering property holds for any set satisfying |2A| ≤ 3|A| - 4 as long as the rather strong density requirement |A| < p/10215 is satisfied. In this talk I will present a version of this statement that allows for sets satisfying |2A| ≤ 2.48|A| - 7 with the more modest density requirement of |A| < p/1010. In doing so, I hope to shed some light on how the methods of rectification and modular reduction go hand-in-hand when proving these types of small doubling covering property both in the cyclic setting and in the integers. This is joint work with Pablo Candela and Oriol Serra.
Ziyang
Gao
IMJ-PRG Paris
Functional transcendence on the universal abelian variety
Abstract.
The main topic of this talk is the mixed Ax-Schanuel theorem for the universal abelian variety. I will explain the statement and sketch its proof. We will start from the analogous statement for abelian varieties over the field of complex numbers.
Raphael
Steiner
Technische Universität Berlin
Majority Colorings of Sparse Digraphs
Abstract.
A majority coloring of a directed graph is a vertex-coloring in which every vertex has the same color as at most half of its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that every digraph has a majority 4-coloring and conjectured that every digraph admits a majority 3-coloring. We verify this conjecture for digraphs with chromatic number at most 6 or dichromatic number at most 3. We obtain analogous results for list coloring: We show that every digraph with list chromatic number at most 6 or list dichromatic number at most 3 is majority 3-choosable. We deduce that digraphs with maximum out-degree at most 4 or maximum degree at most 7 are majority 3-choosable. On the way to these results we investigate digraphs admitting a majority 2-coloring. We show that every digraph without odd directed cycles is majority 2-choosable. We answer an open question posed by Kreutzer et al. negatively, by showing that deciding whether a given digraph is majority 2-colorable is NP-complete. Finally we deal with a fractional relaxation of majority coloring proposed by Kreutzer et al. and show that every digraph has a fractional majority 3.9602-coloring. We show that every digraph D with sufficiently large minimum out-degree has a fractional majority-(2+ε)-coloring. Joint work with Michael Anastos, Ander Lamaison and Tibor Szabó.
Tibor
Szabó
Freie Universität Berlin
Turán numbers, projective norm graphs, quasirandomness
Abstract.
The Turán number of a (hyper)graph H, defined as the maximum number of (hyper)edges in an H-free (hyper)graph on a given number of vertices, is a fundamental concept of extremal graph theory. The behaviour of the Turán number is well-understood for non-bipartite graphs, but for bipartite H there are more questions than answers. A particularly intriguing half-open case is the one of complete bipartite graphs. The projective norm graphs $NG(q,t)$ are algebraically defined graphs which provide tight constructions in the Tur\'an problem for complete bipartite graphs $H=K_{t,s}$ when $s > (t-1)!$. The $K_{t,s}$-freeness of $NG(q,t)$ is a very much atypical property: in a random graph with the same edge density a positive fraction of $t$-tuples are involved in a copy of $K_{t,s}$. Yet, projective norm graphs are random-like in various other senses. Most notably their second eigenvalue is of the order of the square root of the degree, which, through the Expander Mixing Lemma, implies further quasirandom properties concerning the density of small enough subgraphs. In this talk we explore how far this quasirandomness goes. The main contribution of our proof is the estimation, and sometimes determination, of the number of solutions of certain norm equation system over finite fields. Joint work with Tomas Bayer, Tam\'as M\'esz\'aros, and Lajos R\'onyai.
Hannah
Sjöberg
FU Berlin
What is the Euler characteristic of a polytope?
Abstract.
The Euler characteristic of a non-empty (solid) polytope $P$ is the alternating sum of the number of non-empty $i$-dimensional faces of $P$. The Euler-Poincaré formula asserts that this alternating sum is equal to $1$. In this talk we discuss how the Euler characteristic can be constructed as a valuation. The construction has nice applications: we will see that it gives us simple proofs of the Euler-Poincaré formula and of a theorem on the number of regions in a hyperplane arrangement.
Dominic
Bunnett
TU Berlin
and
Marta
Panizzut
TU Berlin
What is algebraic curves and their tropical friends?
Abstract.
We will begin by illustrating stable tropical curves of genus $g$. We will then give a pictorial introduction to smooth algebraic curves of genus $g$ and their degenerations. We will define the moduli space of such curves and show the appearance of tropical curves in its study.
Samuel
Payne
U Texas, Austin
Tropical curves, graph homology, and top weight cohomology of M_g
Miruna-Stefana
Sorea
Leipzig
The shapes of level curves of real polynomials near strict local minima
Abstract.
We consider a real bivariate polynomial function vanishing at the origin and exhibiting a strict local minimum at this point. We work in a neighbourhood of the origin in which the non-zero level curves of this function are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, these level curves may fail to be convex. The aim of this talk is two-fold. Firstly, to study a combinatorial object measuring this non-convexity; it is a planar rooted tree. And secondly, we want to characterise all possible topological types of these objects. To this end, we construct a family of polynomial functions with non-Morse strict local minima realising a large class of such trees.
Malte
Renken
What does the Cyclic Polytope have to do with Terrain Visibility Graphs?
Karola
Mészáros
Cornell University
Encoding polytopal properties with (reductions on) graphs
Abstract.
The flow polytope FG associated to an acyclic graph G is the set of all nonnegative flows of size 1 on the graph G. Fundamental questions about this and other flow polytopes include: (1) what is the polytope’s volume? (2) how many integer points are in it? I will explain how to use a reduction procedure on graphs to answer these question.
Neela
Nataraj
IIT Bombay
Finite element methods for nematic liquid crystals
Abstract.
We consider a system of second order non-linear elliptic partial differential equations that models the equilibrium configurations of a two dimensional planar bistable nematic liquid crystal device. Discontinuous Galerkin finite element methods are used to approximate the solutions of this nonlinear problem with non-homogeneous Dirichlet boundary conditions. A discrete inf-sup condition demonstrates the stability of the discontinuous Galerkin discretization of a well-posed linear problem. We then establish the existence and local uniqueness of the discrete solution of the non-linear problem. An a priori error estimate in the energy norm is derived and the quadratic convergence of Newton’s iterates is illustrated. This is a joint work with Ruma Maity and Apala Majumdar.
Julia
Schaeffer
HU Berlin
Convergence rates for the FEAST algorithm with dPG resolvent discretization
Sven
Jäger
Scheduling stochastic jobs with release dates on a single machine
Liana
Yepremyan
University of Oxford
On the size Ramsey number of bounded powers of bounded degree trees
Abstract.
We say a graph G is Ramsey for a graph H if every red/blue edge-colouring of the edges of G contains a monochromatic copy of H. The size Ramsey number of a graph H is defined to be the minimum number of edges among all graphs which are Ramsey for H. The study of size Ramsey numbers originated by the work of Erdős, Faudree, Rousseau and Schelp from 1970's. This number was studied for graphs including paths, cycles, powers of paths and cycles, trees of bounded degree. For all mentioned graphs it was shown that the size Ramsey number grows linearly in the number of vertices ("is linear"). This line of research was inspired by a question of Beck who asked whether the size Ramsey number is linear for graphs of bounded degree. Later this was disproved by Rödl and Szemerédi. In this talk I will present our recent result showing that fixed powers of bounded degree trees also have linear size Ramsey number. Equivalently, this result says that all graphs of bounded degree and bounded treewidth have linear size Ramsey number. We also obtain off-diagonal version of this result. Many exciting problems remain open. This is joint work with Nina Kam\v{c}ev, Anita Liebenau and David Wood.
Günter
Rote
Freie Universität Berlin
Lattice paths with states, and counting geometric objects via production matrices
Abstract.
We consider paths in the plane governed by the following rules: (a) There is a finite set of states. (b) For each state q, there is a finite set S(q) of allowable "steps" ((i,j),q′). This means that from any point (x,y) in state q, we can move to (x+i,y+j) in state q′. We want to count the number of paths that go from (0,0) in some starting state q0 to the point (n,0) without ever going below the x-axis. There are strong indications that, under some natural technical conditions, the number of such paths is asymptotic to C^n/(√n^3), for some "growth constant" C which I will show how to compute. I will discuss how lattice paths with states can be used to model asymptotic counting problems for some non-crossing geometric structures (such as trees, matchings, triangulations) on certain structured point sets. These problems were recently formulated in terms of so-called production matrices. This is ongoing joint work with Andrei Asinowski and Alexander Pilz.
October 2019
Amy
Wiebe
FU Berlin
Realization Spaces of Polytopes
Abstract.
In this talk I will present a model for the realization space of a polytope which represents a polytope by its slack matrix. This model provides a natural algebraic relaxation for the realization space, and comes with a defining ideal which can be used as a computational engine to answer questions about the realization space. We will see how this model is related to more classical realization space models (representing realizations by Gale diagrams or points of the Grassmannian). I will also show how the relationship of the slack model to the classical models can be used to improve computational efficiency of the slack model.
Max
Gunzburger
Florida State University
Integral equation modeling for anomalous diffusion and nonlocal mechanics
Abstract.
We use the canonical examples of fractional Laplacian and peridynamics equations to discuss their use as models for nonlocal diffusion and mechanics, respectively, via integral equations with singular kernels. We then proceed to discuss theories for the analysis and numerical analysis of the models considered, relying on a nonlocal vector calculus to define weak formulations in function space settings. In particular, we discuss the recently developed asymptotically compatible families of discretization schemes. Brief forays into examples and extensions are made, including obstacle problems and wave problems.
Manfred
Scheucher
Simple Topological Drawings, Rotation Systems, and SAT Solvers
Tien
Tran-Ngoc
HU Berlin
HHO methods for a class of degenerate convex minimization problems
Alp
Müyesser
FU Berlin
Turán-type results for unavoidable subgraphs
Abstract.
We say that a two-coloring of a K2k is k-unavoidable if one color forms either a clique of order k or two disjoint cliques of order k. Bollob\'as conjectured that for any ε and k, there exists an integer R(ε,k) such that any n≥R(ε,k) vertex graph with at least ε{n choose 2} edges in each color contains a k-unavoidable graph. This was proven by Cutler and Montágh, and Fox and Sudakov showed R(ε, k)=(1/ε)θ(k). Here, we obtain several Turán-type variants of this result, such as showing that in any two-colored graph with density 1-(1/ε)O(k) and at least ε fraction of the edge set in each color, there exists a k-unavoidable subgraph. A result with a similar flavour was recently obtained by DeVos et al. who showed that any graph with density above 2/3 whose edges are colored half red and half blue must contain a non-monochromatic triangle. We solve this same problem entirely for all cycles, and up to an additive constant, for all cliques. This is joint work with Boris Bukh and Michael Tait.
Moritz
Schmitt
TU Berlin
What is Hilbert's 17th problem?
Abstract.
At the International Congress of Mathematicians in 1900 in Paris, Hilbert presented his now famous list of 23 problems. Many of these were rather influential for 20th century mathematics. One of these is Hilbert's 17th problem: Given a real polynomial in several variables that is non-negative, can it be represented as a sum of squares of rational functions? Artin answered this questions in 1927 in the affirmative. In this talk we will take a closer look at both Hilbert's problem and Artin's solution, and try to understand how all this relates to the upcoming lecture of M. F. Roy.
Yuri
Bilu
U Bordeaux
Diversity in Parametric Families of Number Fields
Abstract.
Let $X$ be a projective curve over $\mathbb{Q}$ of genus $g$ and $t$ a non-constant $\mathbb{Q}$-rational function of degree $m>1$. For every $n\in \mathbb{N}$, pick $P_n\in X$ with $t(P_n)=n$. Hilbert's Irreducibility Theorem (HIT) says that for infinitely many $n$ the field $\mathbb{Q}(P_n)$ is of degree $m$ over $\mathbb{Q}$. Moreover, this holds for overwhelmingly many $n$: Among the number fields $\mathbb{Q}(P_1), \dots ,\mathbb{Q}(P_n)$ there is only $o(n)$ fields of degree less than $m$. However, HIT does not say how many distinct field there are among $\mathbb{Q}(P_1), ... ,\mathbb{Q}(P_n)$. A 1994 result of Dvornicich and Zannier implies that for large $n$, there are at least $cn/\log n$ distinct among those fields, with $c=c(m,g)>0$. Conjecturally there should be a positive proportion (that is, $cn$) of distinct fields. This conjecture is proved in many special cases in the work of Zannier and his collaborators, but in general, getting rid of the log term seems very hard. We make a little step towards proving this conjecture. While we cannot remove the log term altogether, we can replace it by log n raised to a power strictly smaller than 1. To be precise, we prove that for large $n$ there are at least $n/(\log n)^{1-e}$ distinct fields, where $e=e(m,g)>0$. A joint work with Florian Luca.
Tim
Ricken
Universität Stuttgart
Neue Methoden zur Modellierung von Mehrphasenmaterialien mit Mehrskalenansätzen
- Anwendungsbeispiele aus den Bereichen Materialwissenschaft, Biomechanik und Umwelttechnik
Arturo
Merino
The minimum cost query problem on matroids with uncertainty areas
Benjamin
Hauskeller
Humboldt-Universität zu Berlin
Dynamic Query Evaluation with Sublinear Update Time
Abstract.
Dynamic Query Evaluation considers the problem of evaluating a database query on a database using the following framework: In a preprocessing phase the query and an initial database are used to build a data structure which can enumerate the query result on the current database. Upon updates to the database the data structure is adapted to the new database. Research is then interested in the time needed for the preprocessing, the handling of an update, and the maximal delay between tuples during enumeration. In this talk I will present a new class of queries that can be maintained with linear preprocessing time, sublinear update time and constant delay.
Alexandre
Vigny
Universität Bremen
Query enumeration and nowhere dense classes of graphs
Abstract.
Given a query q and a relational structure D the enumeration of q over D consists in computing, one element at a time, the set q(D) of all solutions to q on D. The delay is the maximal time between two consecutive output and the preprocessing time is the time needed to produce the first solution. Idealy, we would like to have constant delay enumeration after linear preprocessing. Since this it is not always possible to achieve, we need to add restriction to the classes of structures and/or queries we consider. In this talk I will talk about some restrictions for which such algorithms exist: graphs with bounded degree, tree-like structures, conjunctive queries... We will more specifically consider nowhere dense classes of graphs: What are they? Why is this notion relevant? How to make algorithms from these graph properties?
Manfred
Scheucher
Holes and islands in random point sets
Michael
Anastos
FU Berlin
The longest path in a random graph has a scaling limit
Abstract.
Let G(n,p) denote the random graph on n vertices where each edge appears independently with probability p=c/n. It is well known that p=1/n is the threshold for the existence of a giant component. Erdős conjectured that if c > 1 then w.h.p. the length of the longest path of G(n,c/n), denoted by L(c,n), satisfies L(c,n) ≥ l(c)n where l(c) > 0 is independent of n. This was proved by Ajtai, Komlós and Szemerédi and in a slightly weaker form by de la Vega. The lower bound was then improved in a series of papers. In this talk we go a step further and we show that for sufficiently large c, w.h.p. L (c,n)/n tends to some function f(c) as n tends to infinity. In addition we give a method of computing f(c) to arbitrary accuracy. This talk is based on joint work with Alan M. Frieze.
Sophie
Puttkammer
HU Berlin
A modified HHO method to compute guaranteed lower eigenvalue bounds
Julian
Streitberger
HU Berlin
Two Lowest Order MFEM Examples
Christoph
Hertrich
Theoretical Aspects of Neural Networks for Solving Combinatorial Optimization Problems
Otfried
Cheong
Universität Bayreuth
Convex Covers and Translation Covers
Abstract.
In 1914, Lebesgue asked for a convex set of smallest possible area that can contain a congruent copy of every set of diameter one. The same question can be asked for other families T of planar shapes: What is the convex set of smallest possible area that contains a congruent copy of every element of T? Such a set is then called a convex cover for T, and we will see what smallest-area convex covers for some families of triangles look like. A translation cover for a family T of planar shapes is defined similarly: Z is a translation cover for T if every element of T can be translated into Z. Kakeya's celebrated needle problem, first posed in 1917, turns out to be a question about a smallest-area translation cover. We will see that the generalization of Kakeya's problem to other shapes is also a translation cover problem.
Georg
Lehner
FU Berlin
What is the point of pointless topology?
Abstract.
The open sets of a topological space together with union and intersection form a special kind of partially ordered set — a complete lattice in which finite meets distribute over arbitrary joins. These algebraic gadgets are called locales and as it turns out they behave very similar to the classical spaces we know from point set topology — with one major exception: There are interesting locales which have no points at all! We will give a brief glimpse into this pointfree world and hopefully convince you that giving up points is not as bad an idea as one might think.
Dana S.
Scott
Emeritus Carnegie Mellon University, Visiting Scientist/Scholar UC Berkeley
Geometry without points
Liam
Solus
MPI-Leipzig
s-derangement polynomials and some conjectures on lattice polytopes
Abstract.
A thriving industry in combinatorics centers around the investigation of distributional properties of combinatorial generating polynomials such as unimodality and log-concavity. In the context of lattice polytopes, one of the now longest standing conjectures pertaining to this general problem asserts that the (Ehrhart) h^*-polynomial of a reflexive and IDP lattice polytope is unimodal. In 2013, Schepers and Van Langenhoven put forth a series of questions that propose new methods for attacking this conjecture as based on a classic theorem of Betke and McMullen. In this talk, we will introduce a family of combinatorial generating polynomials, called s-derangement polynomials, that allow us to positively answer these questions and conjectures for some well-studied lattice polytopes. Key to these results is the fact that the s-derangement polynomials generalize the classical derangement polynomial in such a way that it inherits many of its desirable properties, including real-rootedness, symmetry, log-concavity, gamma-positivity, and unimodality. This talk is based on joint work with Nils Gustafsson.
Felix
Schröder
Bichains - 7 equivalent formulations
Georgi
Mitsov
HU Berlin
On the well-posedness of a generalized dPG time-stepping methods for the heat equation
Jean-Philippe
Labbé
FU Berlin
What is counting?
Abstract.
Symmetry is a central concept in many areas of mathematics, if not all of them. In this talk, I will present an intriguing numerical coincidence bridging two seemingly unrelated objects — the first one coming from representation theory, the second from discrete geometry — where symmetry could be the culprit. Confirming the guilt of symmetry would shed new light on both objects, and perhaps help to solve some conjecture in discrete geometry.
Dominic
Bunnett
TU Berlin
What is $M_{g,n}$?
Abstract.
$M_{g,n}$ is an algebraic variety which parameterises isomorphism classes of smooth curves of genus g and n marked points. The study of $M_{g,n}$ goes back to Riemann in 1857 and has been an object of study ever since, although the first rigorous construction is due to Mumford in 1965. We give a gentle introduction to the construction of $M_{g,n}$ and the techniques used to study its geometry.
Carlos
Améndola
Technische Universität München
What is estimation for Gaussian models?
Abstract.
The multivariate Gaussian distribution is fundamental in statistics. In this talk I will introduce two methods for estimating parameters: maximum likelihood and method of moments. Then I will present examples of how these apply to Gaussian covariance models and Gaussian mixture models.
Janin
Heuer
Technische Universität Braunschweig
What is a nonnegativity certificate?
Abstract.
Mathematicians have been studying nonnegativity of real polynomials since as early as the 19th century. Nonnegativity certificates are an important tool in these investigations, giving easier to check, sufficient conditions for nonnegativity. In this talk we will motivate the study of nonnegativity by relating it to polynomial optimization. Furthermore, we will define the nonnegativity certificates sums of squares (SOS) and sums of nonnegative circuit polynomials (SONC).
Josué
Tonelli-Cueto
TU Berlin
What is the probabilistic analysis of a condition number?
Abstract.
For a given problem, a condition number is a quantity depending on the data that measure the numerical sensitivity of the data to perturbations. This parameter plays a fundamental role in the complexity analysis of numerical algorithms, both from a run-time and precision control perspective. However, because of this, numerical algorithms tend to have complexity estimates that do not depend solely on the input size. The main philosophy to solve this is to perform a probabilistic analysis of the condition number assuming some reasonable probability distribution of the input. In this talk, we introduce the different ways in which such a probabilistic analysis can be done and the differences between the different approaches.
Raphael
Steiner
Majority Colourings of Sparse Digraphs
September 2019
Piotr
Micek
Graphs without long ladder minors
Sophie
Rehberg
Verallgemeinerte Permutaeder und Hopf-Monoide
Masashi
Sugiyama
RIKEN-AIP
/
University of Tokyo
Introduction of RIKEN Center for Advanced Intelligence Project
Abstract.
RIKEN is one of Japan's largest fundamental-research institutions. The [RIKEN Center for Advanced Intelligence Project (AIP)](https://aip.riken.jp/) was created in 2016 to propose and investigate new machine learning methodologies and algorithms, and apply them to societal problems. AIP covers a wide range of topics from generic AI research (machine learning, optimization, applied math., etc.), goal-oriented AI research (material, disaster, cancer, etc.), and AI-in-society research (ethics, data circulation, laws, etc.). In the first half of my talk, I will briefly introduce our center's activities and collaboration strategies. Then, in the latter half, I will talk about the research activities in my team, i.e., machine learning from imperfect information. Machine learning has been successfully used to solve various real-world problems. However, we are still facing many technical challenges, for example, we want to train machine learning systems without big labeled data and we want to reliably deploy machine learning systems under noisy environments. I will overview our recent advances in tackling these problems.
Jonas
Neukamm
Straight-line-triangle-representstions und ihre Berechnung
Stefan
Felsner
Circle graphs are quadratically chi-bounded
August 2019
Alexandra
Wesolek
Simon Fraser University
Weak rainbow saturation
Abstract.
Rainbow saturation asks for the minimum number of edges in a t-edge-coloured graph G such that it does not contain a rainbow copy of some fixed graph H but adding any missing edge in any colour from [t] completes a rainbow copy of H. It is an extension of the uncoloured saturation problem which was first studied by Erdős, Hajnal and Moon in 1964. Surprisingly, if H=Ks, the complete graph on s vertices, and G is a graph on n vertices, the minimum number of edges that we need is Θ(n log n) while in the uncoloured saturation problem we only need a linear number of edges. In this work we investigate weak versions of rainbow saturation. In the usual rainbow saturation problem we can choose an arbitrary ordering of the missing edges and add all the missing edges back to the graph in that order and in arbitrary colours completing a rainbow copy each time we add an edge. Instead of being able to add the edges in any ordering to the graph, in a weaker setting we ask for only one specific ordering in which we need to be able to add them back. Is the behaviour for the minimum sets in this case still Θ(n log n)? In the uncoloured setting the bound for this weakened version is linear, so the answer needs to be between these bounds. Moreover, we investigate how the problem changes if we are allowed to choose a colour for every edge that we add. What if we can choose the colours and an ordering? We give answers to those questions. This represents joint work with Shagnik Das.
Basudeb
Datta
Indian Institute of Science
Stacked and tight triangulations of manifolds
Abstract.
Walkup constructed a class of triangulated d-manifolds from a stacked d-sphere by adding successive handles. Now, we know that Walkup's class of manifolds are same as the class of stacked manifolds. (A stacked triangulated manifold is the boundary of triangulated manifold whose co-dimension 2 spaces are on the boundary.) Tight triangulated manifolds are generalisations of neighbourly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. A triangulation of a closed 2-manifold is tight (with respect to a field F) if and only if it is F-orientable and neighbourly. Recently, it is shown that a triangulation of a closed 3-manifold is tight (with respect to a field F) if and only if it is F-orientable, neighbourly and stacked. Thus every tight triangulation of closed 3-manifold is strongly minimal. Except few, all the known tight triangulated manifolds are stacked. From some recent works, we know more on tight triangulations. In this talk, we present a survey on the works done on stacked and tight triangulation.
Dongho
Chae
Chung-Ang University, Seoul, South Korea
On the blow-up criterion for the 3D axisymmetric Euler equations
Jörg
Wolf
On the Serrin condition of one velocity component for solutions to theNavier-Stokes equations
Hiệp
Hàn
University of Santiago de Chile
Quasi-random words and limits of word sequences
Abstract.
Words are finite sequences of letters over a finite alphabet and in this talk we address two intimately related topics for this object: quasi-randomness and limit theory.With respect to the first topic we study the notion of uniform distribution of letters over intervals, and in the spirit of the famous Chung-Graham-Wilson theorem we provide a list of word properties which are equivalent to this property. This investigation naturally gives rise to a notion of convergent word sequences, and in the second part of our work we develop a limit theory for such sequences. Via this theory we address several problems such as property testing and finite forcibility and obtain as a byproduct a new model of random word sequences.Joint work with Matias Pavez-Signe and Marcos Kiwi.
Sandro
Roch
Sortieren in Netzwerken aus Stacks und Queues
July 2019
Madeline
Brandt
Berkeley/Leipzig
Matroids and their Dressians
Abstract.
In this talk we will explore Dressians of matroids. Dressians have many lives: they parametrize tropical linear spaces, their points induce regular matroid subdivisions of the matroid polytope, they parametrize valuations of a given matroid, and they are a tropical prevariety formed from certain Plücker equations. We show that initial matroids correspond to cells in regular matroid subdivisions of matroid polytopes, and we characterize matroids that do not admit any proper matroid subdivisions. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range of interesting matroids. If time permits, we will also discuss an ongoing project extending these ideas to flag matroids.
Benjamin
Gardiner
TU Berlin
What is a ranking algorithm?
Abstract.
A search engine is a software, similar to Google, that presents a list of relevant content from a large data set, where criteria for relevance is usually based off of user input. Further, a search engine will usually rank the data according to relevance to maximize utility for the user. A ranking algorithm assigns a numerical rank to each element of its data set, and then returns the sorted list. In this talk we will explore a couple of well known ranking algorithms including Page Rank and HITS (Hyperlink-Induced Topic Search).
Vijay
Vazirani
University of California, Irvine
Matching is as Easy as the Decision Problem, in the NC Model
Abstract.
Is matching in NC, i.e., is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in TCS for over three decades, ever since the discovery of Random NC matching algorithms. Over the last five years, the TCS community has launched a relentless attack on this question, leading to the discovery of numerous powerful ideas. We give what appears to be the culmination of this line of work: An NC algorithm for finding a minimum weight perfect matching in a general graph with polynomially bounded edge weights, provided it is given an oracle for the decision problem. Consequently, for settling the main open problem, it suffices to obtain an NC algorithm for the decision problem. We believe this new fact has qualitatively changed the nature of this open problem.Our result builds on the work of Anari and Vazirani (2018), which used planarity of the input graph critically; in fact, in three different ways. Our main challenge was to adapt these steps to general graphs by appropriately trading planarity with the use of the decision oracle. The latter was made possible by the use of several of the ideadiscovered over the last five years. The difficulty of obtaining an NC perfect matching algorithm led researchers to study matching vis-a-vis clever relaxations of the class NC. In this vein, Goldwasser and Grossman (2015) gave a pseudo-deterministic RNC algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC algorithm with the additional requirement that on the same graph, it should return the same (i.e., unique) perfect matching for almost all choices of random bits. A corollary of our reduction is an analogous algorithm for general graphs. This talk is fully self-contained. Based on the following joint paper with Nima Anari: https://arxiv.org/pdf/1901.10387.pdf
Michelle
Döring
Flip Graphs, Topological Drawings and Separable Permutations
Amiya
Pani
IIT Bombay
On Fractional in Time Evolution Problems: Some Theoretical and Computational Studies
Federico
Castillo
University of Kansas
Deformations of Coxeter permutahedra and Coxeter submodular functions
Abstract.
One way to decompose a polytope is to represent it as a Minkowski of two other polytopes. These smaller pieces are naturally called summands. Starting from a polytope we want to explore the set of all summands. This set can be parametrized by a polyhedral cone, called deformation cone, in a suitable real vector space. We focus on the case where the starting polytope is a Coxeter permutahedron, which is a polytope naturally associated with a root system. This generalizes the type A case which correspond to generalized permutohedra. This is joint work with Federico Ardila, Chris Eur, and Alexander Postnikov.
Julia
Schaeffer
HU Berlin
FEAST spectral approximation using dPG resolvent discretization
Grégoire
Menet
U Grenoble
Integral cohomology of quotients via toric geometry
Abstract.
I will provide some new methods, based on toric blow-ups, to determine the integral cohomology of complex manifolds quotiented by automorphisms groups of prime order. Indeed, quotient singularities can locally be interpreted as toric varieties, and the framework of toric geometry is well adapted to deal with integral cohomology. The original motivation to study this problem was the computation of an important invariant in the context of hyperhähler geometry: the Beauville-Bogomolov form.
Jonas
Israel
On the price of anarchy for flows over time with spillback
Karl
Däubel
An Improved Upper Bound for the Ring Loading Problem
Anita
Liebenau
University of New South Wales, Sydney
Enumerating graphs and other discrete structures by degree sequence
Abstract.
How many d-regular graphs are there on n vertices? What is the probability that G(n,p) has a specific given degree sequence, where G(n,p) is the homogeneous random graph in which every edge is inserted with probability p? Asymptotic formulae for the first question are known when d=o(\sqrt(n)) and when d= \Omega(n). More generally, asymptotic formulae are known for the number of graphs with a given degree sequence, for a wide enough range of degree sequences. From these enumeration formulae one can then deduce asymptotic formulae for the second question. McKay and Wormald showed that the formulae for the sparse case and the dense case can be cast into a common form, and then conjectured in 1990 and 1997 that the same formulae should hold for the gap range. A particular consequence of both conjectures is that the degree sequence of the random graph G(n,p) can be approximated by a sequence of n independent binomial variables Bin(n − 1, p'). In 2017, Nick Wormald and I proved both conjectures. In this talk I will describe the problem and survey some of the earlier methods to showcase the differences to our new methods. I shall also report on enumeration results of other discrete structures, such as bipartite graphs and hypergraphs, that are obtained by adjusting our methods to those settings.
Alp
Müyesser
Carnegie Mellon
What is a positional game?
Abstract.
We introduce a systematic study of Tic-Tac-Toe-like games, starting with concrete examples like Hex, and then moving on to a more abstract treatment via Maker-Breaker games played on arbitrary hyper-graphs. As time permits, we will explore connections with other fields, including algebraic topology, the probabilistic method, Ramsey theory, and computational complexity theory.
Anna-Lisa
Sokol
TU Berlin
What is a random walk in random environment?
Abstract.
The random walk on a graph is a widely known process in natural science. But what happens if every edge of the graph is equipped with a random variable deciding if it is open or closed? Meaning the graph itself becomes random. How does this process behave on large timescales? In this talk I will give an overview of the matter and state some interesting results.
Johannes
Storn
HU Berlin
Topics in Least-Squares and Discontinuous Petrov-Galerkin Finite Element Analysis
Abstract.
The first part of this thesis defense explores the accuracy of solutions to the LSFEM. It combines properties of the underlying partial differential equation with properties of the LSFEM and so proves the asymptotic equality of the error and a computable residual. Moreover, this talk introduces an novel scheme for the computation of guaranteed upper error bounds. While the established error estimator leads to a significant overestimation of the error, numerical experiments indicate a tiny overestimation with the novel bound. The investigation of error bounds for the Stokes problem visualizes a relation of the LSFEM and the Ladyzhenskaya-Babuska-Brezzi (LBB) constant. This constant is a key in the existence and stability of solution to problems in fluid dynamics. The second part of this talk utilizes this relation to design a competitive numerical scheme for the computation of the LBB constant. The third part of the talk investigates the DPG method. This investigation relates the DPG method with the LSFEM. Hence, the results from the first part of this talk extend to the DPG method. This enables precise investigations of existing and the design of novel DPG schemes.
Piotr
Micek
Weak coloring numbers of planar graphs
Takashi
Kumagai
Kyoto U
Anomalous random walk and diffusion in random media
Rekha
Thomas
Univ. Washington
Geometry and Algebra in Optimization
Abstract.
While optimization is a well established field with myriad tools, the use of algebraic methods to tackle optimization problems, beyond those from linear algebra, is relatively new. Algebraic models inherently bring in geometry via the language of algebraic geometry, which studies solutions to polynomial equations and inequalities. Convexity enters naturally as well since optimizing a linear function over a set is the same as optimizing the function over its convex hull. In this talk I will illustrate a sequence of ideas and results that use polynomials to tackle a variety of optimization problems. On the theoretical side, this will touch on theta bodies, lifts of convex sets, tightness of relaxations, and the use of symmetries. If there is extra time, I will talk about some applications to computer vision and combinatorics.
Nina
Kamcev
Monash University
Asymptotic enumeration of regular hypergraphs
Abstract.
We prove an asymptotic formula for the number of k-uniform hypergraphs with a given degree sequence, for a wide range of parameters. In particular, we can asymptotically enumerate d-regular k-graphs of density O(1/k) for 3 ≤ k< n1/10. This extends the recent results for graphs of Liebenau and Wormald.It follows that within our scope, the distribution of the degree sequence of a random k-uniform hypergraph can be approximated by a sequence of independent random variables with the appropriate binomial distribution. This is joint work with Anita Liebenau and Nick Wormald.
Yuri
Bilu
U Bordeaux
Singular units do not exist
Abstract.
It is classically known that a singular modulus (a j-invariant of a CM elliptic curve) is an algebraic integer. Habegger (2015) proved that at most finitely many singular moduli are units, answering a question of Masser (2011). However, his argument, being non-effective, did not imply any bound for the size of these "singular units". I will report on a recent work with Philipp Habegger and Lars Kühne, where we prove that singular units do not exist at all. First, we bound the discriminant of any singular unit by 10^15. Next, we rule out the remaining singular units using computer-assisted arguments.
Manfred
Scheucher
Technische Universität Berlin
On Arrangements of Pseudocircles
Abstract.
Towards a better understanding of arrangements of circles and also to get rid of geometric difficulties, we look at the more general setting of ''arrangements of pseudocircles'' which was first introduced by Grünbaum in the 1970's. An arrangement of pseudocircles is a collection of simple closed curves on the sphere or in the plane such that any two of the curves are either disjoint or intersect in exactly two points, where the two curves cross. In his book, Grünbaum conjectured that every digon-free arrangement of n pairwise intersecting pseudocircles contains at least $2n-4$ triangular cells. We present arrangements to disprove this conjecture and give new bounds on the number of triangular cells for various classes of arrangements. Furthermore, we study the ''circularizability'' of arrangements: it is clear that every arrangement of circles is an arrangement of pseudocircles, however, deciding whether an arrangement of pseudocircles is isomorphic to an arrangement of circles is computationally hard. Using a computer program, we have enumerated all combinatorially different arrangements of up to $7$ pseudocircles. For the class of arrangements of $5$ pseudocircles and for the class of digon-free intersecting arrangements of $6$ pseudocircles, we give a complete classification: we either provide a circle representation or a non-circularizability proof. For these proofs we use incidence theorems like Miquel's and arguments based on continuous deformation, where circles of an assumed circle representation grow or shrink in a controlled way. This talk summarizes results from two articles, which are both joint work with Stefan Felsner: * Arrangements of Pseudocircles: Triangles and Drawings; short version in Proc. GD'17; full version available at arXiv (1708.06449) * Arrangements of Pseudocircles: On Circularizability; short version in Proc. GD'18; full version in DCG: Ricky Pollack Memorial Issue (doi:10.1007/s00454-019-00077-y)
Rekha R.
Thomas
University of Washington
Graph Density Inequalities and Sums of Squares
Abstract.
Many results in extremal graph theory can be formulated as inequalities on graph densities. While many inequalites are known, many more are conjectured. A standard tool to establish an inequality is to write the expression whose nonnegativity needs to be certified as a sum of squares. This technique has had many successes but also limitations. In this talk I will describe new restrictions that show that several simple inequalities cannot be certified by sums of squares. These results extend to the powerful frameworks of flag algebras by Razborov and graph algebras by Lovasz and Szegedy. This is joint work with Greg Blekherman, Annie Raymond, and Mohit Singh.
June 2019
Josué
Tonelli-Cueto
TU Berlin
What is semialgebraic sets in context?
Abstract.
Semialgebraic sets are a quite general class of sets that can be described by reals polynomials and inequalities. In this talk, we will show how semialgebraic sets appear naturally in many different contexts: a) real algebraic geometry, b) discrete geometry, c) robotic arms and d) rigid models of molecules.
Felix
Schröder
Spherical Thrackles and Conway's Conjecture
Neela
Nataraj
IIT Bombay
Morley FEM for a distributed optimal control problem governed by the von Karman equations
Abstract.
In this talk, we consider the distributed optimal control problem governed by the von Karman equations that describe the deflection of very thin plates defined on a polygonal domain of ℝ² with box constraints on the control variable. The talk discusses a numerical approximation of the problem that employs the Morley nonconforming finite element method to discretize the state and adjoint variables. The control is discretized using piecewise constants. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Error estimates in lower order norms for the state and adjoint variables are derived. The lower order estimates for the adjoint variable and a post-processing of control leads to an improved error estimate for the control variable. Numerical results confirm the theoretical results obtained. This is a joint work with Sudipto Chowdhury and Devika Shylaja.
Mara
Nehring
FU Berlin
Online Ramsey Numbers
Abstract.
The Online Ramsey Game is a game on an unbounded number of vertices played by two players, Builder and Painter. Builder builds an edge between two vertices and Painter paints it immediately red or blue. Builder's goal is to build a monochromatic red copy of Km or a monochromatic blue copy of Kn while Painter tries to delay this for as long as possible. The online Ramsey number is the minimum number of edges Builder needs to guarantee a win regardless of Painter's strategy. We will show new lower and upper bounds for the online Ramsey number. Work by David Conlon, Jacob Fox, Andrey Ginshpun and Xiaoyu He.
Ana Maria
Botero
U Regensburg
The convex set algebra and the $b$-Chow ring of toric varieties
Abstract.
We extend McMullen's polytope algebra to the so called convex set algebra. We show that the convex set algebra embeds in the projective limit of the Chow cohomology rings of all smooth toric compactifications of a given torus, with image generated by the classes of all nef toric $b$-divisors. It follows that the convex set algebra can be viewed as a universal ring for intersection theory of nef toric $b$-cocycles on the toric Riemann-Zariski space. We further discuss some applications of this viewpoint towards a combinatorial interpretation of non-archimidean Arakelov theory of toric varieties over discretely valued fields in the sense of Bloch-Gillet-Soulé.
Simona
Boyadzhiyska
Freie Universität Berlin
On counting problems related to (mutually) orthogonal Latin squares
Abstract.
An n×n array with entries in [n] such that each integer appears exactly once in every row and every column is called a Latin square of order n. Two Latin squares L and L' are said to be orthogonal if, for all x,y∈[n], there is a unique pair (i,j) such that L(i,j) = x and L'(i,j) = y; k Latin squares are mutually orthogonal if any two of them are orthogonal.After the question of existence of a combinatorial structure satisfying given properties, a natural and important problem is to determine how many such objects there are. In this talk, we will consider some counting questions related to (mutually) orthogonal Latin squares. We will present an upper bound on the number of ways to extend a set of k mutually orthogonal Latin squares to a set of k+1 mutually orthogonal Latin squares and discuss some applications, comparing the resulting bounds to previously known lower and upper bounds.This talk is based on joint work with Shagnik Das and Tibor Szabó.
Emo
Welzl
ETH Zürich
Connectivity of Triangulation Flip Graphs in the Plane
Abstract.
In a straight line embedded triangulation of a point set P in the plane, removing an inner edge and - provided the resulting quadrilateral Q is convex - adding the other diagonal is called an edge flip. The flip graph has all triangulations as vertices and a pair of triangulations is adjacent, if they can be obtained from each other by an edge flip. This presentation is towards a better understanding of this graph, with emphasis on its connectivity. It is known that every triangulation allows at least n/2-2 edge flips and we show (n/2-2)-vertex connectivity for flip graphs of all P in general position, n:=|P|. Somewhat stronger, but restricted to P large enough, we show that the vertex connectivity is determined by the minimum degree occurring in the flip graph, i.e. the minimum number of flippable edges in any triangulation of P. A corresponding result is shown for so-called partial triangulations, i.e. the set of all triangulations of subsets of P which contain all extreme points of P. Here the flip operation is extended to bistellar flips (edge flip, and insertion and removal of an inner vertex of degree three). We prove (n-3)-edge connectedness for all P in general position and (n-3)-vertex connectedness for n large enough ((n-3) is tight, since there is always a partial triangulation which allows exactly n-3 bistellar flips). This matches the situation known (through the secondary polytope) for regular triangulations (i.e. partial triangulations obtained by lifting the points and projecting the lower convex hull). This is joint work with Uli Wagner, IST Austria.
Joachim
Naumann
On the existence of weak solutions to the Maxwell equations with linear space-time dependent constitutive law
Gari Yamel
Peralta Alvarez
HU Berlin
What is Klein's $j$-invariant?
Abstract.
The $j$-invariant is a special example of a modular form: a complex-valued function on the upper half-plane which behaves “nicely” respect to a group of symmetries. Modular forms have surprising connections with several areas of mathematics, in particular number theory. In this seminar we cover basic definitions of the theory of modular forms making emphasis on its connection with complex elliptic curves, from which the $j$-invariant arise naturally. Finally, we will try to illustrate how the $j$-invariant encodes meaningful information for several mathematical objects.
Katrin
Wendland
U Freiburg
Representations of Moonshine
Jose
Samper
MPI Leipzig
Slicing matroid polytopes
Abstract.
We introduce the notion of a matroid threshold hypergraph: a set system obtained by slicing a matroid basis polytope and keeping the bases on one side. These hypergraphs can be then used to define a class of objects that slightly extends matroids and has a few new technical advantages. For example, it is amenable to several inductive procedures that are out of reach to matroid theory. In this talk we will motivate the study of these families of hypergraphs, relate it to some old open questions and pose some new conjectures. To conclude we will show that the study of these objects helps us find a link between the geometry of the normal fan of the matroid polytope and shellability invariants/activities of independence complexes.
Gaddam
Sharat
IIT Bombay
A Posteriori Error Estimates of Discontinuous Galerkin Methods for the Elliptic Obstacle Problem
Abstract.
This is a continuation of the previous talk titled "Two New Approaches for Solving Elliptic Obstacle Problems Using Discontinuous Galerkin Methods". In this talk, we will construct the discrete Lagrange Multipliers for both the methods(Integral constraints method and Quadrature point constraints method) and also derive the optimal order(with respect to regularity) a priori error estimates. Later part of the talk focuses upon a posteriori error estimates where we will construct error estimators for both the methods and we will have a look at the reliability of the estimators. Finally, we conclude the talk by presenting the numerical experiments checking the reliability and efficiency of the estimators.
Tamás
Mészáros
FU Berlin
Unlabeled compression schemes and corner peelings for ample and maximum classes
Abstract.
I will be presenting recent work by Chalopin, Chepoi, Moran and Warmuth. "We examine connections between combinatorial notions that arise in machine learning and topological notions in cubical/simplicial geometry. These connections enable to export results from geometry to machine learning. Our first main result is based on a geometric construction by Tracy Hall (2004) of a partial shelling of the cross polytope which can not be extended. We use it to derive a maximum class of VC dimension 3 that has no corners. This refutes several previous works in machine learning from the past 11 years. In particular, it implies that all previous constructions of optimal unlabeled sample compression schemes for maximum classes are erroneous.On the positive side we present a new construction of an unlabeled sample compression scheme for maximum classes. We leave as open whether our unlabaled sample compression scheme extends to ample (a.k.a. lopsided or extremal) classes, which represent a natural and far-reaching generalization of maximum classes. Towards resolving this question, we provide a geometric characterization in terms of unique sink orientations of the 1-skeletons of associated cubical complexes."
Giulio
Orecchia
U Rennes
a tropical/monodromy criterion for existence of Néron models
Abstract.
Néron models are central objects to the study of degenerations of abelian varieties over Dedekind schemes. However, over bases of higher dimension, they do not always exist. In this talk I will introduce a criterion, called toric additivity, for an abelian family with semistable reduction to admit a Néron model. It can be expressed in terms of monodromy action on the $ extbackslash$ell-adic Tate module, and, in the case of jacobians of curves, in terms of finiteness of the tropical jacobian.
Guillaume
Sagnol
An unexpected connection between A-optimal designs and the Group Lasso
Matthias
Beck
San Francisco State University
Lonely Runner Polyhedra
Abstract.
We study the Lonely Runner Conjecture, conceived by Wills in the 1960's: Given positive integers n_1, n_2, ..., n_k, there exists a positive real number t such that for all 1 ≤ j ≤ k the distance of tn_j to the nearest integer is at least 1/(k+1). Continuing a view-obstruction approach by Cusick and recent work by Henze and Malikiosis, our goal is to promote a polyhedral ansatz to the Lonely Runner Conjecture. Our results include geometric proofs of some folklore results that are only implicit in the existing literature, a new family of affirmative instances defined by the parities of the speeds, and geometrically motivated conjectures whose resolution would shed further light on the Lonely Runner Conjecture.
Tom
Klose
TU Berlin
What is a regularity structure?
Abstract.
Commonly, the regularity of a function describes how well it is approximated by polynomials. This understanding breaks down when we consider solutions to PDEs perturbed by a highly irregular stochastic object called white noise: Polynomials are simply too crude to be the right approximating quantity in this case, but what is? To answer this question, I will introduce the notion of a regularity structure pioneered by Martin Hairer. We will see that it is a far-reaching generalisation of Taylor series that is robust enough to set up a solution theory for afore-mentioned (stochastic) PDEs.
Raphael
Steiner
Counterexamples to Hedetniemi's Conjecture
Giulia
Codenotti
FU Berlin
The covering minima of lattice polytopes
Abstract.
The covering minima of a convex body were introduced by Kannan and Lovasz to give a better bound in the flatness theorem, which states that lattice point free convex bodies cannot have arbitrarily large width. These minima are similar in flavor to Minkowski's successive minima, and on the other hand generalize the covering radius of a convex body. I will speak about recent joint work with Francisco Santos and Matthias Schymura, where we investigate extremal values of these covering minima for lattice polytopes.
Mauro
Perego
Generalized Moving Least Squares: Approximation Theory and Applications
Abstract.
(joint work with P. Bochev, P. Kuberry, N. Trask) In this talk we present existence and approximation results for the reconstruction of a few classes of linear functionals, including differential and integral functionals, using the Generalized Moving Least Square (GMLS) method. These results extend or specialize classical MLS theoretical results, and rely both on the classic approximation theory for finite elements and on existence/approximation results for scattered data. In particular, we will consider the reconstruction of vector fields in Sobolev spaces and the reconstruction of differential k-forms. We show how these results can be applied to data transfer problems and to design collocation and variational meshless schemes for the solution of partial differential equations.
Tibor
Szabó
FU Berlin
Slow bootstrap percolation
Abstract.
Graph-bootstrap percolation, also known as weak saturation, was introduced by Bollobás in 1968. In this process, we start with initial "infected" set of edges E0, and infect new edges according to a predetermined rule. Given a graph H and a set of previously infected edges Et ⊆ E(Kn), we infect a non-infected edge e if it completes a new copy of H in G=([n], Et ∪ e). A question raised by Bollobás asks for the maximum time the process can run before it stabilizes. Bollobás, Przykucki, Riordan, and Sahasrabudhe considered this problem for the most natural case where H=Kr. They answered the question for r ≤ 4 and gave a non-trivial lower bound for every r ≤ 5. They also conjectured that the maximal running time is o(n2) for every integer r. In a joint work with József Balogh, Gal Kronenberg, and Alexey Pokrovskiy we disprove their conjecture for every r ≥ 6 and we give a better lower bound for the case that r=5 using Behrend's construction of large 3-term arithmetic progression-free sets of integers.
Derk
Frerichs
HU Berlin
On pressure robustness and adaptivity of a Virtual Element Method for the Stokes problem
Abstract.
For the Stokes problem, many of the standard finite element method, e.g., Taylor-Hood, and also the virtual element method (VEM) proposed in [1] fail when it comes to small viscosity parameters ν or when the continuous pressure is complicated. In this talk, a modification of the VEM is presented which makes the method pressure robust, i.e., locking free for very small ν. For this purpose, a standard interpolation into Raviart-Thomas spaces of order k-1 will be employed which allows for a pressure robust modification of the discrete right hand side. In addition, an reliable error estimator is presented that makes adaptive mesh refinement possible. The presented numerical results will round up the presentation. [1] L. B. da Veiga, C. Lovadina, G. Vacca: Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM: M2AN, 51(2). 2017
Antonio
Rojas-Leon
U Sevilla
Local Systems with sporadic monodromy groups
Abstract.
Let $X$ be a curve over a separably closed field $k$ of characteristic $p$ and $F$ an $ extbackslashell$-adic local system on $X$, where $\ell \neq p$. If we view this local system as a representation of $\pi_1(X)$, its Zariski closure is the global monodromy group $G$ of $F$. When $X$ and $F$ are defined over a finite field, this group determines (after a normalization) the distribution of the Frobenius traces of $F$ on the points of $X$ with values over sufficiently large finite extensions of the base field. In general, one expects this group to be as large as allowed for by the geometric properties of $F$. In particular, only in exceptional cases it will be finite. Abhkanyar's conjecture determines which finite groups can appear as monodromy groups of such local systems. In particular, if $G$ is simple finite, it can be the monodromy group of a local system on the affine line in characteristic $p$ if and only if $p$ divides the order of $G$. In this talk we will give some naturally constructed examples of local systems on the affine line and the punctured affine line on small characteristic whose monodromy groups are sporadic finite: the Conway groups Co1, Co2, Co3, the Suzuki group 6.Suz or the McLaughlin group McL. This is joint work with Nicholas M. Katz (Princeton) and Pham H. Tiep (Rutgers).
Patrick
Graf
U Bayreuth
Reflexive differential forms in positive characteristic
Abstract.
Given a differential form on the smooth locus of a normal variety defined over a field of positive characteristic, we discuss under what conditions it extends to a resolution of singularities (possibly with logarithmic poles). Our main result works for log canonical surface pairs over a perfect field of characteristic at least seven. We also give a number of examples showing that our results are sharp in the surface case, and that they fail in higher dimensions. If time permits, we will give applications to the study of the Lipman-Zariski conjecture.
Hao
Huang
Emory University
A spectral proof of Kleitman's diametric theorem
Peter
Krautzberger
What is an ultrafilter?
Abstract.
A historic re-enactment of the very first “What is ...?” Seminar talk, 0% fewer mistakes guaranteed. We'll introduce the basic notions for ultrafilters, discuss fundamental examples and properties and, if time permits, build a simple model of hyperreals.
Marta
Panizzut
TU Berlin
What is Betti numbers?
Abstract.
Betti numbers are nonnegative integers that helps classifying topological spaces. Loosely speaking, the $n$th Betti number counts the number of $n$-dimensional holes. They are defined as ranks of homology groups. In this talk we will introduce them by looking at simplicial homology and focusing on many examples.
Kristin
Shaw
U Oslo
On the topology of real algebraic hypersurfaces
Gaddam
Sharat
IIT Bombay
Two New Approaches for Solving Elliptic Obstacle Problems Using Discontinuous Galerkin Methods
Abstract.
The main aim of the talk is to present two new ways to solve the elliptic obstacle problem by using discontinuous Galerkin finite element methods. In the talk, using the localized behaviour of DG methods, we discuss an optimal order(with respect to regularity) {\em a priori} error estimates, in 3 dimensions. We consider two different discrete sets, one with integral constraints and the other with nodal constraints at quadrature points. The analysis is carried out in a unified setting which holds true for several DG methods using quadratic polynomials.
Cesar
Ceballos
Univ. Wien
The s-weak order and s-permutahedra
Abstract.
The classical weak order on permutations and the permutahedron are well studied objects in algebraic and geometric combinatorics. In this talk, I will present a generalization of these objects, indexed by a sequence of non-negative integers s. The s-permutahedron has a beautiful geometric structure which is conjectured to be realizable as the polyhedral complex induced by a polytopal subdivision of a polytope. We will present a solution to this conjecture in dimensions 2 and 3, discuss some of its combinatorial properties, and present some connections to the v-Tamari lattices of Préville-Ratelle and Viennot. This is joint work with Viviane Pons.
Devika
Shylaja
IIT Bombay
The Hessian Discretisation Method for fourth order elliptic equations
Abstract.
Fourth order elliptic partial differential equations appear in various domains of mechanics. They model for example thin plates deformations as well as 2D turbulent flows through the vorticity formulation of Navier-Stokes equations. Many numerical methods, most of them are finite elements, have been developed over the years to approximate the solutions of these models. In this talk, we will present the Hessian Discretisation Method (HDM), a generic analysis framework that encompasses many numerical methods for fourth-order problems: conforming and nonconforming finite element methods, methods based on gradient recovery operators, and finite volume-based schemes. The principle of the HDM is to describe a numerical method using a set of four discrete objects, together called a Hessian Discretisation (HD): the space of unknowns, and three operators reconstructing respectively a function, a gradient and a Hessian. Each choice of HD corresponds to a specific numerical scheme. The beauty of the HDM framework is to identify four model-independent properties on an HD that ensure that the corresponding scheme converges for a variety of models, linear as well as non-linear.
Fleurianne
Bertrand
and
Carsten
Carstensen
HU Berlin
A posteriori error estimation for HHO-methods - Part II
Anurag
Bishnoi
FU Berlin
A construction for clique-free pseudorandom graphs
Abstract.
A construction of Alon and Krivelevich gives highly pseudorandom Kk-free graphs on n vertices with edge density equal to Θ(n-1/(k-2)). In this talk I will give a construction of an infinite family of highly pseudorandom Kk-free graphs with a higher edge density of Θ(n-1/(k-1)), for all k≥3, using the geometry of quadratic forms over finite fields. (Joint work with Ferdinand Ihringer and Valentina Pepe.)
Quentin
Guignard
ENS Paris
Geometric $\ell$-adic local factors
Abstract.
I will explain how to give a cohomological definition of epsilon factors for $\ell$-adic sheaves over a henselian trait of positive equicharacteristic distinct from $\ell$. The resulting formula is reminiscent of the cohomological construction by Katz of the $\ell$-adic Swan representation, and involves Gabber-Katz extensions as well. These local factors provide a product formula for the determinant of the cohomology of an $\ell$-adic sheaf on a curve over a field of positive characteristic distinct from $\ell$. When the base field is finite, this specializes to the classical theory of Dwork, Langlands, Deligne, and Laumon.
Lutz
Recke
Semigroup approach to initial-boundary value problems for the Maxwell system
May 2019
Michael
Feischl
TU Wien
Baysian FEM
Riccardo
Morandin
TU Berlin
What is a differential-algebraic equation?
Abstract.
The dynamical behavior of physical processes is usually modeled via differential equations. But if the states of the physical system are in some ways constrained, like for example by conservation laws or position constraints, then the mathematical model also contains algebraic equations to describe these constraints. Such systems, consisting of both differential and algebraic equations, are called differential-algebraic systems. In this talk, we introduce linear differential-algebraic equations, both with constant and variable coefficients. In particular, we will present their canonical forms, and we will discuss what do they imply about the existence, uniqueness and smoothness of solutions.
Helena
Bergold
Hyperbolicity cones of elementary symmetric polynomials are spectrahedral
Dietmar
Gallistl
University of Twente
Taylor-Hood discretization of the Reissner-Mindlin plate
Ezra
Waxman
U Prague
Hecke Characters and the $L$-Function Ratios Conjecture
Abstract.
A Gaussian prime is a prime element in the ring of Gaussian integers $\mathbb{Z}[i]$. As the Gaussian integers lie on the plane, interesting questions about their geometric properties can be asked which have no classical analogue among the ordinary primes. Hecke proved that the Gaussian primes are equidistributed across sectors of the complex plane by making use of Hecke characters and their associated $L$-functions. In this talk I will present several applications obtained upon applying the $L$-functions Ratios Conjecture to this family of $L$-functions. In particular, I will present a refined conjecture for the variance of Gaussian primes across sectors, and a conjecture for the one level density across this family.
Daniel
Schmidt
The price of fixed assignments in stochastic extensible bin packing
Felix
Schröder
Technische Universität Berlin
Lower Bounds on the p-centered coloring number
Abstract.
A p-centered coloring is a vertex-coloring of a graph G such that every connected subgraph H of G either receives more than p colors or there is a color that appears exactly once in H. The concept was introduced by Nešetřil and Ossona de Mendez as a local condition for measuring sparsity. We prove lower bounds on the p-centered coloring numbers. For outerplanar graphs, we show that their maximum p-centered coloring number is in Theta(p log p). We have examples of graphs of treewidth k needing (p+k choose k) colors, this matches the upper bound of Pilipczuk and Siebertz. We show that planar graphs may require Omega(p^2 log(p)) colors, while all of them admit a p-centered coloring with O(p^3 log(p)) colors. This improves an O(p^19) bound by Pilipczuk and Siebertz.
William T.
Trotter
Georgia Institute of Technology
Stability Analysis for Posets
Abstract.
Trivially, the maximum chromatic number of a graph on n vertices is n, and the only graph which achieves this value is the complete graph K_n. It is natural to ask whether this result is "stable", i.e., if n is large, G has n vertices and the chromatic number of G is close to n, must G be close to being a complete graph? It is easy to see that for each c>0, if G has n vertices and chromatic number at least n−c, then it contains a clique whose size is at least n−2c. We will consider the analogous questions for posets and dimension. Now the discussion will really become interesting.
Fei
Xue
TU Berlin
What is Minkowski
geometry?
Abstract.
A Minkowski space is a finite-dimensional real Banach space whose unit ball is a centrally symmetric convex body. In this talk, we will introduce some basic concepts of Minkowski spaces, and we will show some main concepts like the length and the area in the two-dimensional Minkowski space in comparison with the Euclidean space.
Manfred
Scheucher
Planar point sets determine many pairwise crossing segments
Horst
Martini
Minkowski geometry from the discrete point of view
Davide
Bolognini
Univ. Bologna
Betti splitting from a topological point of view
Abstract.
Betti splitting is a Mayer-Vietoris like technique to recover the minimal graded free resolution of homogeneous ideals. In this seminar, I will present a topological version of this technique, in the context of simplicial complexes. As an application, I will show some results establishing that the existence of a particular kind of Betti splitting for a triangulation of a closed manifold can express nice topological properties of the underlying space. Several interesting examples will be given. This is a joint work with Ulderico Fugacci.
Fleurianne
Bertrand
and
Carsten
Carstensen
HU Berlin
A posteriori error estimation for HHO-methods
William T.
Trotter
Georgia Institute of Technology
Dimension and Maximum Degree
Abstract.
The maximum degree of a poset is the maximum degree in the associated comparability graph. For an integer k, let f(k) denote the maximum dimension of a poset P with maximum degree k. It was shown in 1991 by Erdős, Kierstead and Trotter that f(k) = Ω(k logk). In 1986, Füredi and Kahn showed that f(k) = O(k log2k). Just in 2018, Scott and Wood made the following subtle but dramatic improvement: f(k) = O(log1+o(1)k). We outline the proof which uses an iterated application of the Lovász Local Lemma.
Sven
Jäger
Approximating Total Weighted Completion Time on Identical Parallel Machines with Precedence Constraints and Release Dates
Davide
Lofano
Technische Universität Berlin
Collapsible vs Contractible
Abstract.
Probably the most studied invariant in Topological Data Analysis is the homology of a space. The usual approach is to triangulate the space and try to reduce it in order to make the computations more feasible. A common reduction technique is that of collapsing. In a collapsing process we perform a sequence of elementary collapses, where at each step we delete a free face and the unique facet containing it. If we are able to reduce a complex to one of its vertices then we say it is collapsible and its homology is trivial. Collapsibility implies that the space is contractible but the converse is not always true, probably the best known example is the Dunce Hat.We are going to explore the difference between these two concepts and look for minimal examples of contractible non collapsible complexes in each dimension and how often they arise.
Matthew
Kahle
Ohio State University
Configuration spaces of hard disks in an infinite strip
Abstract.
This is joint work with Bob MacPherson. We study the configuration space config(n,w) of n non-overlapping disks of unit diameter in an infinite strip of width w. Our main result establishes the rate of growth of the Betti numbers for fixed j and w as n → ∞. We identify three regions in the (j,w) plane exhibiting qualitatively different topological behavior. We describe these regions as (1) a “gas” regime where homology is stable, (2) a “liquid” regime where homology is unstable, and (3) a “solid” regime where homology is trivial. We describe the boundaries between stable, unstable, and trivial homology for every n ≥ 3.
Levent
Doğan
TU Berlin
What is Kronecker coefficients?
Abstract.
The Kronecker coefficients are some natural numbers that naturally arise from the decomposition of the tensor product of two irreducible representations of the symmetric group. Even though they are fundamental in algebra, representation theory and somehow quantum information theory, there are many open problems surrounding the theory of Kronecker coefficients. One such problem is the existence of a combinatorial description of these numbers. In this talk we will briefly describe what Kronecker coefficients are and then we will discuss the complexity of their computation.
Stefan
Felsner
Counting Linear Extensions
Volkmar
Welker
Philipps-Univ. Marburg
Higher dimensional connectivity versus minimal degree of random graphs and minimal free resolutions
Abstract.
We study the clique complex, i.e. simplices are subsets of the vertex sets that form a complete subgraph of a random graph sampled from the Erdős-Renyi model. Motivated by applications in the study of minimal free resolutions of the Stanley-Reisner ring of the clique complex, we study for i≥0 two invariants: the minimal number of vertices that have to be deleted such that the clique complex of the remaining graph has homology in dimension i; and the minimal number of vertices that have to be deleted such that an i-simplex in the clique complex has empty link. Random graph theory says that the two invariants coincide for i=0 and all (Erdős-Renyi) probability regimes. We show the same for a middle density regime in case i=1, one inequality for all i and conjecture equality in general.
Friederike
Hellwig
HU Berlin
An adaptive lowest order dPG-FEM for linear elasticity
Piotr
Micek
Jagellonian University Krakow and TU Berlin
p-centered colorings of planar graphs
Abstract.
A p-centered coloring of a graph G is a coloring of its vertices such that for every connected subgraph H of G either H receives more than p colors or there is a color that appears exactly once in H. Very recently, we have shown that every planar graph admits a p-centered coloring with O(p3 log(p)) colors. This improves an O(p19) bound by Pilipczuk&Siebertz. We also know that some planar graphs require Omega(p2 log(p)) colors. We have tight bounds for outerplanar graphs, stacked triangulations, and graphs of bounded treewidth.Ongoing work with Stefan Felsner and Felix Schröder.
Fei
Xue
Technische Universität Berlin
On the lattice point covering problem in dimension 2
Abstract.
We say that a convex body K has the lattice point covering property, if K contains a lattice point of Z^n in any position, i.e., in any translation and rotation of K. In this talk we discuss the lattice point covering property of some regular polygons in dimension 2.
Boaz
Slomka
Weizmann Institute of Science, Rehovot Israel
On Hadwiger's covering conjecture
Abstract.
A long-standing open problem, known as Hadwiger’s covering problem, asks what is the smallest natural number N(n) such that every convex body in {\mathbb R}^n can be covered by a union of the interiors of at most N(n) of its translates. In this talk, I will discuss some history of this problem and its close relatives, and present more recent results, including a new general upper bound for N(n).
Claire
Voisin
Collège de France
Some Aspects of Algebraic Geometry
Michael
Cuntz
Univ. Hannover
Classification of Weyl groupoids
Abstract.
Finite Weyl groupoids (these are certain simplicial arrangements in a lattice) were completely classified in a series of papers by Heckenberger and myself. However, this classification is based on two computer proofs checking millions of cases. In this talk, I want to report on recent progress in finding a shorter proof. In particular, we prove without using a computer that, up to equivalence, there are only finitely many irreducible finite Weyl groupoids in each rank greater than two.
Kolja
Knauer
Generating k-connected orientations
Philipp
Bringmann
HU Berlin
H¹ vector potentials for solenoidal vector fields with partial boundary conditions
Tuan
Tran
ETH Zürich
Dense induced bipartite subgraphs in triangle-free graphs
Abstract.
The problem of finding dense induced bipartite subgraphs in H-free graphs has a long history, and was posed 30 years ago by Erdős, Faudree, Pach and Spencer. We obtain several results in this direction. First we prove that any H-free graph with minimum degree at least d contains an induced bipartite subgraph of minimum degree at least c_H log d/log log d, confirming (asymptotically) several conjectures by Esperet, Kang and Thomassé. Complementing this result, we further obtain optimal bounds for this problem in the case of dense triangle-free graphs, and we also answer a question of Erdős, Janson, Luczak and Spencer. Joint work with Kwan, Letzter and Sudakov.
Johan
Commelin
U Freiburg
On the cohomology of smooth project surfaces with $p_g = q = 2$ and maximal Albanese dimension
Abstract.
In this talk I will report on a joint project with Matteo Penegini (Genova). The second cohomology of a surface S as mentioned in the title splits up as a sum of two pieces. One piece comes from the Albanese variety. The other piece looks like the cohomology of a K3 surface, which we call a K3 partner X of S. If the surface S is a product-quotient then we can geometrically construct the K3 partner X and an algebraic correspondence that relates the cohomology of S and X. Finally, we prove the Tate and Mumford-Tate conjectures for all surfaces S that lie in the same connected component of the Gieseker moduli space as a product-quotient surface.
Tillmann
Miltzow
Universität Utrecht
Smoothed Analysis of the Art Gallery Problem
Abstract.
In the Art Gallery Problem we are given a polygon P \subset [0,L]^2 on n vertices and a number k. We want to find a guard set G of size k, such that each point in P is seen by a guard in G. Formally, a guard g sees a point p \in P if the line segment pg is fully contained inside the polygon P. The history and practical findings indicate that irrational coordinates are a "very rare" phenomenon. We give a theoretical explanation. Next to worst case analysis, Smoothed Analysis gained popularity to explain the practical performance of algorithms, even if they perform badly in the worst case. The idea is to study the expected performance on small perturbations of the worst input. The performance is measured in terms of the magnitude \delta of the perturbation and the input size. We consider four different models of perturbation. We show that the expected number of bitsto describe optimal guard positions per guard is logarithmic in the input and the magnitude of the perturbation. This shows from a theoretical perspective that rational guards with small bit-complexity are typical. Note that describing the guard position isthe bottleneck to show NP-membership. The significance of our results is that algebraic methods are not needed to solve the Art Gallery Problem in typical instances. This is the first time an ER-complete problem was analyzed by Smoothed Analysis. This is joint work with Michael Dobbins and Andreas Holmsen.
Matthias
Köppe
University of California
Facets of cut-generating functionology
Abstract.
In the theory of valid inequalities for integer point sets in polyhedra, the traditional, finite-dimensional techniques of polyhedral combinatorics are complemented by infinite-dimensional methods, the study of cut-generating functions. I will give an introduction to these methods and will explain their connection to lattice-free convex bodies. I will present recent results involving inverse semigroups of partial maps, obtained jointly with Robert Hildebrand and Yuan Zhou, and highlight some open questions regarding computability and complexity.
Carlos
Maestro Pérez
HU Berlin
What is a g_d^r?
Abstract.
The notion of linear series, or $g^r_d$'s (collections of cuts of a given variety by hyperplanes), is critical in algebraic geometry, where it yields a rich theory in which both classical and modern techniques beautifully come together. In this seminar we discuss some of the basic tools of this theory, and how they provide us with a better understanding of the geometry of algebraic curves. If time permits, a geometric interpretation of the Riemann-Roch theorem involving linear series on curves will be discussed.
Soledad
Villar
Monte Carlo approximation certificates for k-means clustering
Raphael
Steiner
Odd Dijoins and Cut Minors
Emanuel
Hintze
Universelle Wörter mit Jokersymbolen
Alexandru
Constantinescu
FU Berlin
DISCRETE STRUCTURES ON THE CONE OF MINKOWSKI SUMMANDS
Abstract.
The possibilities of splitting a lattice polytope into a Minkowski sum of lattice polytopes reflect the deformation theory of the induced toric singularity. This was proved by Altmann in 1996. Our goal is to reveal a similar correspondence for arbitrary polytopes. To this aim we look at special diagrams of semigroups and define a lattice structure inside the cone of Minkowki summands.
Joonkyung
Lee
Universität Hamburg
On the extremal number of subdivisions
Abstract.
The extremal number ex(n, H) is the maximum number of edges in an H-free graph with n vertices. When H has chromatic number at least three, the asymptotic behaviour of the extremal number is well understood, but when H is bipartite, the function remains mysterious. We give new estimates for the extremal numbers of various bipartite graphs H, especially if H can be obtained by subdividing edges of another graph H' in certain ways. Joint work with David Conlon and Oliver Janzer.
April 2019
Torsten
Mütze
Technische Universität Berlin
Combinatorial generation via permutation languages
Abstract.
In this talk I present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations, which provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an n-element set by adjacent transpositions; the binary reflected Gray code to generate all n-bit strings by flipping a single bit in each step; the Gray code for generating all n-vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an n-element ground set by element exchanges due to Kaye. The first main application of our framework are permutation patterns, yielding new Gray codes for different pattern-avoiding permutations, such as vexillary, skew-merged, X-shaped, separable, Baxter and twisted Baxter permutations etc. We also obtain new Gray codes for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into n rectangles subject to different restrictions. The second main application of our framework are lattice congruences of the weak order on the symmetric group S_n. Recently, Pilaud and Santos realized all those lattice congruences as (n-1)-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. This is joint work with Liz Hartung, Hung P. Hoang, and Aaron Williams.
Kolja
Knauer
Aix-Marseille Université
Tope graphs of (Complexes of) Oriented Matroids
Abstract.
Tope graphs of Complexes of Oriented Matroids fall into the important class of metric graphs called partial cubes. They capture a variety of interesting graphs such as flip graphs of acyclic orientations of a graph, linear extension graphs of a poset, region graphs of hyperplane arrangements to name a few. After a soft introduction into oriented matroids and tope graphs, we give two purely graph theoretical characterizations of tope graphs of Complexes of Oriented Matroids. The first is in terms of a new notion of excluded minors for partial cube, the second is in terms of classical metric properties of certain so-called antipodal subgraphs. Corollaries include a characterization of topes of oriented matroids due to da Silva, another one of Handa, a characterization of lopsided systems due to Lawrence, and an intrinsic characterization of tope graphs of affine oriented matroids. Moreover, we give a polynomial time recognition algorithms for tope graphs, which solves a relatively long standing open question. I will try to furthermore give some perspectives on classical problems as Las Vergnas simplex conjecture in terms of Metric Graph Theory. Based on joint work with H.-J; Bandelt, V. Chepoi, and T. Marc.
Sophia
Elia
FU Berlin
What is Steinitz's theorem?
Abstract.
Steinitz's theorem states that a graph is the graph of a $3$-dimensional polytope if and only if it is simple, planar, and 3-connected. In this What Is seminar we cover the notions in the theorem and the proof of the easier implication. We discuss polytopes and their graphs, and we look at Schlegel diagrams and $d$-connected graphs. We sketch a proof of Balinski's theorem stating that the graph of a $d$-dimensional polytope is $d$-connected.
Felix
Schröder
Bishellable Drawings of K_n
Jeff
Erickson
U Illinois
Variations on a theme of Steinitz
Jörg
Wolf
Local well posedness of the Euler equations in critical generalized Campanato spaces
Giulia
Codenotti
FU Berlin
The covering minima of lattice polytopes
Abstract.
The covering minima of a convex body were introduced by Kannan and Lovasz to give a better bound in the flatness theorem, which states that lattice point free convex bodies cannot have arbitrarily large width. These minima are similar in flavor to Minkowski's successive minima, and on the other hand generalize the covering radius of a convex body. I will speak about recent joint work with Francisco Santos and Matthias Schymura, where we investigate extremal values of these covering minima for lattice polytopes.
Tien
Tran-Ngoc
HU Berlin
An adaptive lowest order dPG-FEM for the Stokes equations with optimal convergence rate
Ander
Lamaison
FU Berlin
Ramsey upper density of infinite graphs
Abstract.
Let H be an infinite graph. In a two-coloring of the edges of the complete graph on the natural numbers, what is the densest monochromatic subgraph isomorphic to H that we are guaranteed to find? We measure the density of a subgraph by the upper density of its vertex set. This question, in the particular case of the infinite path, was introduced by Erdős and Galvin. Following a recent result for the infinite path, we present bounds on the maximum density for other choices of H, including exact values for a wide class of bipartite graphs.
Giulio
Bresciani
FU Berlin
Essential dimension and pro-finite group schemes
Abstract.
The essential dimension of an algebraic group is a measure of the complexity of its functor of torsors, i.e. $H^1(--,G)$. Classically, essential dimension has only been studied for group schemes of finite type. We study the case of pro-finite group schemes, and prove two very general criteria that show that essential dimension is almost always infinite for pro-finite group schemes: thus, it does not provide much information about them. We thus propose a new, natural refinement of essential dimension, the fce dimension. The fce dimension coincides with essential dimension for group schemes of finite type but has a better behaviour otherwise. Over any field, we compute the fce dimension of the Tate module of a torus. Over fields finitely generated over Q, we compute the fce dimension of Z_p and of the Tate module of an abelian variety.
Vincent
Delecroix
LaBRI/MPI Bonn
Dirichlet domain computation for real hyperbolic groups
Abstract.
A real hyperbolic group of dimension d is a discrete subgroup of SO(1,d) (the special orthogonal group of a quadratic form of signature (1,d) seen as the isometry group of the real hyperbolic space of dimension d). The main studied cases are d=2,3 which are the so-called Fuchsian and Kleinian groups. In various situations, one has access to generators of a real hyperbolic group and would like to study its properties (e.g. find relations among the generators). A possible approach for this problem is the construction of a fundamental domain with respect to its action on the hyperbolic space. The Dirichlet fundamental domain is a systematic way to construct one. After introducing the problem, I will provide several motivating examples. Then, I will discuss the algorithmic part which involves the incidence geometry of polyhedral cones.
Sophie
Blasius
Geschichtete Separatoren und ihre Anwendungen
Simona
Boyadzhiyska
FU Berlin
On counting problems related to (mutually) orthogonal Latin squares
Abstract.
After the question of existence of a combinatorial structure satisfying given properties, a natural and important problem is to determine how many such objects there are. In this talk, we will consider some counting questions related to (mutually) orthogonal Latin squares. We will prove an upper bound on the number of ways to extend a set of k mutually orthogonal Latin squares to a set of k+1 mutually orthogonal Latin squares and discuss some applications, comparing the resulting bounds to previously known lower and upper bounds.This talk is based on joint work with Shagnik Das and Tibor Szabó.
Josué
Tonelli-Cueto
Technische Universität Berlin
Condition meets Computational Geometry: The Plantinga-Vegter algorithm case
Abstract.
The Plantinga-Vegter algorithm is a subdivision algorithm that produces an isotopic approximation of implicit smooth curves in the plane (and also of surfaces in the three dimensional space). Despite remarkable practical success of the algorithm, little was known about its complexity. The only existing complexity analysis due to Burr, Gao and Tsigaridas provides worst-case bounds that are exponential both in the degree and the bit size of the input polynomial. Despite being tight, this worst-case bound doesn't explain why the algorithm is efficient in practice.In this talk, we show how condition numbers, combined with techniques from geometric functional analysis, help to solve this issue.This is joint work with Alperen A. Ergür and Felipe Cucker.
Kaie
Kubjas
Sorbonne Université Paris
Nonnegative rank four boundaries
Abstract.
Matrices of nonnegative rank at most r form a semialgebraic set. This semialgebraic set is understood for r=1,2,3. In this talk, I will recall what was previously known about this semialgebraic set and present recent results on the boundaries of the set of matrices of nonnegative rank at most four using notions from the rigidity theory of frameworks. These results are joint work with Robert Krone. In the nonnegative rank three case, all boundaries are trivial or consist of matrices that have only infinitesimally rigid factorizations. For arbitrary nonnegative rank, we give a necessary condition on zero entries of a nonnegative factorization for the factorization to be infinitesimally rigid, and we show that in the case of 5×5 matrices of nonnegative rank four, there exists an infinitesimally rigid realization for every zero pattern that satisfies this necessary condition. However, the nonnegative rank four case is much more complicated than the nonnegative rank three case, and there exist matrices on the boundary that have factorizations that are not infinitesimally rigid. We discuss two such examples.
Katerina
Papagiannouli
HU Berlin
What is support vector machines?
Abstract.
In this talk, we will introduce Support vector machines (SVMs). SVMs is a supervised learning method for classification. We will discuss about a particularly useful family of hypothesis spaces called Reproducing Kernel Hilbert spaces (RKHS) for non-linear classification. Finally, we will illustrate SVMs algorithm for pattern recognition using IRIS dataset.
Hendrik
Schrezenmaier
Universal point sets for planar 3-trees
Kenny
Chiu
FU Berlin
The Number of Convex Polyominoes with Given Height and Width
Abstract.
We give a new combinatorial proof for the number of convex polyominoes whose minimum enclosing rectangle has given dimensions. We also count the subclass of these polyominoes that contain the lower left corner of the enclosing rectangle (directed polyominoes). In addition, we indicate how to sample random polyominoes in these classes.
Andrew
Treglown
University of Birmingham
A degree sequence Komlós theorem
Abstract.
A classical topic is graph theory concerns finding minimum degree conditions that force a given spanning subgraph in a graph. There has also been interest in generalising such results via degree sequence conditions. Komlós' theorem determines the minimum degree that forces an H-tiling in a graph G covering a given proportion of the vertices in G. (An H-tiling is simply a collection of vertex-disjoint copies of H in G.) In this talk we will discuss a degree sequence generalisation of this result. This is joint work with Hong Liu and Joseph Hyde.
Stefan
Felsner
Segment intersection representations of planar graphs
March 2019
Torsten
Mütze
Combinatorial generation via permutation languages
Piotr
Micek
Planar graphs have bounded queue-number
Raphael
Steiner
Universal Point Sets for Planar Graphs
Tibor
Szabó
FU Berlin
List Ramsey numbers
Abstract.
We introduce the list colouring extension of classical Ramsey numbers, investigate when the two Ramsey numbers are equal, and in general, how far apart they can be from each other. We find graph sequences where the two are equal and where they are far apart. For l-uniform cliques we prove that the list Ramsey number is bounded by an exponential function, while it is well-known that the Ramsey number is super-exponential for uniformity at least 3. This is in great contrast to the graph case where we cannot even decide the question of equality for cliques. Joint work with Noga Alon, Matija Bucic, Tom Kalvari, and Eden Kuperwasser.
Manfred
Scheucher
A Generalization of the Erdős-Szekeres Theorem to Arrangements of Pseudocircles
February 2019
Joachim
Naumann
Über Erhaltungssätze für die Maxwellgleichungen V.
Helena
Bergold
A Combinatorial Extension of the Colorful Carathéodory
Tobias
Tschirch
Humboldt-Universität zu Berlin
Offsets of Non-Uniform Rational B-Spline curves for Computer Aided Design
Jan
van den Heuvel
LSE
Partial solutions to Hadwiger's Conjecture
Abstract.
Hadwiger's Conjecture (1943) asserts that every graph without the complete graph Kt+1 as a minor has a proper vertex-colouring using at most t colours. Since the conjecture is stubbornly refusing to be proved, we might look at relaxed versions of it. In the talk we discuss some results around the following question: If we are given t colours to colour a graph without Kt+1-minor, what kind of vertex-colourings can we guarantee with those t colours?
Eugenio
Buzzoni
TU Berlin
What is the Wright-Fisher diffusion?
Abstract.
The Wright-Fisher diffusion — mainly expressed as the unique solution of a stochastic differential equation — is a key object in probabilistic population genetics. We will see what it looks like, defining Brownian motion and stochastic integrals along the way.
Raphael
Steiner
Colouring Non-Even Digraphs
Dongho
Chae
Chung-Ang University, Seoul, South Korea
On the Lioville type theorems for the stationary Navier-Stokes and related equations
Lorenzo
Venturello
Osnabrück
Algebraic invariants of balanced simplicial complexes
Abstract.
Given a simplicial complex which triangulates a certain manifold it is natural to ask what conditions the topology poses on the number of faces in each dimension. Via the Stanley-Reisner correspondence between simplicial complexes and squarefree monomial ideals we can take advantage of tools from commutative algebra, by studying related algebraic invariants such as the Hilbert function and the graded Betti numbers of a certain graded ring. In particular we focus on balanced simplicial complexes, i.e., (d − 1)-dimensional complexes whose graph is d-colorable. After providing the necessary background we will present upper bounds on graded Betti numbers of various objects in this family and discuss possible applications to face enumeration. This is joint work with Martina Juhnke-Kubitzke.
Albert
Cohen
Paris
Optimal non-intrusive methods in high-dimension
Ferdinand
Ihringer
Ghent University
Rank Bounds on the Independence Number
Abstract.
Recently, the breakthrough results by Croot, Lev, and Pach on progression-free sets in ℤ4n and by Ellenberg and Gijswijt on cap sets brought new attention to rank arguments for bounding sets in combinatorial problems. We discuss several traditional applications of rank arguments to bound the independence number of a graph. The examples include the orthogonality graph on {-1,1}n, generalized quadrangles, Hermitian dual polar graphs, and quasisymmetric designs.
Matías
Bender
Sorbonne Université Paris
Solving sparse polynomial systems using Gröbner basis
Abstract.
Solving systems of polynomial equations is one of the oldest and most important problems in computational mathematics and has many applications in several domains of science and engineering. It is an intrinsically hard problem with complexity at least single exponential in the number of variables. However, in most of the cases, the polynomial systems coming from applications have some kind of structure. For example, several problems in computer-aided design, robotics, computer vision, molecular biology and kinematics involve polynomial systems that are sparse that is, only a few monomials have non-zero coefficients. We focus on exploiting the sparsity of the Newton polytopes of the polynomials to solve the systems faster than the worst case estimates. In this talk, I will present a Gröbner basis approach to solve sparse 0-dimensional systems whose input polynomials have different Newton polytopes. Under regularity assumptions, we can have an explicit combinatorial bound for the complexity of the algorithm. This is joint work with Jean-Charles Faugère and Elias Tsigaridas.
Sergio
Cabello
Universität Ljubljana
Computational geometry, optimization and Shapley values
Abstract.
I will discuss three unrelated sets of results combining geometry and algorithms. First we will see classes of graphs defined using the intersection of geometric objects in the plane, and discuss classical optimization problems for them. Then we will consider approximation algorithms for the potato peeling problem: find a maximum-area convex body inside a given polygon. The problem amounts to finding a maximum clique in the visibility graph of random samples of points inside the polygon, and results from stochastic geometry are used to bound the size of the samples. Finally, we will discuss the efficient computation of Shapley values for coalitional games defined by the area of usual geometric objects, such as the convex hull or the minimum axis-parallel bounding box.
Dimitris
Bogiokas
FU Berlin
What is it like to lower the mixed volume?
Abstract.
The study of mixed volumes plays a central role in the Brunn-Minkowski theory, since it combines two fundamental notions of the area: Minkowski sums and volume. In this presentation we introduce the notion of mixed volume and examine some first properties.
Deane
Yang
NYU Courant
Minkowski problems for convex bodies
Nils
Engler
On Cayley Graphs Generated by Permutations and Shifts
Karin
Schaller
FU Berlin
Stringy Invariants of Algebraic Varieties and Lattice Polytopes
Abstract.
We present topological invariants in the singular setting for projective Q-Gorenstein varieties with at worst log-terminal singularities, such as stringy Euler numbers, stringy Chern classes, stringy Hodge numbers, and stringy E-functions. In the toric setting, we give formulae to efficiently compute these stringy invariants. Using these combinatorial expressions and the stringy Libgober-Wood identity, we derive several appealing new combinatorial identities for lattice polytopes. We go on to generalise the famous ‘number 12’ and ‘number 24’ identities which hold far more generally than previously expected.
Sophie
Puttkammer
Humboldt-Universität zu Berlin
Guaranteed Lower Eigenvalue Bounds with the Weak Galerkin FEM
David
Fabian
FU Berlin
Rainbow arithmetic progressions and anti-van der Waerden numbers
Abstract.
Given a colouring of a set in an ambient abelian group, a rainbow arithmetic progression is an arithmetic progression whose elements receive pairwise distinct colours. We are going to consider two related problems that have been studied over the last years: First we ask whether any k-colouring of [n] with equally sized colour classes admits a rainbow arithmetic progression of length k. The second problem is to find the smallest positive integer r, for which every r-colouring of [n] contains a rainbow k-AP. We shall also have a look at the analogous problems when [n] is replaced by a finite abelian group.
Hsueh-Yung
Lin
U Bonn
Algebraic approximations of compact Kähler manifolds of algebraic codimension 1
Abstract.
Let $X$ be a compact Kähler manifold. The so-called Kodaira problem asks whether $X$ has arbitrarily small deformations to some projective varieties. In this talk, we will explain our solution to the Kodaira problem for compact Kähler manifolds of algebraic dimension $a(X) = dim(X) − 1$. This is partly joint work with B. Claudon and A. Höring.
Patrick
Morris
Freie Universität Berlin
Clique tilings in randomly perturbed graphs
Abstract.
A major theme in both extremal and probabilistic combinatorics is to find the appearance thresholds for certain spanning structures. Classical examples of such spanning structures include perfect matchings, Hamilton cycles and H-tilings, where we look for vertex disjoint copies of H covering all the vertices of some host graph G. In this talk we will focus on H-tilings in the case that H is a clique, a natural generalisation of a perfect matching. On the one hand there is the extremal question, how large does the minimum degree of an n-vertex graph G have to be to guarantee the existence of a clique factor in G? On the other hand, there is the probabilistic question. How large does p need to be to almost surely ensure the appearance of a clique factor in the Erdős-Rényi random graph G(n,p)? Optimal answers to these questions were given in two famous papers. The extremal question was answered by Hajnal and Szemerédi in 1970 and the probabilistic question by Johansson, Kahn and Vu in 2008. In this talk we bridge the gap between these two results by approaching the following question which contains the previous questions as special cases. Given an arbitrary graph of some fixed minimum degree, how many random edges need to be added on the same set of vertices to ensure the existence of a clique tiling? We give optimal answers to this question in all cases. Such results are part of a recent research trend studying properties of what is known as the randomly perturbed graph model, introduced by Bohman, Frieze and Martin in 2003. This is joint work with Jie Han and Andrew Treglown.
Karim
Adiprasito
Hebrew University of Jerusalem
Triangulated manifolds, Lefschetz conjectures and the revenge of marriages
Abstract.
Stanley gave us necessary conditions that f-vectors of simplicial polytopes must satisfy by relating the problem to the hard Lefschetz theorem in algebraic geometry, an insurmountably deep and intimidating theorem in algebraic geometry. McMullen conjectured that these conditions are necessary in general, posing before us the problem of proving the hard Lefschetz theorem beyond what algebraic geometers would dream of. I will talk about the revenge of combinatorics, and in particular discuss how Hall's marriage theorem (or rather, one of its proofs) provides a way to this deep algebraic conjecture. Based on arxiv:1812.10454
Tommaso Cornelis
Rosati
HU Berlin
What is a hydrodynamic limit?
Abstract.
We will consider one, a few or an infinite number of particles which move, interact or reproduce randomly. We explain in what sense a law of large numbers and a central limit theorem can hold for such systems. We exploit these structures to (re-)discover deep interactions between probability theory and the analysis of PDEs. We will discuss several prototypical examples and take a look at some rather ugly simulations.
Hans Niklas
Jacob
On the Generalized Middle Levels Problem
January 2019
Joachim
Naumann
Über Erhaltungssätze für die Maxwellgleichungen IV.
Carsten
Lange
FU Berlin
Some volume identities for $W$-permutahedra
Abstract.
Lovasz (2001) constructed a weighted adjacency matrix $A_G$ for the vertex-edge-graph of a given $3$-polytope $P$ containing the origin such that a basis of its kernel yields vertices for $P$ up to a linear transformation. In 2010, Izmestiev extended the construction to polytopes of arbitrary dimension. We study Izmestiev’s construction for generalized $W$-permutahedra $P_W$ were $W$ denotes a finite reflection group. This yields an interesting relation between volumes of faces of the dual polytope $P_W^\Delta$ and the volume of simplices spanned by vertices of $P_W$. Joint works with Ioannis Ivrissimtzis, Shiping Liu and Norbert Peyerimhoff.
Philipp
Bringmann
Humboldt-Universität zu Berlin
Quasi-optimal convergence of adaptive LSFEM for three model problems in 3D
László
Kozma
FU Berlin
Hamiltonicity below Dirac's condition
Abstract.
Dirac's theorem (1952) is a classical result of graph theory, stating that an n-vertex graph (n≥3) is Hamiltonian if every vertex has degree at least n/2. Both the value n/2 and the requirement for every vertex to have high degree are necessary for the theorem to hold. In this work we give efficient algorithms to determine whether a graph is Hamiltonian when either of the two conditions are relaxed. Joint work with Bart Jansen and Jesper Nederlof.
Jan Hendrik
Bruinier
TU Darmstadt
Arithmetic degrees of special cycles and derivatives of Siegel Eisenstein series
Abstract.
Let $V$ be a rational quadratic space of signature $(m,2)$. A conjecture of Kudla relates the arithmetic degrees of top degree special cycles on an integral model of a Shimura variety associated with $SO(V)$ to the coefficients of the central derivative of a Siegel Eisenstein series of genus $m+1$. We report on joint work with Tonghai Yang proving this conjecture for the coefficients of non-singular index $T$ under certain conditions on $T$. To this end we establish some new local arithmetic Siegel-Weil formulas at the archimedean and non-archimedean places.
Fei
Xue
Technische Universität Berlin
On successive minima-type inequalities for the polar of a convex body
Abstract.
The second theorem of Minkowski on successive minima provides optimal upper and lower bounds on the volume of a symmetric convex body in terms of its successive minima. Motivated by conjectures of Mahler and Makai Jr., we study bounds on the volume of a convex body in terms of the successive minima of its polar body. In this talk, we will show the lower bound in the planar case, and the upper bound in the general case. This talk is based on joint work with Martin Henk.
Peter
Gritzmann
Technische Universität München
On dynamic discrete tomography: Constrained flow and multi assignment problems for plasma particle tracking
Abstract.
The talk deals with the problem of reconstructing the paths of a set of points over time, where, at each of a finite set of moments in time the current positions of the points in space are only accessible through a small number of their line X-rays. Such problems originate from the demand of particle tracking in plasma physics and can be viewed as constrained version of min-cost-flow and multi assignment problems. (Joint work with A. Alpers)
Kristin
Shaw
Oslo
Jörg
Wolf
Local regularity of weak solutions of parabolic systems with unbounded VMOcoefficients
Lena
Walter
FU
The Kingman Coalescent as a Density on a Space of Trees
Abstract.
Randomly pick n individuals from a population and trace their genealogy backwards in time until you reach the most recent common ancestor. The Kingman n-coalescent is a probabilistic model for the tree one obtains this way. The common definitions are stated from a stochastic point of view. The Kingman Coalescent can also be described by a probability density function on a space of certain trees. For the space of phylogenetic trees Ardila and Klivans showed, that this space is closely related to the Bergman fan of the graphical matroid of the complete graph. Using its fine fan structure, similar things can be said about the space we are considering. I will describe the density and report on work in progress with Christian Haase about relations of population genetics and polyhedral and algebraic geometry.
Guillaume
Sagnol
First order methods for convex optimization
Anurag
Bishnoi
FU Berlin
Non-intersecting Ryser hypergraphs
Abstract.
A famous conjecture of Ryser states that every r-partite hypergraph has vertex cover number at most r − 1 times the matching number. In recent years, hypergraphs meeting this conjectured bound, known as r-Ryser hypergraphs, have been studied extensively. It was proved by Haxell, Narins and Szabó that all 3-Ryser hypergraphs with matching number ν > 1 are essentially obtained by taking ν disjoint copies of intersecting 3-Ryser hypergraphs. In this talk we will see new infinite families of r-Ryser hypergraphs, for any given matching number ν > 1, that do not contain two vertex disjoint intersecting r-Ryser subhypergraphs.
Torsten
Mütze
Technische Universität Berlin
On symmetric chains and Hamilton cycles
Abstract.
The n-cube is the poset obtained by ordering all subsets of {1,2,...,n} by inclusion. A symmetric chain is a sequence of subsets Ak⊆Ak+1⊆…⊆An-k with |Ai|=i for all i=k,…,n-k, and a symmetric chain decomposition, or SCD for short, of the n-cube is a partition of all its elements into symmetric chains. There are several known descriptions of SCDs in the n-cube for any n≥1, going back to works by De Bruijn, Aigner, Kleitman and several others. All those constructions, however, yield the very same SCD. In this talk I will present several new constructions of SCDs in the n-cube. Specifically, we construct five pairwise edge-disjoint SCDs in the n-cube for all n≥90, and four pairwise orthogonal SCDs for all n≥60, where orthogonality is a slightly stronger requirement than edge-disjointness. Specifically, two SCDs are called orthogonal if any two chains intersect in at most a single element, except the two longest chains, which may only intersect in the unique minimal and maximal element (the empty set and the full set). This improves the previous best lower bound of three orthogonal SCDs due to Spink, and is another step towards an old problem of Shearer and Kleitman from the 1970s, who conjectured that the n-cube has ⌊n/2⌋+1 pairwise orthogonal SCDs. We also use our constructions to prove some new results on the central levels problem, a far-ranging generalization of the well-known middle two levels conjecture (now theorem), on Hamilton cycles in subgraphs of the (2n+1)-cube induced by an even number of levels around the middle. Specifically, we prove that there is a Hamilton cycle through the middle four levels of the (2n+1)-cube, and a cycle factor through any even number of levels around the middle of the (2n+1)-cube. This talk is based on two papers, jointly with Sven Jäger, Petr Gregor, Joe Sawada, and Kaja Wille (ICALP 2018), and with Karl Däubel, Sven Jäger, and Manfred Scheucher, respectively.
Maria
Bras Amorós
Universitat Rovira i Virgili Tarragona
On numerical semigroups
Abstract.
A numerical semigroup is a subset of the positive integers (N) together with 0, closed under addition, and with a finite complement in N∪{0}. The number of gaps is its genus. Numerical semigroups arise in algebraic geometry, coding theory, privacy models, and in musical analysis. It has been shown that the sequence counting the number of semigroups of each given genus g, denoted (ng)g≥0, has a Fibonacci-like asymptotic behavior. It is still not proved that, for each g, ng+2 ≥ ng+1 + ng or, even more simple, ng+1 ≥ ng. We will explain some classical problems on numerical semigroups as well as some of their applications to other fields and we will explain the approach of counting semigroups by means of trees.
Florian
Nie
TU Berlin
What is a competing species model?
Abstract.
We will introduce a certain model arising from mathematical population genetics – the competing species model using the theory of Stochastic Differential Equations. Using this we will discuss how the different terms in the model may affect the evolution of the modelled population. In particular, we are interested in how real life biological effects can be represented through these terms and what kind of mathematical (and biological) questions one can answer in the model.
Raphael
Steiner
On Woodall's Conjecture, Cyclic Base Orderings, Fractional Arboricity and Pseudosphere Arrangements
Sylvie
Méléard
CMAP, École Polytechnique
Stochastic dynamics for adaptation and evolution of microorganisms
Jorge Alberto
Olarte Parra
FU Berlin
The moduli space of Harnack curves
Abstract.
Harnack curves are a special family of plane real algebraic curves which are characterized by having amoebas that behave in the simplest possible way. Given a lattice polygon, we will show how to compute the moduli space of Harnack curves which have that polygon as Newton polygon. This space can be embedded into the moduli space of pointed tropical curves. Through this embedding, the moduli space of Harnack curves has a compactification which essentially is the secondary polytope of the Newton polygon. We show how in this compactification lattice subdivisions of the Newton polygon correspond to collections of Harnack curves that can be glued together using Viro’s patchworking.
Tien
Tran-Ngoc
Humboldt-Universität zu Berlin
Non-standard discretizations of a class of relaxed minimization problems
Jan
Corsten
LSE
Erdős-Rothschild problem for five and six colours
Abstract.
Given positive integers n,r,k, the Erdős-Rothschild problem asks to determine the largest number of r-edge-colourings without monochromatic k-cliques a graph on n vertices can have. In the case of triangles, i.e. when k=3, the solution is known for r = 2,3,4. We investigate the problem for five and six colours.
Paola
Frediani
U Pavia
On the geometry of some special subvarieties contained in the Torelli locus
Abstract.
Almost all the examples of special subvarieties of $A_g$ contained in the Torelli locus are given by families of Jacobians of Galois covers of the projective line or of elliptic curves. They satisfy a sufficient condition that I will explain. I will first show that this condition is also necessary in the case of double covers of elliptic curves. In fact I will prove that the bielliptic locus is not totally geodesic for $g>3$. Finally I will discuss the geometry of some examples of Galois covers of elliptic curves yielding special subvarieties of $A_3$, which are fibered in totally geodesic curves and hence contain countably many Shimura curves.
Carlos
Amendola
Technische Universität München
Max-Linear Graphical Models via Tropical Geometry
Abstract.
Motivated by extreme value theory, max-linear graphical models have been recently introduced and studied as an alternative to the classical Gaussian or discrete distributions used in graphical modeling. We present max-linear models naturally in the framework of tropical geometry. This perspective allows us to shed light on some known results and to prove others with algebraic techniques. In particular, we rephrase parameter estimation for max-linear models in terms of cones in the tropical eigenspace fan. This is joint work with Claudia Klüppelberg, Steffen Lauritzen and Ngoc Tran.
Penny
Haxell
University of Waterloo
Algorithms for independent transversals vs. small dominating sets
Abstract.
An independent transversal (IT) in a vertex-partitioned graph G is an independent set in G consisting of one vertex in each partition class. There are several known criteria that guarantee the existence of an IT, of the following general form: the graph G has an IT unless the subgraph GS of G, induced by the union of some subset S of vertex classes, has a small dominating set. These criteria have been used over the years to solve many combinatorial problems. The known proofs of these IT theorems do not give efficient algorithms for actually finding an IT or a subset S of classes such that GS has a small dominating set. Here we present appropriate weakenings of such results that do have effective proofs. These result in algorithmic versions of many of the original applications of IT theorems. We will discuss a few of these here, including hitting sets for maximum cliques, circular edge colouring of bridgeless cubic graphs, and hypergraph matching problems.
Simona
Boyadzhiyska
FU Berlin
What is interval graphs, interval orders, and their friends?
Abstract.
Interval graphs and interval orders are two classes of discrete structures that arise naturally in many real-world problems. They find applications in scheduling, archaeology, genetics, psychology, and circuit design, among others. In this talk, we will give a short introduction to the theory of interval graphs and orders. In particular, we will discuss the connection between these two types of structures, how they can be characterized, and why they are important from both a theoretical and a practical point of view. We will conclude by mentioning some special cases and generalizations.
Manfred
Scheucher
On Erdős-Szekeres Type Questions in $R^d$
Thomas
Kahle
OVGU Magdeburg
The geometry of gaussoids
Abstract.
Gaussoids are combinatorial structures that encode independence in probability and statistics, just like matroids encode independence in linear algebra. We show that the gaussoid axioms of Lnenicka and Matus are equivalent to compatibility with certain quadratic relations among principal and almost-principal minors of a symmetric matrix. This approach facilitates insights into symmetry and realizability of gaussoids as well as several extensions (like oriented, positive, and valuated gaussoids). (based on joint work with T. Boege, A. D'Ali, and B. Sturmfels)
Péter Pál
Pach
BME
Polynomial Schur's theorem
Abstract.
We consider the Ramsey problem for the equation x+y=p(z), where p is a polynomial with integer coefficients. Under the assumption that p(1)p(2) is even we show that for any 2-colouring of ℕ the equation x+y=p(z) has infinitely many monochromatic solutions. Indeed, we show that the number of monochromatic solutions with x,y,z∈ {1,2,\dots,n} is at least n2/d^3-o(1), where d=deg p.On the other hand, when p(1)p(2) is odd, that is, when p attains only odd values, then there might not be any monochromatic solution, e.g., this is the case when we colour the integers according to their parity. We give a characterization of all 2-colourings avoiding monochromatic solutions to x+y=p(z).This is a joint work with Hong Liu and Csaba Sándor.
Renjie
Lyu
U Amsterdam
A result of algebraic cycles on cubic hypersurfaces
Abstract.
The Chow group of algebraic cycles of a smooth projective variety is an important subject in algebraic geometry, which in general, is too massive to grasp. In this talk, we show that the Chow group of a smooth cubic hypersurface $X$ can be recovered by the algebraic cycles of its Fano variety of lines $F(X)$. It generalizes $M$. Shen’s previous work of 1-cycles on cubics. The proof relies on some birational geometry concerning the Hilbert square of cubic hypersurfaces recently studied by E. Shinder, S. Galkin and C. Voisin. As applications, when $X$ is a complex smooth 4-fourfold, the result we obtained could prove the integral Hodge conjecture for 1-cycles on the polarised hyper-Kähler variety $F(X)$. In the arithmetic aspect, C. Schoen addressed the integral analog of the Tate conjecture, which is predicted to be true for 1-cycles of any smooth projective variety defined over finite fields. We will show how to use our result to prove this conjecture for 1-cycles on the Fano variety $F(X)$ if $X$ is a smooth cubic 4-fold over a finitely generated field.
Ardalan
Khazraei
Bonn
Cost-distance Steiner trees
Abstract.
In the well-known Steiner Tree problem, the objective is to connect a set of terminals at the least total cost. We can further constrain the problem by specifying upper bounds for the distance of each terminal to a chosen root terminal. Further, using the Lagrangianr elaxation of this restriction, we can penalize large distances in the objective function rather than applying strict distance constraints. We arrive at a special case of the so-called Cost-Distance Steiner Tree Problem in which we have a single weight function on the edges. In this talk, I will present several results from my master's thesis that concern the Cost-Distance Steiner Tree Problem. The NP-hardness of the Cost-Distance Steiner Tree Problem trivially follows from the fact that the regular Steiner Tree problem is the special case where we set demand weights (Lagrange multipliers) of the terminals to zero. I therefore proceed to prove that the problem remains NP-hard in three restricted cases that do not contain the regular Steiner Tree Problem as a special case. Then I improve on a previous constant-factor approximation for the single-weighted case by using a hybrid method of an approximate Steiner tree with a shortest-path tree replacing sections of the tree where path segments are used by many terminals with demand weights summing to higher than a tunable threshold. I also use a similar approach to extend Arora's dynamic-programming method for the two-dimensional geometric Steiner Tree Problem to the case with the penalizing terms in the objective function.
Péter Pál
Pach
Budapest University of Technology and Economics
The polynomial method and the cap set problem
Abstract.
In this talk we will look at a new variant of the polynomial method which was first used to prove that sets avoiding 3-term arithmetic progressions in groups like Z4n (Croot, Lev and myself) and Z3n (Ellenberg and Gijswijt) are exponentially small (compared to the size of the group).We will discuss lower and upper bounds for the size of the extremal subsets, including some recent bounds found by Elsholtz and myself. We will also mention some further applications of the method, for instance, the solution of the Erdős-Szemerédi sunflower conjecture.
December 2018
Laura
Colmenarejo
MPI Leipzig
Signatures of paths: an algebraic perspective.
Abstract.
Coming from stochastic analysis, the signature of a path is the collection of all the iterated integrals of the path. It can be seen in terms of tensors or as formal power series in words, which make them more relevant in other areas such as algebraic geometry or combinatorics. In this talk, I would like to look at the signatures of paths from an algebra perspective. For that, we will look at the work done by C. Améndola, P. Friz, and B. Sturmfels about the variety defined by the signature of piecewise linear paths, as well as the work done by F. Galuppi about the variety of rough paths. As a continuation, I would like to present our work no the variety defined by the signature of axis paths. This is a joint work with F. Galuppi and M. Michalek.
Christoph
Spiegel
UPC
Intervals in the Hales-Jewett Theorem
Abstract.
The Hales–Jewett Theorem states that any r–colouring of [m]n contains a monochromatic combinatorial line if n is large enough. Shelah’s proof of the theorem implies that for m = 3 there always exists a monochromatic combinatorial lines whose set of active coordinates is the union of at most r intervals. I will present some recent findings relating to this observation. This is joint work with Nina Kamcev.
Christian
Liedtke
TU München
Rigid rational curves in positive characteristic
Abstract.
Rational curves are central to higher-dimensional algebraic geometry. If a rational curve “moves” on a variety, then the variety is uniruled and in characteristic zero, this implies that the variety has negative Kodaira dimension. Over fields of positive characteristic, varieties can be inseparably uniruled without having negative Kodaira dimension. However, I will show in my talk that in the case that a rational curve moves on a surface of non-negative Kodaira dimension, then this rational curve must be “very singular”. In higher dimensions, there is a similar result that is more complicated to state. I will also give examples that show the results are optimal. This is joint work with Kazuhiro Ito and Tetsushi Ito.
Marie
Brandenburg
Freie Universität Berlin
Product-Mix Auctions, Competitive Equilibrium and Lattice Polytopes
Abstract.
In 2007, when the credit crisis began, the Bank of England approached the economist Paul Klemperer with the need of a new auction design that would enable them to give liquidity to all banks in an economically efficient way. Since then, Klemperer’s Product-Mix Auction design became more and more relevant, with particular interest in when competitive equilibrium exists, that is, when a set of (indivisible) goods can be split such that exactly one bid of each bidder is fulfilled. Originally being a question of economics, this can be translated into a question of specific lattice polytopes that rely on underlying graphs. I will present the Product-Mix Auction setting, the translation to polytopes and first results on when competitive equilibrium exists.
Shagnik
Das
Freie Universität Berlin
Randomly perturbed Ramsey problems
Abstract.
The combinatorial properties much-loved Erdős-Rényi random graph G(n,p), which has n vertices and whose edges are present independently with probability p, have been comprehensively studied in the decades since its introduction. In recent years, much research has been devoted to the randomly perturbed graph model, introduced in 2003 by Bohman, Frieze and Martin. In this talk we shall present and motivate this new model of random graphs, and then focus on the Ramsey properties of these randomly perturbed graphs. More precisely, given a pair of graphs (F,H), we ask how many random edges must be added to a dense graph G to ensure that any two-colouring of the edges of the perturbed graph has either a red copy of F or a blue copy of G. This question was first studied in 2006 by Krivelevich, Sudakov and Tetali, who answered it in the case of F being a triangle and H being a clique. We generalise these results, considering pairs of larger cliques, and, should the audience be willing (but even otherwise), shall take a quick look at some of the ideas required in our proofs. This is joint work with Andrew Treglown (Birmingham).
Alex
Nietner
FU Berlin
What is tensor networks, topological order and entanglement renormalization?
Abstract.
In this introductory talk we will look at many body physics by means of the partition function as its central object. Starting with an introduction to tensor networks as a general formalism we will use the language of tensor networks in order to get an intuition for the partition function and hence many body physics. The notion of equivalence classes of tensor networks will lead us directly to a notion of equivalence classes of many body systems usually referred to as phases. As a special case we will dig deeper into the equivalence class with respect to topological moves (Pachner moves) leading directly to gapped topological phases. Combining the properties of tensor networks and topological moves we will introduce the entanglement renormalization scheme as a method to detect topological phases. We will conclude the talk with some general remarks on renormalization, regularization and discretization.
Sara
Zemljič
The Sierpinski product of graphs
Sylvie
Paycha
Are locality and renormalisation reconcilable?
Michael
Joswig
TU Berlin
Cluster partitions and fitness landscapes of the Drosophila fly microbiome
Abstract.
Beerenwinkel et al.(2007) suggested studying fitness landscapes via regular subdivisions of convex polytopes. Building on their approach we propose cluster partitions and cluster filtrations of fitness landscapes as a new mathematical tool. In this way, we provide a concise combinatorial way of processing metric information from epistatic interactions. Using existing Drosophila microbiome data, we demonstrate similarities with and differences to the previous approach. As one outcome we locate interesting epistatic information where the previous approach is less conclusive. Joint works with Holger Eble, Lisa Lamberti and Will Ludington.
Jozef
Skokan
LSE
The k-colour Ramsey number of odd cycles via non-linear optimisation
Abstract.
For a graph G, the k-colour Ramsey number Rk(G) is the least integer N such that every k-colouring of the edges of the complete graph KN contains a monochromatic copy of G. Let Cn denote the cycle on n vertices. We show that for fixed k > 2 and n odd and sufficiently large, Rk(Cn) = 2k-1(n-1) + 1. This generalises a result of Kohayakawa, Simonovits and Skokan and resolves a conjecture of Bondy and Erdős for large n. We also establish a surprising correspondence between extremal k-colourings for this problem and perfect matchings in the hypercube Qk. This allows us to in fact prove a stability-type generalisation of the above. The proof is analytic in nature, the first step of which is to use the Regularity Lemma to relate this problem in Ramsey theory to one in convex optimisation. This is joint work with Matthew Jenssen.
Marco
D'Addezio
FU Berlin
Finiteness of perfect torsion points of an abelian variety and $F$-isocrystals
Abstract.
I will report on a joint work with Emiliano Ambrosi. Let $k$ be a field which is finitely generated over the algebraic closure of a finite field. As a consequence of the theorem of Lang-Néron, for every abelian variety over $k$ which does not admit any isotrivial abelian subvariety, the group of $k$-rational torsion points is finite. We show that the same is true for the group of torsion points defined on a perfect closure of $k$. This gives a positive answer to a question posed by Hélène Esnault in 2011. To prove the theorem we translate the problem into a certain question on morphisms of $F$-isocrystals. Then we handle it by studying the monodromy groups of the $F$-isocrystals involved.
Ander
Lamaison
Freie Universität Berlin
Ramsey density of infinite paths
Abstract.
In a two-colouring of the edges of the complete graph on the natural numbers, what is the densest monochromatic infinite path that we can always find? We measure the density of a path by the upper asymptotic density of its vertex set. This question was first studied by Erdös and Galvin, who proved that the best density is between 2/3 and 8/9. In this talk we settle this question by proving that we can always find a monochromatic path of upper density at least (12+sqrt(8))/17=0.87226…, and constructing a two-colouring in which no denser path exists. This represents joint work with Jan Corsten, Louis DeBiasio and Richard Lang.
Marijn
Heule
University of Texas, Austin
Everything's Bigger in Texas: "The Largest Math Proof Ever"
Abstract.
Progress in satisfiability (SAT) solving has enabled answering long-standing open questions in mathematics completely automatically resulting in clever though potentially gigantic proofs. We illustrate the success of this approach by presenting the solution of the Boolean Pythagorean triples problem. We also produced and validated a proof of the solution, which has been called the ``largest math proof ever''. The enormous size of the proof is not important. In fact a shorter proof would have been preferable. However, the size shows that automated tools combined with super computing facilitate solving bigger problems. Moreover, the proof of 200 terabytes can now be validated using highly trustworthy systems, demonstrating that we can check the correctness of proofs no matter their size.
Antonio
Macchia
FU Berlin
Binomial edge ideals of bipartite graphs
Abstract.
Binomial edge ideals are ideals generated by binomials corresponding to the edges of a graph. They generalize the ideals of 2-minors of a generic matrix with two rows and arise naturally in Algebraic Statistics. We give a combinatorial classification of Cohen-Macaulay binomial edge ideals of bipartite graphs providing an explicit construction in graph-theoretical terms. In the proof we use the dual graph of an ideal, showing in our setting the converse of Hartshorne’s Connectedness theorem. As a consequence, we prove for these ideals a Hirsch-type conjecture of Benedetti-Varbaro. This is a joint work with Davide Bolognini and Francesco Strazzanti.
Rui
Ma
Humboldt-Universität zu Berlin
Adaptive mixed finite element methods for non-selfadjoint
indefinite second-order elliptic PDEs with optimal rates
Abstract.
This talk establishes the convergence of adaptive mixed finite element methods for second-order linear non-selfadjoint indefinite elliptic problems in three dimensions with piecewise Lipschitz continuous coefficients. The error is measured in the L^2 norm of the flux variable and then allows for an adaptive algorithm with collective Dörfler marking. The axioms of adaptivity apply to this setting and guarantee the rate optimality for Raviart-Thomas and Brezzi-Douglas-Marini finite elements of any order for sufficiently small initial mesh-sizes and bulk parameter. Particular attention is laid out for the multiply connected polyhedral bounded Lipschitz domain and the quasi-interpolation of Nédélec finite elements.
Alexandra
Wesolek
FU Berlin
Bootstrap percolation in Ore-type graphs
Abstract.
In the r-neighbour bootstrap process on a graph G we start with an initial set of infected vertices A0 ⊆ V(G) and a new vertex gets infected as soon as it has r infected neighbours. We call A0 percolating if an infection of A0 infects all of our graph. Under which conditions on G can we find a small percolating set A0? In general, the denser our graph is the easier it is to infect new vertices as the initially infected vertices share potentially more neighbours. If v(G) ≤ r then we need to infect at least r vertices initially to infect all of our graph otherwise we will not be able to infect any new vertices during the bootstrap process. Gunderson showed that if a graph G on n vertices has minimum degree δ(G) ≤ ⌊ n/2 ⌋ + r - 3 then we can always find a percolating set of size r (if r ≤ 4 and n is big enough). How much can we decrease the minimum degree conditions if we are initially allowed to infect r+k vertices for k ∈ ℕ? What if we consider more general graphs where the sum of the degrees of any two non-adjacent vertices x and y is deg(x) + deg(y) ≤ D? This is more general because if a graph G has δ(G) = D/2 then for any two vertices in G it holds that deg(x) + deg(y) ≤ D. In this talk we give conditions on D=D(r,k) that guarantee a percolating set of size r + k, answering both open questions at once for small enough k=k(r).
Jürg
Kramer
HU Berlin
Sup-norm bounds of automorphic forms
Abstract.
In our talk we will talk about new approaches for establishing optimal sup-norm bounds for Maass forms.
Petr
Gregor
Charles University, Prague
Incidence colorings of subquartic graphs and Cartesian products
Abstract.
Two incidences (u,e) and (v,f) of vertices u, v and edges e, f (respectively) are adjacent if u=v, or e=f, or uv is one of edges e, f. An incidence coloring of a graph G is a coloring of its incidences such that adjacent incidences have distinct colors. We show that every graph of maximal degree 4 has an incidence coloring with 7 colors. Furthermore, we present sufficient conditions for Cartesian product graphs to have incidence colorings with Delta+2 colors where Delta is the maximal degree. In particular, we confirm a conjecture of Pai et al. on incidence colorings of hypercubes. Joint work with B. Lužar and R. Soták.
László
Kozma
Freie Universität Berlin
Self-adjusting data structures: trees and heaps
Abstract.
Binary search trees (BSTs) and heaps are perhaps the simplest implementations of the dictionary and the priority queue data types. They are among the most extensively studied structures in computer science, yet, many basic questions about them remain open. What is the best strategy for updating a BST in response to queries? Is there a single strategy that is optimal for every possible scenario? Are self-adjusting trees ("splay trees", Sleator, Tarjan, 1983) optimal? In which cases can we improve upon the logarithmic worst-case cost per operation? Our understanding of heaps is even more limited. Fibonacci heaps (and their relatives) are theoretically optimal in a worst-case sense, but they perform operations outside the "pure" comparison-based heap model (in addition to being complicated in practice). Is there a simple, "self-adjusting" alternative to Fibonacci heaps? Is there, by analogy to BSTs, a heap that can adapt to regularities in the input? Are the two problems related? In my talk I will present some old and new results pertaining to this family of questions.
November 2018
Max
Krause
TU Berlin
What is the Alexander polynomial of a knot?
Abstract.
In the 1920s, J. W. Alexander discovered the first polynomial invariant for knots. It would take several more decades for topologists to realize the full potential of this construction, which has inspired several similar invariants and now is a cornerstone of modern knot theory. In this talk we will see three different ways of calculating the Alexander polynomial, discuss the connections between them.
András
Stipsicz
Alfréd Rényi Institute of Mathematics
Invariants of knots
Raphael
Steiner
Flip Distances between Graph Orientations
Daniel
Ciripol
Jena
Computing Convex Hulls of Trajectories
Abstract.
Motivated by the problem of designing chemical reactors we study convex hulls of trajectories generated by polynomial dynamical systems. Real algebraic curves constitute a special case. We numerically determine whether the convex hull of such a curve is forward closed. That is, if the dynamics started from any point within the hull generate trajectories remaining inside the convex hull. To this end, we compute polyhedral approximations of the convex hull. We give results on the limiting behavior of those approximations. From the polyhedral approximations we derive information about stratified families of faces on the boundary of the convex hull. These stratifications help to partition the boundary into regions of inward and outward pointing dynamics. Experiments have been conducted for dimensions up to 4.
Hong
Liu
Warwick
Enumeration in arithmetic setting
Johannes
Storn
Humboldt-Universität zu Berlin
Advanced analysis of the DPG method
Abstract.
The functional analytical framework of the discontinuous Petrov-Galerkin (DPG) method bases on three hypotheses. Carstensen’s, Demkowicz’s, and Gopalakrishnan’s 2016 paper introduces a general guideline to verify the first two hypotheses. This talk modifies their approach. This modification improves existing results and allows for the design of well-posed DPG methods for parabolic and hyperbolic problems.
Andrew
Newman
TU Berlin
Small simplicial complexes with large torsion in homology
Abstract.
From the work of Kalai on ℚ-acyclic complexes it is known that for d ≥ 2 a d-dimensional simplicial complex on n vertices can have enormous torsion, on the order of exp(Θd(nd)), in its (d-1)st homology group. However, explicit constructions of complexes which realize this enormous-torsion phenomenon have been somewhat rare. Therefore, in this talk we consider and affirmatively answer the following inverse question: Given a finite abelian group of order m, can one always construct a d-dimensional simplicial complex on Θd((log m)1/d) vertices which realizes the given group in its (d - 1)st homology group?
Philipp
Reichenbach
TU Berlin
Vector bundles on elliptic curves and an associated Tannakian category
Abstract.
In 1957 Atiyah classified all vector bundles on elliptic curves for an algebraically closed ground field. Moreover, for the characteristic 0 case he completely described the multiplicative structure, i.e. the behavior of the tensor product. In this talk we review the essential results due to Atiyah and will interpret them in the light of Tannakian categories. Namely, allowing only morphisms of vector bundles on elliptic curves that respect the Harder-Narasimhan filtration leads to a neutral Tannakian category. For characteristic 0 we discuss some properties of the corresponding affine group scheme and give complete classifications of certain Tannakian sub-categories. Finally, some known results for the characteristic $p$ case are stated and questions for future research will be formulated.
Sebastian
Siebertz
Humboldt-Universität zu Berlin
First-order interpretations of sparse graph classes
Abstract.
The notions of bounded expansion and nowhere denseness capture uniform sparseness of graph classes and render various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce structurally bounded expansion and structurally nowhere dense graph classes, dfined as first-order interpretations of bounded expansion and nowhere dense graph classes. As a first step towards their algorithmic treatment, we provide a characterization of structurally bounded expansion classes via low shrubdepth decompositions, a dense analogue of low treedepth decompositions. We prove that structurally nowhere dense graph classes are vc-minimal.
Marcin
Pilipczuk
University of Warsaw
The square root phenomenon: subexponential algorithms in sparse graph classes
Abstract.
While most NP-hard graph problems remain NP-hard when restricted to planar graphs, they become somewhat simpler: they often admit algorithms with running time exponential in the square root of the size of the graph (or some other meaningful parameter). This behavior has been dubbed "the square root phenomenon". For many problems, such an algorithm follows from the theory of bidimensionality relying on a linear dependency on the size of the largest grid minor and treewidth of a planar graph. In recent years we have seen a number of other algorithmic techniques, significantly extending the boundary of the applicability of the "square root phenomenon". In my talk I will survey this area and highlight main algorithmic approaches.
Andrew
Newman
TU Berlin
The integer homology threshold in random simplicial complexes
Abstract.
A classic result of Erdős and Rényi is that the connectivity threshold in their random graph model is at (log n)/n. That is if one samples a graph G from the model G(n, p) with p less than (log n)/n then almost certainly G is disconnected, while p > (log n)/n implies that G almost certainly is connected. As graph connectivity is a homological property, it generalizes nicely to higher-dimensional simplicial complexes as homological connectivity, the vanishing of all homology groups in positive codimension. As such, the threshold for homological connectivity has been studied in the Linial--Meshulam model of random simplicial complexes. A paper of Meshulam and Wallach from 2006 develops the cocycle counting technique to establish the threshold for homological connectivity over a fixed finite field for random simplicial complexes. However, the question of homological connectivity over the integers remained open. In this talk I will give an overview of the Linial--Meshulam model and then discuss joint work with Elliot Paquette to adapt the cocycle counting technique to establish the threshold for homological connectivity over the integers.
Derk
Frerichs
Humboldt-Universität zu Berlin
Introduction to the Virtual Element Method
Abstract.
This talk introduces the virtual element method that can be seen as an extension of finite element methods to polygonal and polyhedral meshes. The first part illustrates the core ideas and some implementation aspects in the discretisation of the Poisson model problem. The second part concerns an outlook to mixed problems, in particular the Stokes problem. The content of this talk is based on the papers "Basic principles of Virtual Element Methods", and "The Hitchhiker's Guide to the Virtual Element Method" by L. Beirao da Veiga, Franco Brezzi, Andrea Cangiani, Gianmarco Manzini, L. D. Marini, and Allessandro Russo.
Sophia
Elia
FU Berlin
A comparison of Quotient Posets
Abstract.
Given an equivalence relation on the elements of a poset, place a new relation on the equivalence classes of elements. If this new relation is reflexive, antisymmetric, and transitive, then a quotient poset is created. We survey different types of quotient posets and see how they compare. In particular, we look at quotient posets that arise from lattice congruences, order-preserving group actions, and as images of surjective maps. We characterize quotient posets that correspond to a certain relation on equivalence classes by defining a sequence related to transitivity. Then we look at an application of quotient posets arising from order-preserving group actions. Let Hom(P,n) be the set of order-preserving maps from a poset P to a chain with elements. Stanley showed that |Hom(P,n)| is given by a polynomial in . Given an order-preserving group action on P, there is an induced group action on Hom(P,n). Katharina Jochemko showed that the number of maps in Hom(P,n)up to equivalence under the induced group action is also polynomial in using quotient posets. We touch on an alternative approach to the proof through the Ehrhart theory of order polytopes.
Jean-Philippe
Labbé
Freie Universität Berlin
(At least) three hard problems behind the multiassociahedron
Abstract.
The associahedron has a natural extension called the multiassociahedron. It is a vertex-decomposable simplicial sphere (in other words, a "combinatorially nice" sphere). Since at least 2003, people tried to construct simplicial polytopes, whose boundary is this simplicial complex, with limited success. In this talk, I will reveal (at least) three combinatorial "known-to-be-challenging" computational problems which should be indispensably faced to solve this problem.
Jörg
Rambau
Universität Bayreuth
Optimal Diplomacy
Abstract.
Picture yourself in a committee numerically evaluating a scientific proposal that you find worth funding: A rating of "0" means "easily achievable but not at all innovative", whereas "1" means "very innovative but totally unachievable". In both cases, funding is not recommended. In contrast, "1/2" means "innovative and achievable", in other words: worth funding. All intermediate values are possible. Any rating that is closer to "1/2" than to "1/4" and "3/4" is considered a vote for funding. The proposal passes if 50% of the members support funding, i.e., rate the proposal between "1/2 - 1/8" and "1/2 + 1/8". Now, there are 10 meetings ahead of you. You have an idea how the opinions of the committee members develop. How should your statements look like in the meetings one through ten if you want to have eventually as many supporters of the proposal as possible? This is an instance of the "Optimal Diplomacy Problem" (ODP), introduced by Hegselmann, König, Kurz, Niemann, and Rambau in 2010, published in 2015. How do opinions interact? How is the dynamics of opinions modeled mathematically? What does it mean to "influence others" in this dynamical system? How difficult is it to find optimal diplomacies? How can one compute or at least narrow down optimal diplomacies? What happens if not all informations about committee members are known to the diplomat? In this talk we will discuss our findings, techniques, and open questions based on the arguably most influential model: the Bounded-Confidence model by Hegselmann and Krause. We draw on joint work with Andreas Deuerling, Rainer Hegselmann, Stefan König, Julia Kinkel, Sascha Kurz, and Christoph Niemann.
Manfred
Scheucher
Using SAT Solvers in Combinatorics and Geometry
Marta
Panizzut
TU Berlin
Local Dressians Of Matroids
Abstract.
The tropical Grassmannian is a rational polyhedral fan parametrizing realizable (d − 1)-dimensional tropical linear spaces in the tropical projective space TP^(n−1). These are contractible polyhedral complexes arising from the tropicalization of (d − 1)-dimensional linear spaces in the projective space P^(n−1). Herrmann, Jensen, Joswig and Sturmfels introduced the Dressian Dr(d, n), an outer approximation of the tropical Grassmannian which parametrizes all (d − 1)-dimensional tropical linear spaces in TP^(n−1). Moreover, they remarked that a stratification based on matroids can be described on the Dressian, motivating a definition of local Dressian Dr(M) of a matroid M. Dressians and local Dressian will be the main characters of the talk. After introducing the main concepts, I will focus on the two fan structures that these objects are endowed with: one coming from the Plücker relations, and one as a subfan of the secondary fan of the matroid polytope. Generalizing a result of Herrmann et al., I will show that the two fan structures coincide. The proof is based on a careful analysis of the subdivisions induced on the 3-dimensional skeleton of the matroid polytope. This is a joint work with Jorge Alberto Olarte and Benjamin Schröter.
Christoph
Hertrich
Scheduling a Proportionate Flow Shop of Batching Machines
Carsten
Carstensen
Humboldt-Universität zu Berlin
HLO methodology
Thorsten
Herrig
HU Berlin
Fixed points and entropy of endomorphisms on complex tori
Abstract.
We investigate the fixed-point numbers and entropies of endomorphisms on complex tori. Motivated by an asymptotic perspective that has turned out in recent years to be so fruitful in Algebraic Geometry, we study how the number of fixed points behaves when the endomorphism is iterated. In this talk I show that the fixed-points function can have only three kinds of behaviour, and I characterize them in terms of the analytic eigenvalues. An interesting follow-up question is to determine criteria to decide of which type an endomorphism is. I will provide such criteria for simple abelian varieties in terms of the possible types of endomorphism algebras. The gained insight into the occurring eigenvalues can be applied to questions about the entropy of an endomorphism. I will give criteria for an endomorphism to be of zero or positive entropy and answer the important question whether the entropy can be the logarithm of a Salem number.
Till
Fluschnik
Technische Universität Berlin
Fractals for Kernelization Lower Bounds
Abstract.
Kernelization is a key concept in fixed-parameter algorithmics for polynomial-time preprocessing of NP-hard problems. Herein one seeks to preprocess any instance of a parameterized problem with parameter k to an “equivalent” instance (the so-called (problem) kernel) with size upper-bounded by some function (preferably by some polynomial) in k. Ten years ago, with the development of the so-called composition technique, first NP-hard parameterized problems were proven to presumably admit no polynomial-size problem kernel. Since then the line of Research concerning "limits of kernelization" received a lot of attention. We contribute to this line and present a new technique exploiting triangle-based fractal structures (so called T-fractals) for extending the range of applicability of compositions. Our technique makes it possible to prove new no-polynomial-kernel results for a number of graph problems dealing with some sort of cuts, hereby answering an open question stated in 2009. Our T-fractals---due to their fractal structure---admit easy-to-prove useful properties exploitable for constructing compositions. They also apply for planar as well as directed variants of the basic problems and also apply to both edge and vertex-deletion problems. This is joint work with Danny Hermelin, André Nichterlein, and Rolf Niedermeier. (Journal version appeared in SIAM Journal on Discrete Mathematics 2018.)
Christoph
Berkholz
Humboldt-Universität zu Berlin
A comparison of algebraic and semi-algebraic proof systems
Abstract.
In this lecture I will give an introduction to algebraic and semi-algebraic methods for proving the unsatisfiability of systems of real polynomial equations over the Boolean hypercube. The main goal of this talk is to compare the relative strength of these approaches using methods from proof complexity. On the one hand, I will focus on the static semi-algebraic proof systems Sherali-Adams and sum-of-squares (a.k.a. Lasserre), which are based on linear and semi-definite programming relaxations. On the other hand, I will discuss polynomial calculus, which is a dynamic algebraic proof system that models Gröbner basis computations. I am going to present two recent results on the relative strength between algebraic and semi-algebraic proof systems:¹ The first result is that sum-of-squares simulates polynomial calculus: any polynomial calculus refutation of degree d can be transformed into a sum-of-squares refutation of degree 2d and only polynomial increase in size. In contrast, the second result shows that this is not the case for Sherali-Adams: there are systems of polynomial equations that have polynomial calculus refutations of degree 3 and polynomial size, but require Sherali-Adams refutations of large degree and exponential size. ¹) this work was presented at STACS 2018; a preprint is available at https://eccc.weizmann.ac.il/report/2017/154/
Felix
Schröder
On the extendability of non-degenerate box configurations to tilings
Raman
Sanyal
The discrete and the continuous side of valuations – same same but different
Alex
Fink
Queen Mary University, London
The Tutte polynomial via lattice point enumeration
Abstract.
I will explain how to recover the Tutte polynomial of a matroid from an Ehrhart-style polynomial which counts lattice points in Minkowski sums of simplices and its base polytope. The key ingredient is a polyhedral interpretation of activity; along the way, this will give us a regular subdivision whose cells naturally encode Dawson's activity partition. I will also talk about its generalisation to polymatroids: in this setting, finding a bivariate activity invariant was a question of Tamás Kálmán, who constructed the univariate activity invariant in his work on enumerating spanning trees of hypergraphs. This work is joint with Amanda Cameron (Eindhoven).
Friederike
Hellwig
Humboldt-Universität zu Berlin
Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods
Patrick
Morris
FU Berlin
On the discrepancy of permutation families
Thirupathi
Gudi
IISc Bangalore
Patch-wise local projection stabilized methods for convection-diffusion problem
Abstract.
In this talk, we discuss on two stabilized methods incorporating local projections on the patches of the basis functions of finite element spaces. The underlying finite element spaces can be either the conforming finite element space or the classical nonconforming finite element space. Numerical experiments illustrating the effect of the stabilization will be presented. This is joint work with Dr. Asha K. Dond.
Peter
Jossen
ETH Zürich
Exponential Periods and Exponential Motives
Abstract.
I begin by explaining Nori's formalism and how to use it to construct abelian categories of motives. Then, following ideas of Katz and Kontsevich, I show how to construct a tannakian category of "exponential motives" by applying Nori's formalism to rapid decay cohomology, which one thinks of as the Betti realisation. This category of exponential motives contains the classical mixed motives à la Nori. We then introduce the de Rham realisation, as well as a comparison isomorphism with the Betti realisation. When k = IQ, this comparison isomorphism yields a class of complex numbers, "exponential periods", which includes special values of the gamma and the Bessel functions, the Euler-Mascheroni constant, and other interesting numbers which are not expected to be periods of classical motives. In particular, we attach to exponential motives a Galois group which conjecturally governs all algebraic relations among their periods.
Alexander
van der Grinten
Universität zu Köln
Scalable Katz Ranking Computation
Abstract.
Network analysis defines a number of centrality measures to identify the most central nodes in a network. Fast computation of those measures is a major challenge in algorithmic network analysis. Aside from closeness and betweenness, Katz centrality is one of the established centrality measures. We consider the problem of computing rankings for Katz centrality. In particular, we propose upper and lower bounds on the Katz score of a given node. While previous approaches relied on numerical approximation or heuristics to compute Katz centrality rankings, we construct an algorithm that iteratively improves those upper and lower bounds until a correct Katz ranking is obtained. For a certain class of inputs, this yields an optimal algorithm for Katz ranking computation. Furthermore, Experiments demonstrate that our algorithm outperforms both numerical approaches and heuristics with speedups between 1.5x and 3.5x, depending on the desired quality guarantees. Specifically, we provide efficient parallel CPU and GPU implementations of the algorithms that enable near real-time Katz centrality computation for graphs with hundreds of millions of nodes in fractions of seconds.
Henning
Meyerhenke
Humboldt-Universität zu Berlin
Algorithms for Large-scale Network Analysis
Abstract.
Network centrality measures indicate the importance of nodes (or edges) in a network. In this talk we will discuss a few popular measures and algorithms for computing complete or partial node rankings based on these measures. These algorithms are implemented in NetworKit, an open-source framework for large-scale network analysis, on which we provide an overview, too. One focus of the talk will be on techniques for speeding up a greedy (1-1/e)-approximation algorithm for the NP-hard group closeness centrality problem. Compared to a straightforward implementation, our approach is orders of magnitude faster and, compared to a heuristic proposed by Chen et al., we always find a solution with better quality in a comparable running time in our experiments. Our method Greedy++ allows us to approximate the group with maximum closeness centrality on networks with up to hundreds of millions of edges in minutes or at most a few hours. In a comparison with the optimum, our experiments show that the solution found by Greedy++ is actually much better than the theoretical guarantee.
Thirupathi
Gudi
IISc Bangalore
The obstacle problem
Raphael
Steiner
Metric Colourings of Graphs
October 2018
Tamás
Mészáros
FU Berlin
Sample compression schemes
Abstract.
In this talk we concentrate on two fundamental concepts of classical machine learning theory: learning and compression. Motivated by the fact that if a concept class admits a small sample compression scheme then it is efficiently learnable (Littlestone and Warmuth), we propose to study the relationship between the size of sample compression schemes and the combinatorial notion of VC dimension. In particular, we propose several directions to tackle the compression scheme conjecture which states that the smallest possible size of a sample compression scheme is linear in the VC dimension of the underlying concept class.
Nero
Budur
KU Leuven
Absolute sets and the Decomposition Theorem
Abstract.
The celebrated Monodromy Theorem states that the eigenvalues of the monodromy of a polynomial are roots of unity. In this talk we give an overview of recent results on local systems giving a generalization of the Monodromy Theorem. We end up with a conjecture of André-Oort type for special loci of local systems. If true in general, it would provide a simple conceptual proof for all semi-simple perverse sheaves of the Decomposition Theorem, assuming only the geometric case of perverse sheaves constructed from the constant sheaf (Beilinson-Bernstein-Deligne-Gabber). We prove the conjecture in rank one. Thus we have a new proof of the Decomposition Theorem for perverse sheaves constructed from rank one local systems. Joint work with Botong Wang.
Jan
Goedgebeur
Ghent University
Obstructions for 3-colouring graphs with one forbidden induced subgraph
Abstract.
For several graph classes without long induced paths there exists a finite forbidden subgraph characterization for k-colourability. Such a finite set of minimal obstructions allows to provide a “no-certificate" which proves that a graph is not k-colourable. We prove that there are only finitely many 4-critical P6-free graphs, and give the complete list that consists of 24 graphs. In particular, we obtain a certifying algorithm for 3-colouring P6-free graphs, which solves an open problem posed by Golovach et al. (Here, P6 denotes the induced path on six vertices.) Our result leads to the following dichotomy theorem: if H is a connected graph, then there are finitely many 4-critical H-free graphs if and only if H is a subgraph of P6. This answers a question of Seymour. The proof of our main result involves two distinct automatic proofs, and an extensive structural analysis by hand. In this talk we will mainly focus on the algorithmic results by presenting a new algorithm for generating all minimal forbidden subgraphs to k-colourability for given graph classes. This algorithm (combined with new theoretical results) has been successfully applied to fully characterise all forbidden subgraphs for k-colourability for various classes of graphs without long induced paths. (This is joint work with Maria Chudnovsky, Oliver Schaudt and Mingxian Zhong.)
Barbara
Jung
HU Berlin
What is the secret of zeta function?
Abstract.
The Riemann zeta function, in its simplicity, is the key to a bunch of well-known theorems in algebra and number theory. We will introduce the zeta function in its various shapes and forms, and discover these connections. Finally, we will have a glance at the Riemann hypothesis, find evidence for its correctness and shed light on some of its consequences.
Folkmar
Bornemann
TUM
Finite Size Effects: Random Matrices, Quantum Chaos and Riemann Zeros
Gaku
Liu
MPI Leipzig
Cubical Pachner moves and cobordisms of immersion
Abstract.
We study various analogues of theorems from PL topology for cubical complexes. In particular, we characterize when two PL homeomorphic cubulations are equivalent by Pachner moves by showing the question to be equivalent to the existence of cobordisms between immersions of hypersurfaces. This solves a question and conjecture of Habegger and Funar. The main tool is a theorem to show that any cubical PL decomposition of a disk is regular after some cubical stellar subdivision; this extends a result of Morelli. Joint work with Karim Adiprasito.
Neil
Olver
Generalized flow, the net present value problem, and an open question in arithmetic computation
Roman
Glebov
Ben Gurion University
Perfect Matchings in Random Subgraphs of Regular Bipartite Graphs
Abstract.
Consider the random process in which the edges of a graph G are added one by one in a random order. A classical result states that if G is the complete graph K2n or the complete bipartite graph Kn,n, then typically a perfect matching appears at the moment at which the last isolated vertex disappears. We extend this result to arbitrary k-regular bipartite graphs G on 2n vertices for all k=Ω(n). Surprisingly, this is not the case for smaller values of k. We construct sparse bipartite k-regular graphs in which the last isolated vertex disappears long before a perfect matching appears. Joint work with Zur Luria and Michael Simkin.
Giulio
Codogni
U Roma Tre
Gauss map, singularities of the theta divisor and trisecants
Abstract.
The Gauss map is a finite rational dominant map naturally defined on the theta divisor of an irreducible principally polarised abelian varieties. In the first part of this talk, we study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. Thanks to this analysis, we obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus. We spell out this bound in several examples, and we use it to understand the local structure of isolated singular points. We further define a stratification of the moduli space of ppavs by the degree of the Gauss map. In dimension four, we show that this stratification gives a weak solution of the Schottky problem, and we conjecture that this is true in any dimension. This is a joint work with S. Grushevsky and E. Sernesi. In the second part of this talk, we will study the relation between the Gauss map and trisecant of the Kummer variety. Fay's trisecant formula shows that the Kummer variety of the Jacobian of a smooth projective curve has a four dimensional family of trisecant lines. We study when these lines intersect the theta divisor of the Jacobian, and prove that the Gauss map of the theta divisor is constant on these points of intersection, when defined. We investigate the relation between the Gauss map and multisecant planes of the Kummer variety as well. This is a joint work with R. Auffarth and R. Salvati Manni.
Akiyoshi
Tsuchiya
Osaka University
Polyhedral characterizations of perfect graphs
Abstract.
Perfect graphs are important objects in graph theory. The perfect graphs include many important families of graphs, and serve to unify results relating colorings and cliques in those families. One of the most famous and most important results is the strong perfect graph theorem conjectured by Claude Berge and proved by Chudnovsky, Robertson and Thomas. This theorem characterizes perfect graphs. Our interest is to give other characterizations of perfect graphs. In this talk, we construct several lattice polytopes arising from a finite simple graph and characterize when the graph is perfect in terms of the lattice polytopes. This talk is based on joint work with Takayuki Hibi and Hidefumi Ohsugi.
Alexander
Fink
Queen Mary University London
Stiefel tropical linear spaces
Abstract.
As we tell our undergraduates, if K is a field, then there are a great number of different ways to describe a linear subspace of K^n. If the base is an algebraic object with less structure than a field, linear algebra becomes more subtle, and some of these descriptions cease to agree. One such setting is tropical geometry. Tropical geometers have reached consensus as to what the "correct" notion of tropical linear subspace is (one way to get it is by a vector of determinants). My subject will be one of the "wrong" descriptions, namely row spaces of matrices, which only produces a subset of the tropical linear spaces. Applications include generalisations of Mason's results from the '70s on presentations of transversal matroids, and a construction in the new area of tropical ideal theory. This work is variously joint with Felipe Rinc\'on, Jorge Alberto Olarte, and Jeffrey and Noah Giansiracusa.
Jörg
Wolf
Local regularity of weak solutions of parabolic systems with unbounded VMOcoefficients
Felipe
Rincón
University of Oslo
William
Trotter
Two Fundamentally Important Problems for Planar Posets
September 2018
Stefan
Lendl
Recoverable Robust Discrete Optimization
Raman
Sanyal
Goethe-Universität Frankfurt
Cone Angles, Gram's relation, and zonotopes
Abstract.
Euler’s Polyhedron Formula and its generalization, the Euler-Poincare formula, is a cornerstone of the combinatorial theory of polytopes. It states that the number of faces of various dimensions of a convex polytope satisfy a linear relation and it is the only linear relation (up to scaling). Similarly, Gram’s relation generalizes the fact that the sum of (interior) angles at the vertices of a convex $n$-gon is $(n-2)π$. In dimensions $3$ and up it is possible to associate an angle to faces of all dimensions and summing angles over faces of fixed dimension gives rise to the interior angle vectors of polytopes. Gram’s relation is the unique linear relation (up to scaling) satisfied by the angle vectors of polytopes. Interestingly, Gram’s relation is independent of the notion of angle. To make this precise, we will consider generalizations of “angles” in the form of cone valuations. It turns out that the associated generalized angle vectors still satisfy Gram’s relation and, surprisingly, it is the only linear relation, independent of the underlying cone valuation! Behind the scenes a beautiful interplay of discrete geometry and algebraic combinatorics is at work that we will try to explain, starting from the beginning. This is joint work with Spencer Backman and Sebastian Manecke.
August 2018
Manfred
Scheucher
L-shaped Point Set Embeddings of Trees
Raphael
Steiner
Star Dichromatic Number
Hendrik
Schrezenmaier
On grounded L-graphs and their relatives
July 2018
Marcel
Milich
Kreise für planare Matchings
Martin
Balko
On Erdős-Szekeres-type problems for k-convex point sets
Aaron
Williams
Gray Codes and Universal Cycles: Thinking Locally instead of Globally
Daniele
Bartoli
University of Perugia
Permutation polynomials over finite fields
Abstract.
Let q = p^h be a prime power. A polynomial f(x) in Fq[x] is a permutation polynomial (PP) if it is a bijection of the finite field Fq into itself. On the one hand, each permutation of Fq can be expressed as a polynomial over Fq. On the other hand, particular, simple structures or additional extraordinary properties are usually required by applications of PPs in other areas of mathematics and engineering, such as cryptography, coding theory, or combinatorial designs. Permutation polynomials meeting these criteria are usually difficult to find. A standard approach to the problem of deciding whether a polynomial f(x) is a PP is the investigation of the plane algebraic curve Cf : (f(x) − f(y))/(x − y) = 0; in fact, f is a PP over Fq if and only if Cf has no Fq-rational point (a, b) with a != b. In this talk, we will see applications of the above criterion to classes of permutation polynomials, complete permutation polynomials, exceptional polynomials, Carlitz rank problems, the Carlitz conjecture.
Monique
Laurent
CWI Amsterdam
Ordering data with combinatorial graph algorithms
Francisco
Criado
TU Berlin
The Hirsch conjecture for simplicial spheres
Abstract.
We introduce topological prismatoids, a combinatorial abstraction of the (geometric) prismatoids used in the recent counter-examples to the Hirsch Conjecture. We show that the "strong d-step Theorem" that allows to construct non-Hirsch polytopes from prismatoids of large width still works at this combinatorial level. Then, using metaheuristic methods on the flip graph, we construct four combinatorially different "non-d-step" 4-dimensional topological prismatoids with 14 vertices. This implies the existence of 8- dimensional spheres with 18 vertices which do not satisfy the Hirsch bound, which is smaller that the previously known examples by Mani and Walkup (24 vertices, dimension 11). Our non-Hirsch spheres are shellable but we do not know whether they are polytopal.
Andreas
Wiese
A (5/3 + ε)-Approximation for Unsplittable Flow on a Path: Placing Small Tasks into Boxes
Jeroen
Schillewaert
University of Auckland, NZ
Combinatorial methods in finite geometry
Abstract.
I will discuss a few different combinatorial techniques to study and characterise special classes of incidence structures (ovoids, spreads, maximal arcs,...) in finite geometry
Javier
Fresán
École Polytechnique
Hodge Theory of Kloosterman Connections
Abstract.
Broadhurst and Roberts recently studied the L-functions associated with symmetric powers of Kloosterman sums and conjectured a functional equation after extensive numerical computations. By the work of Yun, these L-functions correspond to “usual” motives over Q which, in low degree, are known to be modular. For the purpose of computing the Hodge numbers or relating the L-functions to periods, it is however more convenient to change gears and work with exponential motives. I will construct the relevant motives and show how the irregular Hodge filtration allows one to explain the gamma factors at infinity in the functional equation, as well as to get lower bounds for the p-adic valuations of Frobenius eigenvalues. It is a joint work with Claude Sabbah and Jeng-Daw Yu.
Stephen
Kobourov
Multi-Level Steiner Tree
Akihiro
Higashitani
Kyoto Sangyo University
Finding a new universal condition in Ehrhart theory
Abstract.
Recently, Balletti and I proved that for the h^*-polynomial h_0^*+h_1^*t+... of a lattice polytope, if we assume h_3^*=0, then (h_1^*, h_2^*) satisfies (i) h_2^*=0; or (ii) h_1^* (iii) (h_1^*,h_2^*)=(7,1). These conditions derive from Scott's theorem (1976), who characterized the possible h^*-polynomials of 2-dimensional lattice polytopes, and Scott's theorem is also essentially valid for lattice polytopes with degree at most 2 (Treutlein (2010)). On the other hand, we proved that it also holds under the assumption h_3^*=0. Since the assumption h_3^*=0 is independent of both dimension and degree of polytopes, we call the conditions (i), (ii), (iii) universal. In this talk, towards finding a new universal condition, we investigate the possibility for the polynomial h_0^*+h_1^*t+... to be the h^*-polynomial of some lattice polytope under the assumption that some of h_i^*'s vanish.
Necati Alp
Muyesser
CMU
Ramsey-type results for balanced graphs
Abstract.
Consider G, a 2-coloring of a complete graph on n vertices, where both color classes have at least \ep fraction of all the edges. Fix some graph H, together with a 2-coloring of its edges. By H^c, we denote the same graph with the colors switched. How large does n have to be so that G necessarily contains one of H or H^c as a subgraph? Call the smallest such integer, if it exists, R_\ep(H). We completely characterize the H for which R_\ep(H) is finite, discuss some quantitative bounds, and consider some related problems. Based on joint works with Matthew Bowen and Ander Lamaison.
Jörg
Wolf
Local well posedness of the Euler equations in generalized Campanato spaces. The critical case.
June 2018
Stefan
Felsner
Variationen über "Crossing Families"
Felipe
Cucker
On the homology of semialgebraic sets
Giulia
Codenotti
FU Berlin
Separation-type combinatorial invariants for triangulations of manifolds
Abstract.
In this talk, I propose a combinatorial invariant defined for simplicial complexes as an alternative way to obtain lower bounds on the size of triangulations of manifolds. The advantage of this approach lies within its simplicity: the invariants under investigation are defined by counting connected components and/or homological features of subcomplexes. As an application I obtain a linear identity for triangulated 4-manifolds with a fixed f-vector, unifying previously known lower bounds on the number of faces needed to triangulate a) simply connected 4-manifolds, and b) 4-manifolds of type 'boundary of a handlebody'. Moreover, based on the same construction, I present a conjectured upper bound for 4-manifolds with prescribed vector of Betti-numbers. This talk is based on joint work with Francisco Santos and Jonathan Spreer.
Frieder
Smolny
Multiscale optimization of logistics networks
Alexey
Pokrovskiy
ETH Zürich
Ryser's Conjecture & diameter
Abstract.
Ryser conjectured that the vertices of every r-edge-coloured graph with independence number i can be covered be (r - 1)i monochromatic trees. Recently Milicevic conjectured that moreover one can ensure that these trees have bounded diameter. We'll show that the two conjectures are equivalent. As a corollary one obtains new results about Milicevic's Conjecture.
Sophie
Puttkammer
Humboldt-Universität zu Berlin
A local a posteriori approximation error estimate for the
companion operator
Lukas
Gehring
Humboldt-Universität zu Berlin
Adaptive Mesh Refinement via Newest Vertex Bisection in n Dimensions - Part III
Linda
Kleist
Convexity increasing morphs of planar graphs
Daniele
Boffi
U Pavia
Finite element approximation of the Maxwell eigenvalue problem
Guillaume
Sagnol
Design of Optimal Experiments with Model Uncertainty
Benny
Sudakov
ETH Zürich
Rainbow structures, Latin squares & graph decompositions
Abstract.
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares. Since then rainbow structures were the focus of extensive research and found applications in design theory and graph decompositions. In this talk we discuss how probabilistic reasoning can be used to attack several old problems in this area. In particular we show that well known conjectures of Ryser, Hahn, Ringel, Graham-Sloane and Brualdi-Hollingsworth hold asymptotically. Based on joint works with Alon, Montgomery, and Pokrovskiy.
Lukas
Gehring
Humboldt-Universität zu Berlin
Adaptive Mesh Refinement via Newest Vertex Bisection in n Dimensions - Part II
Georgi
Mitsov
Humboldt-Universität zu Berlin
Discontinuous Petrov-Galerkin Methods for the Time-Dependent Maxwell Equations
Alberto Navarro
Garmendia
Uni Zürich
Recent advancements on the Grothendieck-Riemann-Roch theorem
Abstract.
Grothendieck's Riemann-Roch theorem compares direct images at the level of K-theory and the Chow ring. After its initial proof at the Borel-Serre report, Grothendieck aimed to generalise the Riemann-Roch theorem at SGA 6 in three directions: allowing general schemes not necessarily over a base field, replacing the smoothness condition on the schemes by a regularity condition on the morphism, and avoiding any projective assumption either on the morphism or on the schemes. After the coming of higher K-theory there was also a fourth direction, to prove the Riemann-Roch also between higher K-theory and higher Chow groups. In this talk we will review recent advancements in these four directions during the last years. If time permits, we will also discuss refinements of the Riemann-Roch formula which takes into account torsion elements.
Joachim
Naumann
Über Erhaltungssätze für die Maxwellgleichungen III.
Matthias
Eller
Georgetown University
Das Huygenssche Prinzip für Hyperbolische Systeme
Marco
D'Addezio
FU Berlin
What is a scheme?
Abstract.
In this talk I will introduce schemes and I will explain why and how they were discovered in the 50's.
Felix
Schröder
Stand der Wissenschaft zur Borsuk-Vermutung und Max-Distance-Graphen
BMS Friday
Grothendieck
Moritz Kerz (U Regensburg): Negative algebraic K-theory, Carlos Simpson (U Nice): From the nonabelian Grothendieck period conjecture to the structure of surfaces uniformized by the ball
Dietmar
Gallistl
University of Twente
Numerical approximation of planar oblique derivative problems in nondivergence form
Tien
Tran-Ngoc
Humboldt-Universität zu Berlin
Non-standard Discretisation of a Class of Degenerate Convex Minimisation Problems
John
Bamberg
UWA Perth
Hemisystem-like objects in finite geometry
Abstract.
Beniamino Segre showed in his 1965 manuscript 'Forme e geometrie hermitiane, con particolare riguardo al caso finito' that there is no way to partition the points of the Hermitian surface H(3,q^2) into lines, when q is odd. Moreover, Segre showed that if there is an m-cover of H(3,q^2), a set of lines covering each point m times, then m=(q+1)/2; half the number of lines on a point. Such a configuration of lines is known as a hemisystem and they give rise to interesting combinatorial objects such as partial quadrangles, strongly regular graphs, and imprimitive cometric Q-antipodal association schemes. This talk will be on developments in the field of hemisystems of polar spaces and regular near polygons and their connections to other interesting combinatorial objects. No background in finite geometry will be assumed.
Giuseppe
Ancona
Uni Strasbourg
On the standard conjecture of Hodge type for abelian fourfolds
Abstract.
Let S be a surface and V be the Q-vector space of divisors on S modulo numerical equivalence. The intersection product defines a non degenerate quadratic form on V. We know since the Thirties that it is of signature (1,d-1), where d is the dimension of V. In the Sixties Grothendieck conjectured a generalization of this statement to cycles of any codimension on a variety of any dimension. In characteristic zero this conjecture is an easy consequence of Hodge theory but in positive characteristic almost nothing is known. Instead of studying these quadratic forms at the archimedean place we will study them at p-adic places. It turns out that this question is more tractable. Moreover, using a classical product formula on quadratic forms, the p-adic result will give us non-trivial informations on the archimedean place. For instance, we will prove the original conjecture for abelian fourfolds.
Joachim
Naumann
Über Erhaltungssätze für die Maxwellgleichungen II.
Lukas
Gehring
Humboldt-Universität zu Berlin
Adaptive Mesh Refinement via Newest Vertex Bisection in n Dimensions - Part VI
Lukas
Gehring
Humboldt-Universität zu Berlin
Adaptive Mesh Refinement via Newest Vertex Bisection in n Dimensions - Part V
Seungchan
Ko
University of Oxford
Finite element approximation of incompressible chemically reacting
non-Newtonian fluids
Anna
von Pippich
TU Darmstadt
An analytic class number type formula for PSL2(Z)
Abstract.
For any Fuchsian subgroup Γ ⊂ PSL2(R) of the first kind, Selberg introduced the Selberg zeta function in analogy to the Riemann zeta function using the lengths of simple closed geodesics on Γ\H instead of prime numbers. In this talk, we report on a formula that determines the special value at s = 1 of the derivative of the Selberg zeta function for Γ = PSL2(Z). This formula is obtained as an application of a generalized Riemann--Roch isometry for the trivial sheaf on Γ\H, equipped with the Poincaré metric. This is joint work with Gerard Freixas.
Alejandro
López
FU Berlin
What is control theory?
Abstract.
I will give a short survey on control theory, some of the many ways in which it can be approached and its applications in the real world. After this the talk will focus on Pyragas control, a special technique used to modify the stability of particular elements within a dynamical system.
Fatiha
Alabau-Boussouira
U Lorraine / LJLL at Sorbonne U
Control theory and some applications on inverse problems and degenerate equations
May 2018
Francisco
Santos
Universidad de Cantabria
A Polyhedral Method for Sparse Systems with many Positive Solutions
Abstract.
We investigate a version of Viro's method for constructing polynomial systems with many positive solutions, based on regular triangulations of the Newton polytope of the system. The number of positive solutions obtained with our method is governed by the size of the largest positively decorable subcomplex of the triangulation. Here, positive decorability is a property that we introduce and which is dual to being a subcomplex of some regular triangulation. Using this duality, we produce large positively decorable subcomplexes of the boundary complexes of cyclic polytopes. As a byproduct we get new lower bounds for the maximal number of positive solutions of polynomial systems with prescribed numbers of monomials and variables. We also study the asymptotics of these numbers and observe a log-concavity property. This talk is based on joint work with Frédéric Bihan and Pierre-Jean Spaenlehauer.
Philipp
Bringmann
Humboldt-Universität zu Berlin
Quasi-optimal convergence of adaptive LSFEM in 3D
Malte
Renken
FU Berlin
Finding Disjoint Connecting Subgraphs in Surface-embedded Graph
Abstract.
Given a graph $G$, a set $T \subseteq V(G)$ of terminal vertices and a partition of $T$ into blocks, the problem "Disjoint Connecting Subgraphs" is to find a set of vertex-disjoint subgraphs of G, each covering exactly one block. While NP-hard in general, Robertson and Seymour showed that the problem is solvable in polynomial time for any fixed size of $T$. Generalizing a result of Reed, we give an $O(n \log(n))$ algorithm for when $G$ is embedded into an arbitrary but fixed surface.
Barbara
Jung
HU Berlin
The arithmetic self intersection number of the Hodge bundle on A2
Abstract.
The Hodge line bundle ω on the moduli stack A2, metrized by the L2-metric, can be identified with the bundle of Siegel modular forms, metrized by the logarithmically singular Petersson metric. Intersection theory for line bundles on arithmetic varietes, developed by Gillet and Soulé and generalized to the case of line bundles with logarithmically singular metrics by Burgos, Kramer, and Kühn, states a formula for the arithmetic self intersection number ϖ4 in terms of integrals of Green currents over certain cycles on the complex fibre of A2, and a contribution from the finite fibres. The computation of these ingredients for ϖ4 can be tackled by regarding the space A2 from different points of view. We will specify these viewpoints and sketch the associated methods for obtaining the explicit value of ϖ4.
Stefan
Sauter
Universität Zürich
Estimating the effect of data simplification for elliptic PDEs
Uwe
Schauz
The Combinatorial Nullstellensatz and List Edge Colorings of Graphs
Christian
Stump
TU Berlin
Counting inversions and descents of random elements in finite Coxeter groups
Abstract.
In this talk I will report on Mahonian and Eulerian probability distributions given by inversions and descents in general finite Coxeter groups. I will provide uniform formulas for the mean values and variances in terms of Coxeter group data in both cases. I will also provide uniform formulas for the double-Eulerian probability distribution of the sum of descents and inverse descents. I will finally establish necessary and sufficient conditions for general sequences of Coxeter groups of increasing rank for which Mahonian and Eulerian probability distributions satisfy central and local limit theorems. This talk is based on joint work with Thomas Kahle.
Anurag
Bishnoi
FU Berlin
A generalization of Chevalley-Warning and Ax-Katz via polynomial substitutions
Roberto
Laface
TU München
Picard numbers of abelian varieties in all characteristics
Abstract.
I will talk about the Picard numbers of abelian varieties over C (joint work with Klaus Hulek) and over fields of positive characteristic. After providing an algorithm for computing the Picard number, we show that the set Rg of Picard numbers of g-dimensional abelian varieties is not complete if g ≥ 2, that is there exist gaps in the sequence of Picard numbers seen as a sequence of integers. We will also study which Picard numbers can or cannot occur, and we will deduce structure results for abelian varieties with large Picard number. In characteristic zero we are able to give a complete and satisfactory description of the overall picture, while in positive characteristic there are several pathologies and open questions yet to be answered.
Kaja
Wille
Zerlegungen des Hyperwürfels in symmetrische Ketten
Christine
Bessenrodt
Leibniz Universität Hannover
Symmetries and tensor products
Nevena
Palić
FU Berlin
Grassmann polytopes and amplituhedra
Abstract.
The amplituhedra arise as images of the totally nonnegative Grassmannians by projections that are induced by linear maps. They were introduced in Physics by Arkani-Hamed & Trnka (Journal of High Energy Physics, 2014) as model spaces that should provide a better understanding of the scattering amplitudes of quantum field theories. The topology of the amplituhedra has been known only in a few special cases, where they turned out to be homeomorphic to balls. The amplituhedra are special cases of Grassmann polytopes introduced by Lam (Current developments in mathematics 2014, Int. Press). In this talk I will present results from the paper *Some more amplituhedra are contractible*, which is a joint work with Pavle V.M. Blagojević, Pavel Galashin and Günter M. Ziegler, and in which we show that some further amplituhedra are homeomorphic to balls, and that some more Grassmann polytopes and amplituhedra are contractible.
Rui
Ma
Humboldt-Universität zu Berlin
Guaranteed lower bounds for eigenvalues of elliptic operators in any dimension
Robin
de Jong
Uni Leiden
The Néron-Tate heights of cycles on abelian varieties
Abstract.
Given a polarized abelian variety A over a number field and an effective cycle Z on A, there is naturally attached to Z its so-called Néron-Tate height. The Néron-Tate height is always non-negative, and behaves well with respect to multiplication-by-N on A. Its good properties have led and still lead to applications in number theory. We discuss formulas for the Néron-Tate heights of some explicit cycles. First we focus on the tautological cycles on jacobians, where we find a new proof of the Bogomolov conjecture for curves. Then we focus on the symmetric theta divisors on a general principally polarized abelian variety. Here we find an explicit relation with the Faltings height. The latter part is based on joint work with Farbod Shokrieh.
Felix
Schröder
Zerlegungsgleichheit und die Dehn'sche Invariante
Johannes
Storn
Humboldt-Universität zu Berlin
On the analysis of discontinuous Petrov-Galerkin methods
Milos
Stojakovic
University of Novi Sad
Semi-random graph process
Abstract.
We introduce and study a novel semi-random multigraph process, described as follows. The process starts with an empty graph on n vertices. In every round of the process, one vertex v of the graph is picked uniformly at random and independently of all previous rounds. We then choose an additional vertex (according to a strategy of our choice) and connect it by an edge to v. For various natural monotone increasing graph properties P, we give tight upper and lower bounds on the minimum (extended over the set of all possible strategies) number of rounds required by the process to obtain, with high probability, a graph that satisfies P. Along the way, we show that the process is general enough to approximate (using suitable strategies) several well-studied random graph models. Joint work with: Omri Ben-Eliezer, Dan Hefetz, Gal Kronenberg, Olaf Parczyk and Clara Shikhelman.
Emre
Sertöz
MPI Leipzig
Computing and using periods of hypersurfaces
Abstract.
The periods of a complex projective manifold X are complex numbers, typically expressed as integrals, which give an explicit representation of the Hodge structure on the cohomology of X. Although they provide great insight, periods are often very hard to compute. In the past 20 years, an algorithm for computing the periods existed only for plane curves. We will give a different algorithm which can compute the periods of any smooth projective hypersurface and can do so with much higher precision. As an application, we will demonstrate how to reliably guess the Picard rank of quartic K3 surfaces and the Hodge rank of cubic fourfolds from their periods.
Joachim
Naumann
Über Erhaltungssätze für die Maxwellgleichungen I.
Manfred
Scheucher
Arrangements of Pseudocircles: Triangles, Drawings, and Circularizability
Emmanuel Jean
Candès
Stanford U
Sailing Through Data: Discoveries and Mirages
Jonathan
Kliem
FU Berlin
Discretizing Measure Partition Results
Abstract.
In this talk I will explain how to discretize convex measure partition results. It is a common approach to model colored points in R^d by measures. From measure partition results, e.g. the Ham Sandwich Theorem, one likes to obtain results on convex partitions of colored points. This progress has been inefficient so that even equipartition results could not be transferred except for trivial cases. A recent paper by Blagojević, Rote, Steinmeyer, and Ziegler uses a Network Flow to simultaneously equipartition d colors in R^d. I will show how this method can be generalized. This talk is based on joint work with Pavle V. M. Blagojević, Nevena Palić, Johanna K. Steinmeyer, and Günter M. Ziegler.
Christoph
Hertrich
Scheduling a Proportionate Flowshop of Batching Machines
Friederike
Hellwig
Humboldt-Universität zu Berlin
Optimal convergence rates for adaptive lowest-order dPG methods
April 2018
Jorge Alberto
Olarte
FU Berlin
What is a matroid?
Abstract.
Matroids are rich combinatorial structures that can be seen as a general notion of independence. However, matroids are particular in that they have dozens of different cryptomorphic definitions. Therefore they appear underlying in many mathematical objects and applications can been found in several different fields including algebra, geometry, graph theory, model theory and optimization. In this talk we will briefly describe the main concepts of matroid theory as well as explaining how to abstract matroids from matrices and graphs.
Patricia
Franz
Zur Komplexität von Sechseckskontaktdarstellungen
Louis J.
Billera
Cornell U
In pursuit of a white whale: On the real linear algebra of vectors of zeros and ones
Louis
Billera
Cornell Univ.
Unbalanced collections and Farkas complexes
Abstract.
We will define and discuss unbalanced collections (PSS systems of Björner) both from the perspective of posets (the order complex of each such collection) and complexes (the simplicial complex of all such collections). In the first case, they are all shellable balls with the same f-vector. In the second, they form a homotopy sphere. The latter leads to a general construction of something we call the Farkas complex of an arrangement of hyperplanes or oriented matroid. The first perspective is due to Anders Björner; the second represents ongoing joint work with Florian Frick. A few examples of other Farkas complexes will be examined, including a family of complexes considered by Klee and Novik. Farkas complexes come complete with a description of their “missing” faces, leading to a way of describing their face rings.
Patrick
Morris
FU Berlin
Tilings in randomly perturbed hypergraphs
Abstract.
In 2003, Bohman, Frieze and Martin introduced a random graph model called the perturbed model. Here we start with some $\alpha>0$ and an arbitrary graph $G$ of minimum degree $\alpha n$. We are then interested in the threshold probability $p=p(n)$ for which $G \cup G(n,p)$ satisfies certain properties. That is, for a certain property, for example Hamiltonicity, we can ask what is the minimum probability $p(n)$, such that for \emph{any} $n$-vertex graph $G$ of minimum degree $\alpha n$, $G \cup G(n,p)$ has this property with high probability. This model has been well studied and the threshold probability has been established for various properties. One key property is the notion of having a $H$-tiling for some fixed graph $H$. By this, we mean a union of disjoint copies of $H$ in $G\cup G(n,p)$ that covers every vertex exactly once. This generalisation of a perfect matching is fundamental and was studied in the setting of perturbed dense graphs by Balogh, Treglown and Wagner in 2017. In this talk, we look to extend this problem to the setting of hypergraphs. This is work in progress and joint with Wiebke Bendenknecht, Jie Han, Yoshiharu Kohayakawa and Guilherme Mota.
Daniel
Peterseim
Universität Augsburg
Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials
Shagnik
Das
FU Berlin
Colourings without monochromatic chains
Abstract.
In 1974, Erdős and Rothschild introduced a new kind of extremal problem, asking which n-vertex graph has the maximum number of monochromatic-triangle-free red/blue edge-colourings. While this original problem strengthens Mantel’s theorem, recent years have witnessed the study of the Erdős-Rothschild extension of several classic combinatorial theorems. In this talk, we seek the Erdős-Rothschild extension of Sperner’s Theorem. More precisely, we search for the set families in 2^{[n]} with the most monochromatic-k-chain-free r-colourings. Time and interest permitting, we shall present some results, sketch some proofs, and offer many open problems. This is joint work with Roman Glebov, Benny Sudakov and Tuan Tran.
Ulf
Kühn
Uni Hamburg
Multiple q-zeta values and period polynomials
Abstract.
We present a class of q-analogues of multiple zeta values given by certain formal q-series with rational coefficients. After introducing a notion of weight and depth for these q-analogues of multiple zeta values we will state a dimension conjecture for the spaces of their weight- and depth-graded parts, which have a similar shape as the conjectures of Zagier and Broadhurst-Kreimer for multiple zeta values.
Jörg
Wolf
On the global well-posedness of the Navier-Stokes equations in a subspace of L^2_loc
Stefan
Felsner
A routing problem in arrangements with applications
Ander
Lamaison
FU Berlin
Ramsey density of infinite paths
Gonzalo
Gonzalez de Diego
Universität Duisburg-Essen
Approximating the Cauchy stress tensor in hyperelasticity
Abstract.
In this presentation, a least squares finite element method (LSFEM) is presented for the first order system of hyperelasticity defined over the deformed configuration in order to approximate the Cauchy stress tensor. Unlike the first Piola-Kirchhoff stress tensor, the Cauchy stress tensor is symmetric, a property intimately related to the conservation of angular momentum. With this work, we wish to explore the possibility of imposing the symmetry of the stress tensor, strongly or weakly, in non-linear elasticity. Firstly, an overview of a LSFEM for hyperelasticity (over the reference configuration) by (Müller et al., 2014) is presented. Secondly, we address the question of under which conditions can a first order system over the deformed configuration be considered. Then, the former LSFEM is extended to the deformed configuration; we introduce a Gauss-Newton method for solving the non-linear minimization problem and show that, under small strains and stresses, the least-squares functional represents an a posteriori error estimator. Finally, we display numerical results for two test cases. These results indicate that this LSFEM is capable of giving reliable results even when the regularity assumptions from the analysis are not satisfied.
Felix
Schröder
On a crossing lemma for multigraphs
Manfred
Scheucher
Partial Least-Squares Point Matching under Translations
March 2018
Hendrik
Schrezenmaier
Faster Algorithms for some Optimization Problems on Collinear Points
Manfred
Scheucher
Minimal Geometric Graph Representations of Order Types
Amy
Wiebe
University of Washington
Slack Ideals of Polytopes
Abstract.
In this talk we discuss a new tool for studying the realization spaces of polytopes, namely the slack ideal associated to the polytope. These ideals were first introduced to study PSD rank of polytopes, and their structure also encodes other important polytopal properties, gives us a new way to understand important concepts such as projective uniqueness, and suggests connections with the study of other algebraic and combinatorial objects (toric ideals and graphs, for example).
Nadine
Raasch
Kontaktdarstellungen planarer Graphen mit Fünfecken
February 2018
Penny
Haxell
University of Waterloo
Chromatic index of random multigraphs
Sarka
Necasova
Weak-strong uniqueness in fluid structure interaction problem
Stefan
Felsner
Alternating permutations, Euler numbers and some bijections
Mohsen
Sadeghi
FU Berlin
What is molecular dynamics?
Abstract.
Molecular dynamics has been around since the 60's, and has gained ever increasing popularity as the method of choice for doing computer experiments on molecular systems. Despite the immensity of its forms and applications, the method at its core is built upon rather simple concepts. In this talk, we will look behind the curtain of a generic molecular dynamics simulation, and investigate its main components. We will talk about dynamics of particle systems, forcefields, and integrators, and look at the challenges we face in making the connection between molecular dynamics simulations and the physical world.
Jörg
Wolf
Localization of Gronwall's argument with applications to the Euler equations
Héctor
Andrade Loarca
TU Berlin
What is an inverse problem?
Abstract.
What is an inverse problem? Is it just another kind of problem? In this talk I will present an intuitive approach to inverse problems as well as a rigorous mathematical way to define them and some techniques to solve them, known as regularization theory. I will also focus on inverse problems applied in imaging science, showing some examples and implementations of typical algorithms. At the end if the time is enough I would like to give an small introduction to deep learning techniques in inverse problem regularization.
Carola-Bibiane
Schönlieb
U Cambridge
Model based learning in imaging
Linda
Kleist
Area-universality of triangulations
Anna Maria
Hartkopf
FU Berlin
Martin
Skutella
On the Complexity of Instationary Gas Flows
Andrei
Asinowski
FU Berlin
Enumeration of lattice paths with forbidden patterns
Yuri
Bilu
U Bordeaux
Equations with singular moduli: effective aspects
Abstract.
A singular modulus is the j-invariant of an elliptic curve with complex multiplication. André (1998) proved that a polynomial equation F(x,y)=0 can have only finitely many solutions in singular moduli (x,y), unless the polynomial F(x,y) is 'special' in a certain precisely defined sense. Pila (2011) extended this to equations in many variables, proving the André-Oort conjecture on C^n. The arguments of André and Pila were non-effective (used Siegel-Brauer). I will report on a recent work by Allombert, Faye, Kühne, Luca, Masser, Pizarro, Riffaut, Zannier and myself about partial effectivization of these results.
Dongho
Chae
Chung-Ang University, Seoul, South Korea
On the blow-up of the Euler equations
Hendrik
Schrezenmaier
Universal slope sets for 1-bend planar drawings
Gustavo
Jasso
Univ. Bonn
Triangulations of cyclic polytopes and simplicial combinatorics
Abstract.
Motivated by applications in higher category theory, Dyckerhoff and Kapranov introduced the notion of a d-Segal set in 2012. These are a class of simplicial sets satisfying a property defined in terms of triangulations of d-dimensional cyclic polytopes. In this talk I will give an elementary introduction to d-Segal sets and relate them to the combinatorics of outer horns, which are specific simplicial subsets of the standard simplices. This is part of joint work in progress with Tobias Dyckerhoff (Bonn).
Anurag
Bishnoi
FU Berlin
Zeros of a polynomial in a finite grid
Xavier
Roulleau
U Aix-Marseille
Irrationality of cubic threefolds by their reduction mod 3
Abstract.
A smooth cubic threefold X is unirational: there exists a dominant rational map f : P^3 → X. Clemens and Griffiths proved that a smooth complex cubic X is irrational, i.e. the degree of such f is always > 1. That was the first counter-example to the Lüroth Problem. The difficult part of their proof was to show that the intermediate Jacobian of X (which is an Abelian variety canonically attached to X) is not the Jacobian of a curve. In this talk we will prove that result anew for a generic cubic, by elementary methods: reduction mod p and point counting. This is a joint work with Dimitri Markouchevitch.
Alexander
Müller
FU Berlin
What is a homotopy invariant?
Abstract.
We will introduce and explain the fundamental notions of homotopy theory underlying all of algebraic topology. Path components, homotopy groups of spheres, Betti numbers: homotopy invariants are diverse and powerful tools to understand spaces of all flavours.
Hélène
Barcelo
MSRI/UC Berkeley
Discrete Notions of Homotopy and Homology
Martin
Skrodzki
FU Berlin
Combinatorial and Asymptotical Results on the Neighborhood Grid
Abstract.
In 2009, Joselli et al introduced the Neighborhood Grid data structure for fast computation of neighborhood estimates in point clouds. Even though the data structure has been used in several applications and shown to be practically relevant, it is theoretically not yet well understood. The purpose of this talk is to present a polynomial-time algorithm to build the data structure. Furthermore, it is investigated whether the presented algorithm is optimal. This investigations leads to several combinatorial questions for which partial results are given. The talk follows our current ArXiv paper: https://arxiv.org/abs/1710.03435
Matthias
Schymura
FU Berlin
Some remarks on the reverse isodiametric problem
Abstract.
Motivated by an old question of Makai Jr. on the thinnest non-separable arrangement of convex bodies, we study a reverse form of the classical isodiametric inequality. More precisely, we are interested in estimating the maximal isodiametric quotient that an affine image of a given convex body can have. After an early solution of Behrend (1937) in the plane, the problem seemed to be forgotten and is open starting from three dimensions. In the talk, we shall first explain the connection to densities of non-separable arrangements, and then illustrate how concepts such as Löwner-position, well-distributed point configurations on the sphere, and measures of orthogonality of orthonormal matrices naturally appear in this context. As a result we obtain the currently best asymptotic solution to the reverse isodiametric problem. Joint (ongoing) work with Bernardo González Merino.
Tibor
Szabó
FU Berlin
Exploring the projective norm graphs
January 2018
Antareep
Mandal
HU Berlin
Uniform sup-norm bound for the average over an orthonormal basis of Siegel cusp forms
Abstract.
This talk updates our progress towards obtaining the optimal sup-norm bound for the average over an orthonormal basis of Siegel cusp forms by relating it to the discrete spectrum of the heat kernel of the Siegel-Maass Laplacian. In particular, we study the Maass operators and generalize the relation between the cusp forms and the Maass forms in higher dimensions along with presenting an interesting trick to obtain the heat kernel corresponding to the Maass Laplacians by adapting the 'method of images' used to obtain the heat kernel corresponding to the Laplace-Beltrami operator.
Volodymyr
Derkach
Donetsk National University, Vinnytsa, Ukraine
Friedrichs type inequality and an inverse problem for wave equation with partial data
Tamara
Fastovska
V.N. Karazin Kharkiv University, Ukraine
Rational decay rates and attractors for fluid-structure interaction systems
Bartosz
Walczak
Discrepancy of axis-parallel boxes
Mario
Kummer
MPI Leipzig/TU Berlin
Interlacing Ehrhart Polynomials of Reflexive Polytopes
Abstract.
The symmetric edge polytope is a lattice polytope associated to a graph. The Ehrhart polynomial of a symmetric edge polytope often has a remarkable property: All their roots have real part -1/2. One of the first positive results was the case of complete (1,n)-bipartite graphs proved independently by Kirschenhofer et al. and by Bump et al in the context of analytic number theory. We prove several conjectures confirming that certain families of polynomials have real part -1/2. The main ingredient is the theory of interlacing polynomials.
Rico
Raber
and
Khai Van
Tran
Stochastic Machine Scheduling, Gammoids and Time-Expanded Networks
Dániel
Korándi
EPFL
Rainbow saturation and graph capacities
Alexander
Danilin
Immanuel Kant Baltic Federal University, Kaliningrad, Russia
Determination of diffractors in elastic media
Joachim
Naumann
HU Berlin
Über einige Eigenschaften schwacher Lösungen des Maxwell-Gleichungen II.
Hyeong-Ohk
Bae
Ajou University, Suwon, South Korea
The interaction of particles and fluid
Marek
Gluza
FU Berlin
What is Maxwell's equations?
Abstract.
The talk is an invitation to Maxwell's equations, a set of four partial differential equations that describe the classical electromagnetic phenomena. In physics, they were a milestone with huge impact and influence on subsequent physics such as the theory of special relativity and the quantum gauge field theories, like the Standard Model of particle physics. In engineering, the numerical solution to Maxwell's equations allows important technical advances, such as the improvement of the building blocks of electronic devices and our communication channels (which allows us to watch cats over Internet in always better quality). In this talk, I will introduce electromagnetic fields in the Euclidean space and the Maxwell's equations governing them, explain their physical meaning and show how one can derive that the light is an electromagnetic wave from Maxwell's equations. So wonder no more about the fundamental laws making it all possible, let's see together what are the Maxwell's equations!
Marlis
Hochbruck
KIT
On the numerical solutions of linear Maxwell's equations
Henning
Heinrich
Approaching area-universality
Gaku
Liu
MPI Leipzig
A counterexample to the extension space conjecture for realizable oriented matroids
Abstract.
The extension space conjecture of oriented matroid theory states that the space of all one-element, non-loop, non-coloop extensions of a realizable oriented matroid of rank $d$ has the homotopy type of a sphere of dimension $d-1$. We disprove this conjecture by showing the existence of a realizable uniform oriented matroid of high rank and corank 3 with disconnected extension space. The talk will not assume any prior knowledge of oriented matroids.
Aaron
Bernstein
Online Bipartite Matching with Amortized O(log^2 N) Replacements
Tamás
Mészáros
FU Berlin
Boolean dimension and tree-width
Walter
Gubler
U Regensburg
Non-archimedean Monge-Ampère equations
Abstract.
We study non-archimedean volumes, a tool which allows us to control the asymptotic growth of small sections of big powers of a metrized line bundle. We prove that the nonarchimedean volume is differentiable at a continuous semipositive metric and that the derivative is given by integration with respect to a Monge-Ampère measure. Such a differentiability formula had been proposed by M. Kontsevich and Y. Tschinkel. In residue characteristic zero, it implies an orthogonality property for non-archimedean plurisubharmonic functions which allows us to drop an algebraicity assumption in a theorem of S. Boucksom, C. Favre and M. Jonsson about the solution to the non-archimedean Monge-Ampère equation. We will also present a similar result in positive equicharacteristic assuming resolution of singularities.
Leonid
Pestov
Immanuel Kant Baltic Federal University, Kaliningrad, Russia
Construction of a speed of sound from a part of a boundary (BC-method)
Joachim
Naumann
HU Berlin
Über einige Eigenschaften schwacher Lösungen des Maxwell-Gleichungen I.
András
Tóbiás
TU Berlin
What is large deviations?
Abstract.
The field of large deviations deals with rare events in probability theory. It considers events whose probability tends to zero exponentially fast. In many cases, the exponential rate of decay can be identified precisely, and the rate function helps understanding the rare event itself. It shows the most likely way how the event is realized, and it can often be used for proving a law of large numbers. Thanks to these properties, large deviations are used, e.g., for Markov chains, statistical mechanics, SDEs, random walks in a random environment or extremal combinatorics. In this talk, I will introduce large deviations starting with the example of coin tosses, I will present Cramér's theorem and sketch some applications of large deviation theory.
Stefan
Felsner
Introduction to Placements of Rooks
Moritz
Firsching
FU Berlin
The classification of polytopal 3-spheres with 9 vertices into polytopes and nonpolytopes
Abstract.
Altshuler and Steinberg classified all 1336 types of 3-dimensional polytopal spheres with 8 vertices into 1294 combinatorial types of 4-dimensional polytopes with 8 vertices and 42 non-polytopal spheres. We examine how an analogues classification of 3-dimensional polytopal spheres can be obtained in order to answer the question: 'How many combinatorial types of 4-dimensional polytopes with 9 vertices exists?'. We also discuss if rational coordinates can be given for those types.
Torsten
Muetze
TU Berlin
Sparse Kneser graphs are Hamiltonian
December 2017
Simona
Boyadzhiyska
FU Berlin
Interval orders with restrictions on the interval lengths
Christoph
Spiegel
UPC Barcelona
On a question of Sárkozy and Sós
Davide Cesare
Veniani
Uni Mainz
Recent advances about lines on quartic surfaces
Abstract.
The number of lines on a smooth complex surface in projective space depends very much on the degree of the surface. Planes and conics contain infinitely many lines and cubics always have exactly 27. As for degree 4, a general quartic surface has no lines, but Schur's quartic contains as many as 64. This is indeed the maximal number, but a correct proof of this fact was only given quite recently. Can a quartic surface carry exactly 63 lines? How many can there be on a quartic which is not smooth, or which is defined over a field of positive characteristic? In the last few years many of these questions have been answered, thanks to the contribution of several mathematicians. I will survey the main results and ideas, culminating in the list of the explicit equations of the ten smooth complex quartics with most lines.
Lara
Skuppin
TU Berlin
What is the Teichmüller space of a surface?
Abstract.
The aim of this talk is to give an introduction to Teichmüller space. Roughly speaking, Teichmüller space is a space of geometric surfaces sharing the underlying topological surface. An important property is its relation to the so-called moduli space. This presentation has a connection to Anna Wienhard's colloquium talk. It will contain a review of concepts related to the topology of an orientable compact surface, such as its genus and fundamental group, and concepts related to its geometry, such as hyperbolic structures, conformal structures and the uniformization theorem. Teichmüller space parametrizes these structures. Further, it is related to representations of the fundamental group as a discrete group of isometries of the hyperbolic plane. As final outlook, we will discuss some results relating to Teichmüller spaces.
Anna
Wienhard
U Heidelberg
A tale of rigidity and flexibility: discrete subgroups of higher rank Lie groups
Aaron
Bernstein
and
Frieder
Smolny
Incremental Cycle Detection and Topological Sort, Distance-preserving graph contractions
Martin
Skrodzki
FU Berlin
Combinatorial and Asymptotical Results on the Neighborhood Grid
Morten
Risager
U Kopenhagen
Arithmetic statistics of modular symbols
Abstract.
Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols. We prove a refined version of this conjecture.
Georg
Loho
TU Berlin
What is monomial tropical conesfor multicriteria optimization?
Abstract.
We introduce a special class of tropical cones which we call `monomial tropical cones'. They arise as a helpful tool in the description of discrete multicriteria optimization problems. After an introduction to tropical convexity with an emphasis on these particular tropical cones, we explain the algorithmic implications. We finish with connections to commutative algebra.
Patrick
Schnider
Measures of depth for multiple points
Henri
Mühle
TU Dresden
The Alternating Group and Noncrossing Partitions
Abstract.
We consider the alternating group generated by the set of all three-cycles. By orienting the corresponding Cayley graph 'away from the identity', we obtain a natural partial order on the alternating group. It turns out that this partial order is in fact a subword order, in which every interval admits a braid group action on its set of maximal chains, the Hurwitz action. The purpose of this talk is to study this poset from an enumerative and a structural point of view. We outline how noncrossing partitions enter the picture, and how they help us to produce beautiful formulas. Moreover, we enumerate orbits under Hurwitz action. If time permits, we present further generalizations of this construction. This is joint work with Philippe Nadeau.
Karl
Däubel
A Comparison-Based Approach to Spanners and Contractions
Lutz
Warnke
Georgia Institute of Technology
Paking nearly optimal Ramsey R(3,t) graphs
Bart
De Bruyn
Ghent University
Old and new results on extremal generalized polygons
Efthymios
Sofos
MPI Bonn
An Erdős-Kac law for local solubility in families of varieties
Abstract.
A famous theorem due to Erdős and Kac states that the number of prime divisors of an integer N behaves like a normal distribution. In this talk we consider analogues of this result in the setting of arithmetic geometry, and obtain probability distributions for questions related to local solubility of algebraic varieties. This is joint work with Daniel Loughran.
Hendrik
Schrezenmaier
Planar graphs as L-intersection graphs
November 2017
Thomas
Torsney-Weir
Universität Wien
Slicing multi-dimensional spaces
Abstract.
I will discuss our work with Torsten Möller on investigating methods of visualizing multi-dimensional spaces using interactive one and two dimensional slices. Visualizing multi-dimensional spaces on a two-dimensional screen can be done either by projection or slicing the space. In medical visualization often shows axial, medial, and sagittal slices of a volume around a particular focus point (location in the body). Slices directly visualize an object and it is easy to measure distances in the horizontal and vertical directions. I will show how we addressed the issue of picking a focus point. In addition, I will show a number of examples including comparing regression models, function spaces, and regular polygons.
Sven
Jäger
Generalizing the Kawaguchi-Kyan bound to stochastic parallel machine scheduling
Ander
Lamaison
FU Berlin
The random strategy in Maker-Breakers
Ezra
Waxman
U Tel Aviv
Angles of Gaussian primes
Abstract.
Fermat showed that every prime p=1 mod 4 is a sum of two squares: p=a^2+b^2, and hence such a prime gives rise to an angle whose tangent is the ratio b/a. Hecke proved that these Gaussian primes are equidistributed across sectors of the complex plane, by making use of (infinite) Hecke characters and their associated L-functions. In this talk I will present a conjecture, motivated by a random matrix model, for the variance of Gaussian primes across sectors, and discuss ongoing work about a more refined conjecture that picks up lower-order-terms. I will also introduce a function field model for this problem, which will yield an analogue to Hecke's equidistribution theorem. By applying a recent result of N. Katz concerning the equidistribution of 'super even' characters (the function field analogues to Hecke characters), I will provide a result for the variance of function field Gaussian primes across sectors; a computation whose analogue in number fields is unknown beyond a trivial regime.
Mario
Kummer
MPI Leipzig
What is a spectrahedron?
Abstract.
Spectrahedra are a central object in convex algebraic geometry. We will present some basic properties.
Hendrik
Schrezenmaier
Planar graphs as L-contact graphs
Claus
Scheiderer
U Konstanz
Spectrahedral shadows
Rainer
Sinn
FU Berlin
Matrix completion problems from a geometric point of view
Abstract.
We will see several matrix completion problems for matrices with real entries and discuss questions that are specific to the case of real entries from a geometric point of view. I will present answers for special families of examples based on joint work with Daniel Bernstein and Greg Blekherman.
Miriam
Schlöter
Earliest Arrival Transshipments in Networks With Multiple Sinks
Raphael
Steiner
Neumann-Lara-flows and the Two-Colour-Conjecture
Afshin
Goodarzi
FU Berlin
Face numbers of shellable nonpure complexes
Abstract.
The f-triangle of a simplicial complex is an integer array whose entries determine the number of faces of complexes with a given dimension and co-dimension. The f-triangle is a finer invariant than the ordinary f-vector. If the simplicial complex is pure (all the maximal faces have the same dimension), then the f-triangle encodes the same information as the f-vector. In this talk, we discuss a numerical characterization of the f-triangles of shellable non-pure complexes. This result generalizes the classical Macaulay-Stanley theorem to the nonpure case. (This talk is based on a joint work with K. Adiprasito and A. Björner.)
Guillaume
Sagnol
The Price of Fixed Assignments in Stochastic Extensible Bin Packing
Patrick
Morris
FU Berlin
Random Steiner Triple systems
Monika
Eisenmann
TU Berlin
What is a finite element method?
Abstract.
Partial differential equations appear in many applications, therefore it is of great importance to have efficient approximation concepts. For a wide range of problems the finite element method can be used to obtain numerical approximations. In this talk, I will explain the idea of this method on a simple one dimensional example problem.
Bartosz
Walczak
Extending Partial Representations of Trapezoid Graphs
Ilaria
Perugia
U Vienna
Trefftz finite elements
Anurag
Bishnoi
FU Berlin
Spectral methods in extremal combinatorics
Martin
Lüdtke
U Frankfurt
A birational anabelian reconstruction theorem for curves over algebraically closed fields
Abstract.
The question central to birational anabelian geometry is how strongly a field $K$ is determined by its absolute Galois group $G_K$. According to a conjecture of Bogomolov, if $K$ is the function field of a variety of dimension at least 2 over an algebraically closed field, it can by fully recovered from $G_K$. In dimension 1, however, $G_K$ is a profinite free group of rank equal to the cardinality of the base field, containing therefore no information about $K$. We show that a complete reconstruction is possible if one knows in addition how $G_K$ is embedded into the group of field automorphisms fixing only the base field.
Carlos
Améndola
TU Berlin
What is algebraic statistics?
Abstract.
Algebraic statistics is a relatively new field that has developed rapidly in recent years, using algebraic-geometric methods to provide new tools to analyze and solve statistical problems. In this talk I will present examples that illustrate the exciting interplay between algebra and statistics. No background in either of these areas will be assumed!
Stefan
Felsner
More about Circle- and Pseudocirclearrangements
Jorge Alberto
Olarte
FU Berlin
Pure flag simplicial complexes and the Erdős-Ko-Rado-property
Abstract.
We launch the thorough study of the pure Erdős-Ko-Rado property of simplicial complexes. Specifically, we identify two fundamental conditions on the maximal faces of a pure simplicial complex to satisfy the Erdős-Ko-Rado property, namely flagness and the absence of boundary ridges. We establish a tower of increasingly ambitious conjectures stating, in the most general form, that these properties are sufficient to establish the pure-EKR property in arbitrary dimension. We then prove this conjecture in dimensions two and three. This is joint work with Paco Santos, Jonathan Spreer and Christian Stump
October 2017
Jannes
Münchmeyer
HU Berlin
What is a random graph?
Abstract.
Graphs are a crucial model in many parts of science nowadays. As often the structure of the underlying graphs is not fully known, the field of random graphs has hugely gained significance. This talk will give an introduction on random graphs. It will also present the main stochastic models for random graphs and their properties.
Paul
Wancura
Rechteckszerlegungen auf Punktmengen
Steffen
Lauritzen
University of Copenhagen
Graphical models
Robert
Löwe
TU Berlin
Secondary fans and secondary polyhedra of punctured Riemann surfaces
Abstract.
A famous construction of Gelfand, Kapranov and Zelevinsky associates to each finite point configuration A subset R^d a polyhedral fan, which stratifies the space of weight vectors by the combinatorial types of regular subdivisions of A. That fan arises as the normal fan of a convex polytope. In a completely analogous way we associate to each hyperbolic Riemann surface R with punctures a polyhedral fan. Its cones correspond to the ideal cell decompositions of R that occur as the horocyclic Delaunay decompositions which arise via the convex hull construction of Epstein and Penner. Similar to the classical case, this secondary fan of R turns out to be the normal fan of a convex polyhedron, the secondary polyhedron of R.
Andrew
Treglown
Birmingham
The complexity of perfect matchings and packings in dense hypergraphs
Enlin
Yang
U Regensburg
Characteristic class and the ε (epsilon) factor of an étale sheaf
Abstract.
In this talk, we will briefly recall the definition and the properties of singular support and characteristic cycle of a constructible complex on a smooth variety. The theory of singular support is developed by Beilinson, which is motivated by the theory of holonomic D-modules. The characteristic cycle is constructed by Saito. In a joint work with Umezaki and Zhao, we prove a conjecture of Kato-Saito on a twist formula for the epsilon factor of a constructible sheaf on a projective smooth variety over a finite field. In our proof, Beilinson and Saito's theory plays an essential role.
Christoph
Standke
Zu Kodierung und Kreisbarkeit von Pseudokreisarrangements
Linda
Kleist
On the Monge-Kontorovich transportation problem and Voronoi diagrams
Hendrik
Schrezenmaier
Applications of a topological lemma in combinatorial geometry (circle packing, floorplans)
September 2017
Torsten
Mütze
A short proof of the middle levels theorem
Stephan
Tillmann
University of Sydney
Computing trisections of 4–manifolds
Abstract.
Gay and Kirby recently generalised Heegaard splittings of 3-manifolds to trisections of 4-manifolds. A trisection describes a 4-dimensional manifold as a union of three -dimensional handlebodies. The complexity of the 4-manifold is captured in a collection of curves on a surface, which guide the gluing of the handelbodies. After defining trisections and giving key examples and applications, I will describe an algorithm to compute trisections of 4-manifolds using arbitrary triangulations as input. This results in the first explicit complexity bounds for the trisection genus of a 4–manifold in terms of the number of pentachora (4-simplices) in a triangulation. This is joint work with Mark Bell, Joel Hass and Hyam Rubinstein.
July 2017
Stefan
Felsner
The chip firing game
Irem
Portakal
FU Berlin
What is Gröbner basis doing in Sudoku?
Abstract.
For most mathematicians, it is a difficult task to answer the question "What are you studying?", or even harder "What is it good for?". Algebraic geometry is one of the most abstract research areas in mathematics, and these kind of questions leave us speechless. But luckily, almost everyone knows what Sudoku is. It turns out that one can use algebraic geometry to solve these games via the so-called Gröbner bases. We will introduce this technique and apply it to some examples.
Linda
Kleist
Towards the Harary Hill conjecture
Olena
Naboka
National Technical University "Kharkov Polytechnic Institute", Ukraine
Synchronization in the systems of nonlinear coupled Berger plates
Olena
Sukhorukova
Donetsk National University, Vinnytsa, Ukraine
A-regular-A-singular factorizations of generalized J-inner matrix functions
Charles
Vial
Universität Bielefeld
Distinguished models of intermediate Jacobians
Abstract.
Let X be a smooth projective variety defined over a subfield K of the complex numbers. It is natural to ask whether the complex abelian variety that is the image of the Abel-Jacobi map defined on algebraically trivial cycles admits a model over K. I will show that it admits a unique model making the Abel-Jacobi map equivariant with respect to the action of the automorphism group of the complex numbers fixing K. We offer three applications. First, we show that such a model is a derived-invariant for smooth projective varieties defined over K. Second, we answer a question of Mazur by showing that this model over the base field K is dominated by the Albanese variety of a product of components of the Hilbert scheme of X. Third, we recover a result of Deligne on complete intersections of Hodge level one. This is joint work with Jeff Achter and Yano Casalaina-Martin.
Sebastian
Hensel
HU Berlin
Local boundedness of local suitable weak solutions of the Navier-Stokes equations
Robert
Lasarzik
TU Berlin
What is Onsager's conjecture?
Abstract.
In this talk, the equations of motion for an idealized fluid are motivated from basic laws of physics. I will comment on some properties of solutions to this equations, like existence, uniqueness and conservation of energy, and link them to Onsager's conjecture.
Camillo
De Lellis
U Zurich
The Onsager Theorem and folding papers
Paul
Breiding
TU Berlin
What is a condition number?
Abstract.
In this talk I will motivate and introduce a fundamental notion in numerical analysis — the condition number. Condition numbers serve as a measure of how sensitive a function is with respect to perturbations in the input. I will give an introductory example demonstrating that small changes in the input data may lead to large deviations in the output, even for seemingly simple problems such as solving linear equations. Using this example, I will give the definition of condition number and then go over to define condition numbers à la Shub and Smale within a more general framework.
Dmytro
Strelnikov
Donetsk National University, Vinnytsa, Ukraine
Boundary triples for integral systems
Viktoria
Filatova
Immanuel Kant Baltic Federal University, Kaliningrad, Russia
A numerical solution of the ultrasound medical tomography problem
Katy
Beeler
FU Berlin
Polytopes with few coordinate values
Abstract.
(0,1)-polytopes are defined by their vertex coordinates taking values 0 or 1. In this talk we will present two generalizations of (0,1)-polytopes called (0,1,a)- and (0,1,a_i)-polytopes. Interesting examples of these polytopes arise already in dimension three, with combinatorial types that cannot be realized as rational polytopes. We will discuss the enumeration in lower dimensions that produced these examples and then examine the tractability of a classification in higher dimensions. We will also discuss various combinatorial properties of these polytopes.
Matthias
Rost
Virtual Network Embedding Approximations: Leveraging Decomposable LP Formulations and Randomized Rounding
Volodymyr
Derkach
Donetsk National University, Vinnytsa, Ukraine
Boundary triples and Weyl functions of symmetric operators
Leonid
Pestov
Immanuel Kant Baltic Federal University, Kaliningrad, Russia
On the boundary rigidity problem of the Riemannian manifolds
June 2017
Josué
Tonelli-Cueto
TU Berlin
What is BPP, RP and the other probabilistic complexity classes?
Abstract.
When one faces problems to solve, randomness can be used in order obtain faster answers at the cost of some uncertainty. Probabilistic complexity classes capture the different ways in which this can be done. In this talk, we introduce the basic probabilistic complexity classes, their interrelations and we illustrate by outlining the solution to concrete problems.
Avi
Wigderson
IAS Princeton
Randomness
Jean-Philippe
Labbé
FU Berlin
Some bounds on the number of faces of flag homology spheres (joint work with Eran Nevo)
Abstract.
In this talk, we discuss enumerative and structural properties of certain simplicial complexes called 'flag homology spheres'. A simplicial complex Δ, it is called 'flag' when it is equal to its clique complex, and Δ is an 'homology sphere' when the link of every face in Δ has the homology of a sphere (over some field). The study of f-vectors of flag homology spheres (or their better suited γ-vectors) is motivated by the g-conjecture, and the Charney--Davis conjecture, which asserts the non-negativity of a certain linear combination of the entries of the f-vector. We present bounds on some entries of the γ-vector supporting a conjecture of Nevo--Petersen. The techniques used give two further results 1) relating f-vector of flag balanced simplicial complexes and γ-vectors of flag homology spheres, and 2) an analog to Perles' Theorem on k-skeleta of flag homology spheres. This is joint work with Eran Nevo (Hebrew University of Jerusalem).
Aaron
Bernstein
An improved deterministic algorithm for dynamic single source shortest paths
Vera
Nosikova
Immanuel Kant Baltic Federal University, Kaliningrad, Russia
Application of the Reverse Time Migration (RTM) procedure in ultrasound tomography, numerical modeling
Arnau
Padrol
Paris 6
Extension complexity of polytopes with few vertices (or facets)
Abstract.
The number of combinatorial types of d-polytopes with up to d+4 vertices (or facets) grows superexponentially with d. However, only quadratically many can have realizations with extension complexity smaller than d+4; that is, arise as a linear projection of a higher-dimensional polytope with less than d+4 facets. These are easy to classify into finitely many families and the exact extension complexity of each realization is easy to decide. The classification involves (generalized) Gale transforms and an upper bound on the number of polytopes with few vertices and facets.
Julie
Meißner
MST under Uncertainty in Theory and Experiments
Hendrik
Schrezenmaier
Maximum-area triangle in a convex polygon
Thomas Valentin
Mikosch
U Copenhagen
Richard von Mises and the development of modern extreme value theory
Florian
Frick
Cornell University
Geometry and Combinatorics of Fair Division
Abstract.
Sperner's lemma is a combinatorial version of Brouwer's fixed point theorem and states that if the vertices of a triangulation of an n-simplex are colored by n+1 colors following certain simple rules, then there is a face of the triangulation that exhibits all n+1 colors on its vertices. This talk focuses on extensions of Sperner's lemma and related results and applications to problems of fair division. In particular, this approach yields an algorithm for how to fairly divide rent such that n frugal roommates with subjective preferences will not be envious of each other even if the preferences of one roommate are unknown. This is joint work with Megumi Asada, Kelsey Houston-Edwards, Frédéric Meunier, Vivek Pisharody, Maxwell Polevy, David Stoner, Ling Hei Tsang, and Zoe Wellner.
Iryna
Ryzhkova-Gerasimova
V.N. Karazin Kharkiv University, Ukraine
Asymptotic dynamics of a class of fluid-plate interactive models
Marcin
Lara
FU Berlin
What is Iwasawa theory?
Abstract.
Iwasawa theory is the study of arithmetic objects over infinite towers of number fields. The main example is the extension of $\mathbb{Q}$ by the $p$-power roots of unity. The theory origins in the following insight of Iwasawa: instead of working with a fixed finite Galois extension and modules under its Galois group, it is often easier to describe every Galois module in an infinite tower of fields at once. This talk will be an introduction to Iwasawa theory.
Sujatha
Ramdorai
U British Columbia
Galois representations and Iwasawa theoretic invariants
Stefan
Felsner
Maximum crossing number of cycles and bipartite graphs
Jonathan
Spreer
FU Berlin
Combinatorial invariants, tightness, and a lower bound for triangulated 3-manifolds
Abstract.
In 1987, Kalai proved that stacked spheres of dimension at least three are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result does not extend to dimension two. In this talk, I will present a characterisation of stacked 2-spheres using what is called the separation index. This characterisation of stacked 2-spheres is then applied to settle the 3-dimensional case of a conjecture by Lutz, Sulanke and Swartz stating that 'tight-neighborly triangulated manifolds are tight' -- essentially by reproving a lower bound for triangulated 3-manifolds. This is joint work with Benjamin Burton, Basudeb Datta, and Nitin Singh.
May 2017
Shagnik
Das
Berlin
Some of my favourite problems from Sanya
Robert
Weismantel
Proximity results and faster algorithms for Integer Programming using the Steinitz Lemma
Alexander
Danilin
Immanuel Kant Baltic Federal University, Kaliningrad, Russia
Numerical solutions of the problem of detecting uitraweak defractors in a complex acoustic medium
Linda
Kleist
Area-universal rectangular layouts
Manfred
Scheucher
SageMath - Eine Kurzeinführung
Alfio
Quarteroni
EPFL Lausanne
Taking Mathematics to the Heart
Marta
Panizzut
TU Berlin
Linear Systems on Graphs
Abstract.
Baker and Norine introduced a combinatorial theory of linear systems on graphs in analogy with the one on algebraic curves. The interplay between the theories is given by specialization of linear systems from curves to dual graphs of degenarations. As shown by Baker, through specialization the rank of a linear system can only increase. In this talk I will present results on linear systems on complete graphs and complete bipartite graphs. This is based on an ongoing project with Filip Cools (KU Leuven) and Michele D’Adderio (ULB).
Patrick
Jähnichen
HU Berlin
What is variational inference?
Abstract.
In my talk I will give a (very) brief introduction into Bayesian probabilistic models. In general, we are interested in the posterior distribution over the latent variables in such a model. I will explain how we can turn this estimation problem into an optimization one and how to make use of properties of the exponential family of distributions to derive an elegant solution to it.
Hendrik
Schrezenmaier
Optimal Coding and Sampling of Triangulations
Klaus-Robert
Müller
TU Berlin
Machine learning and AI for the sciences – towards understanding
Jacinta
Torres
MPI Leipzig
April 2017
Giulia
Codenotti
FU Berlin
What is triangulated spheres and Pachner moves?
Abstract.
In the world of discrete geometry, somewhere between combinatorics and geometry, Pachner moves (or bistellar flips) are an important tool to "build up" complex triangulations from more simple ones, while preserving topological properties. I will show with examples how Pachner moves pop up in old and new questions, spanning topics from combinatorics to topology.
Nevena
Palic
FU Berlin
Introduction to the oriented matroid Grassmannian
Abstract.
Oriented matroid Grassmannian, also called MacPhersonian, is a combinatorial analogue of the real Grassmannian. It was introduced in early 90's by MacPherson and it was firstly used by Gelfand and MacPherson to give a combinatorial formula for Pontrjagin classes. Since then, there were attempts to understand the MacPhersonian, but still not much is known about its topology. The main open question remains, whether the MacPhersonian and the Grassmannian are homotopy equivalent. In this talk I will define the oriented matroid Grassmannian and look into some of its properties. It will be easy to understand and the questions from the audience are welcome.
Hong
Liu
University of Warwick
Edges not in any monochromatic copy of a fixed graph.
Stefan
Felsner
Flipping Edges of Triangulations
Alexander
Fauck
HU Berlin
What is Hamiltonian dynamics?
Abstract.
Classical mechanical systems are systems where the time evolution depends only on position and momentum. One can describe them with a vector field on the so called phase space — the Hamiltonian vector field. The studies of the dynamics of such a vector field (for example the motion of our planetary system or the 3-body problem) motivated the notion of symplectic and contact structures. In my talk I will explain some of these concepts and describe how they arise from the classical setup.
Till
Miltzow
The Art Gallery Problem is $\exists \mathbb{R}$-complete
Michael
Hutchings
UC Berkeley
Symplectic embeddings of cubes
Michael
Dobbins
Stretching weighted pseudoline arrangements
Francisco
Santos Leal
Universidad de Cantabria
Classification of empty 4-simplices, part 2
Abstract.
This talk is related to the one I gave in February in this same seminar, but it will be self-contained and largely non-overlapping with part 1. I will report on joint work (partially in progress) with O. Iglesias and M. Blanco, aimed at completing the full classification of empty 4-simplices. An empty simplex is a lattice d-polytope with no other lattice point than its d+1 vertices. They are the “building-blocks” of lattice polytopes, in the sense that every lattice polytope can be triangulated into empty simplices, and they are also important in algebraic geometry since they correspond to the “terminal singularities” that arise in minimal model programs. The complete classification of empty 3-simplices was done by White (1964), but in dimension 4 only partial results were known so far. Our approach relies on the recent complete classification of lattice hollow 3-polytopes by Averkov, Krumpelmann and Weltge and follows the following scheme: 1) empty 4-simplices of width 1 form a 3-parameter family quite easy to classify, à la White. (To relate this to what follows, observe that these empty 4-simplices are the ones that project to a hollow 1-polytope Q, the unit segment). 2) empty 4-simplices of width at least three are finitely many and we have classified them completely by proving an explicit upper bound of 5058 for their (normalized) volume and by enumerating all empty 4-simplices up to that bound. There are 179 of width three and a single one of width four. 3) empty 4-simplices of width two come in three types: 3.1) Those that project to a hollow 2-polytope Q. Since Q itself must have width at least two, it must be the second dilation of a unimodular triangle. We show that empty 4-simplices projecting to it form two 2-parameter families. 3.2) Those that project to a hollow 3-polytope Q, but not to a hollow 2-polytope. We show that, apart of finitely many exceptions for P (with volume bounded by 100) Q is necessarily a triangular bipyramid. Looking at the AKW classification we find out that there are exactly 52 possibilities for Q. Each of them corresponds to an infinite 1-parameter family of hollow 4-simplices, which contains either none or infinitely many empty 4-simplices. We are working on deciding which is the case for each of them (current status is 3 produce none, 38 produce infinitely many, and 11 are still open). 3.3) Those that do not project to a hollow 3-polytope. These are finitely many and we are trying to classify them along the same lines of case (2), except our current volume bounds are not good enough to certify that we have a complete classification. As a by-product, we confirm the classification of “stable quintuples” conjectured by Mori, Morrison and Morrison (1989) and proved by Bober (2009). In our language, these quintuples correspond to the infinite families of empty 4-simplices that project to a *primitive* lower dimensional hollow polytope Q (meaning that vertices of Q integer-affinely span the lattice). These include the 3-parameter family of case (1), one of the two 2-parameter families of case (3.1), and 29 of the 52 1-parameter families of case (3.2). The other 1 + 23 families have to be considered new “non-primitive stable quintuples”.
Jan
Corsten
London School of Economics
Grid Ramsey Problem.
March 2017
Linda
Kleist
Area-universality and $\forall\exists\mathbb R$ (\FER)-hardness
Adam Zsolt
Wagner
University of Illinois at Urbana-Champaign
Families with few k-chains
Hendrik
Schrezenmaier
Morphing Schnyder drawings of plane triangulations
Lauri
Loiskekoski
FU Berlin
Separation in the graph of a simple polytope
Abstract.
Separation in graphs is related to graphs being expanders. A conjecture by Kalai generalizes the planar separation theorem to simple polytopes. It states that the graph of a simple d-polytope can be separated to two roughly equal parts by removing O(n^((d-2)/(d-1))) vertices. We provide a counterexample to this conjecture. This is joint work with Günter Ziegler.
Jean-Philippe
Labbé
FU Berlin
Computing with Polytopes -- Progress Report
Abstract.
This seminar is going to give an overview of the progress made during the 2 weeks long Sage Days workshop in Olot (Spain) on softwares involving polyhedral computations: LattE, normaliz, polymake, Sage, etc.
Manfred
Scheucher
A superlinear lower bound on the number of 5-holes
February 2017
Sven
Jäger
Worst case bound of the LRF rule for minimizing total weighted completion time on identical parallel machines
Viktor
Tkachenko
Almost periodic evolution systems with impulse action at state-dependent moment
Janine
Felten
Sortieren mit partieller Information
Georg
Loho
TU Berlin
What is linear programming?
Abstract.
Linear programming is a special optimization problem which is widely applicable for solving real-world problems. It has a rich discrete geometric structure. Furthermore, there are still several open complexity questions concerning the algorithms to solve linear programs. In this talk we will give a geometric intuition for the problem. We present the simplex method which is the major tool to solve linear programs.
Kolja
Knauer
On lattice path matroid polytopes: integer points and Ehrhart polynomial
Florian
Kohl
FU Berlin
Joachim
Mahnkopf
Universität Wien
On local constancy of dimension of slope subspaces of automorphic forms
Abstract.
We prove a higher rank analogoue of a Conjecture of Gouvea-Mazur on local constancy of dimension of slope subspaces of automorphic forms for reductive groups having discrete series. The proof is based on a comparison of Bewersdorff's elementary trace formula for pairs of congruent weights and does not make use of p-adic Banach space methods or rigid analytic geometry.
Sarca
Necasova
Rigorous derivation of the equations describing objects called "accretion disc"
Lasse
Hinrichsen
FU Berlin
What is a (discontinuous) Galerkin finite element method?
Abstract.
Many physical phenomena are modeled by partial differential equations (PDEs) that cannot be solved analytically anymore. Thus our aim is to compute a — hopefully — reliable approximation of the solution to such a PDE. One of the basic steps is to transfer the infinite-dimensional problem to something that computers — with finite memory — can handle. In this "What is..." talk, we will get an idea how the Galerkin Finite Element Method (FEM) serves this purpose. In the end, we shall also see a special variant, namely the Discontinuous Galerkin (DG) method, and discuss its uses.
Veit
Wiechert
Orthogonal tree decompositions and chromatic number
Endre
Süli
U Oxford
Discontinuous Galerkin Finite Element Methods: Stability, Accuracy, Adaptivity
Francisco
Santos
Classification of empty 4-simplices
Abstract.
An empty simplex is a lattice $d$-polytope with no other lattice point than its $d+1$ vertices. In this talk I will explain what is known about empty 4-simplices. In particular, I will comment on three proofs of the fact that all but finitely many (equivalence classes of) empty 4-simplices have width at most two: - the nice (but incomplete) proof by Barile, Bernardi, Borisov, and Kantor ('On empty lattice simplices in dimension 4.', Proc. Am. Math. Soc. 139, 2011, 4247-4253). - the proof by Blanco, Haase, Hofmann, and Santos (arXiv:1607.00798) - the proof by Santos and Iglesias (in preparation, a poster was presented in http://www.mi.fu-berlin.de/en/math/groups/tubdiscmath/Events/Lattice_Poly_Workshop/index.html#Posters)
Frank
Mousset
ETH Zürich
Covering sparse graphs by monochromatic cycles
Bas
Edixhoven
University of Leiden
Group schemes out of birational group laws
Abstract.
In his construction of the jacobian variety of a smooth projective algebraic curve C over a field, Weil first showed that the gth symmetric power of C (with g the genus of C) has a birational group law, and then that this birational group law extends uniquely to a group variety. In the 1960's, Weil's extension result was generalised to schemes by Michael Artin in SGA 3, and used for the construction of reductive group schemes and of Neron models of abelian varieties. At the occasion of the re-edition of SGA 3, I had a look at Artin's article, and it seemed to me that it was better to change the approach to the problem. The main improvement is to construct the group scheme as a sub sheaf of an fppf sheaf of relative birational maps. The fact that relative birational maps admit fppf descent seems to be new. As a consequence, some finiteness conditions in Artin's article are no longer needed. Joint work with Matthieu Romagny: http://arxiv.org/abs/1204.1799 Appeared in Panor. Synthèses, 47, Soc. Math. France, Paris, 2015.
Leon
Sering
A Combinatorial Upper Bound on the Length of Twang Cascades
Stefan
Felsner
Triangles in Arrangements
Günter M.
Ziegler
FU Berlin
January 2017
Josué
Tonelli-Cueto
TU Berlin
What is the length of a potato?
Abstract.
It's clear what someone means when E refers to the volume or the area of a potato, but if this same person starts to talk about its length, a sensible doubt about the meaningfulness of this concept arises. In this talk we will see how the length of a potato is a meaningful concept by introducing the so-called intrinsic volumes and reviewing some of their main properties. Note: Title stolen from one article of Schanuel.
Monika
Ludwig
TU Vienna
Geometric classification
Nikolai
Kalischek
Topologische Zeichnungen bipartiter Graphen
Romanos
Malikiosis
TU Berlin
Christopher
Kusch
FU Berlin
Random Strategies are nearly optimal for generalised van der Waerden games
Sebastián
Herrero
Chalmers Tekniska Högskola/University of Gothenburg
Asymptotic distribution of Hecke points over Cp
Abstract.
[see here]
Shagnik
Das
FU Berlin
What is randomness good for?
Abstract.
We all know that randomness has several useful applications, ranging from quantum physics to cryptography to experimental design. What, though, does it offer the poor pure mathematician, who cannot afford a quantum computer, has nothing to hide, and has even less interest in real-world phenomena? In this talk we will give a partial answer to this question by demonstrating the use of the probabilistic method in extremal combinatorics. No prior knowledge of combinatorics is necessary, although some familiarity with colours — in particular, red and blue — could be useful.
Tom
Trotter
Trees and Circle Orders
Theodosios
Douvropoulos
University of Minnesota
Lyashko-Looijenga morphisms and geometric factorizations of a Coxeter element
Abstract.
A common theme in Combinatorics is an unconditional love for the symmetric group. We like to investigate structural and numerological properties of various objects associated with it. It often happens that such objects and phenomena can be generalized to the other (complex) reflection groups as well. This is the world of Coxeter Combinatorics. A problem that goes back to Hurwitz and the 19th century is to enumerate (reduced) factorizations of the long cycle (12..n)∈ Sn into factors from prescribed conjugacy classes. In the reflection groups case, it corresponds to enumerating factorizations of a Coxeter element. Bessis gave a beautiful geometric interpretation of such factorizations by using a variant of the Lyashko-Looijenga (LL) map, a finite morphism coming from Singularity theory. We extend some of Bessis' and Ripoll's work and use the LL map to enumerate the so called "primitive factorizations" of a Coxeter element c. That is, factorizations of the form c=w⋅ t1⋯ tk, where w belongs to a prescribed conjugacy class and the ti's are reflections.
William T.
Trotter
Georgia Institute of Technology
Dimension and Cut Vertices
Yiannis
Petridis
University College London
Lattice point problems in hyperbolic spaces
Abstract.
[see here]
Dietrich
Stoyan
TU Bergakademie Freiberg
Stochastic Geometry in Nature, Arts and Design
Evan
DeCorte
McGill University, Canada
The spectral method for geometric colouring problems
Abstract.
Consider the graph H(d) whose vertex set is the hyperbolic plane, where we join two points with an edge when their distance is equal to d. Asking for the chromatic number of this graph is the hyperbolic analogue to the famous Hadwiger-Nelson problem. One has the lower bound of 4 for all d>0, as in the Euclidean case. Using spectral methods, we prove that with the additional requirement that the colour classes be measurable, one needs at least 6 colours to properly colour H(d) when d is sufficiently large. This is joint work with Konstantin Golubev. We’ll begin the talk by discussing a few problems about the chromatic number and measurable chromatic number of geometric distance graphs, with the aim of surveying the spectral method.
Tibor
Szabó
FU Berlin
The Oberwolfach problems
Matthew
Jenssen
London School of Economics
Exact Ramsey numbers of odd cycles via nonlinear optimisation
Peter
Bürgisser
TU Berlin
Towards a probabilistic Schubert calculus
Abstract.
Hermann Schubert developed in the 19th century a calculus for answering enumerative questions in algebraic geometry, e.g., 'How many lines intersect four curves of degrees d_1,...,d_4 in three-dimensional space in general position?'. In his 15th problem, Hilberts asked for a rigorous foundation of Schubert's enumerative calculus, which led to important progress in algebraic geometry and topology (intersection theory of the Grassmannians). However, Schubert calculus only yields the typical number of complex solutions. Is there a meaningful way to speak about the typical number of REAL solutions? We shall outline a way to do so, by assuming that the given objects (the four curves in the above example) are randomly rotated and to inquire about the expected number of real solutions. The approach blends ideas from real algebraic geometry with integral geometry and the theory of random polytopes. (Joint work with Antonio Lerario.)
Bartosz
Walczak
Jagiellonian University
Coloring curves that cross a fixed curve.
December 2016
Martin
Skrodzki
FU Berlin
What is formal mathematics?
Abstract.
To call something Formal Mathematics seems to be redundant. However, the field has its own merits, some of which shall be exemplified in the talk. We will take some initial steps to clarify what we understand by the notion of "proof". Having done this, we go on to formally prove a part of de Morgan's law. Furthermore, the "Donkey Sentence" will raise our awareness for difficulties in formal proofs. Finally, we will briefly get into the topic of Automated Theorem Provers and consider the famous proof of the irrationality of sqrt(2), formalized in Isabelle.
Christoph
Benzmüller
FU Berlin
Computational Metaphysics: The Virtues of Formal Computer Proofs Beyond Maths
Jaroslaw
Grytczuk
Jagiellonian University
/
Warsaw University of Technology
Majority Choosability of Digraphs.
Linda
Kleist
Equidissections of polygons
Jörg
Wolf
Recent regularity results concerning the Navier-Stokes equations
Hendrik
Schrezenmaier
Squarability of rectangle arrangements
Afshin
Goodarzi
FU Berlin
On codimension embedding of simplicial complexes
Abstract.
It is known since before 1930 that a d-dimensional simplicial complex is embeddable into the Euclidean (2d+1)-space. In his 1932 article, van Kampen showed that this result is the best possible, by presenting d-dimensional complexes that do not embed into the Euclidean (2d)-space. The major question concerning embeddings of the complexes is: 'Given a d-dimensional simplicial complex X and an integer k between d and 2d. When does X embed in the Euclidean k-space?'. The cases when k=2d or k=d+1 are probably the most intensively investigated cases: - When k=2d (and d is different from 2), the problem is decidable. Actually, there exists an algorithm based on van Kampen's ideas that solve this problem in a polynomial time. - When k=d+1 and d>3, the problem is not even algorithmically decidable. This was shown by Matousek, Tancer and Wagner in 2009 using a significant result of Novikov on sphere recognition. In this talk, which is based on a work with Anders Björner, we give a homological obstruction to codimension one embedding, i.e. k=d+1 case. Some combinatorial corollaries in low dimensions will be presented.
Gwenaël
Joret
Université Libre de Bruxelles
Orthogonal graph decomposition.
Ania
Otwinowska
U Paris-Sud, Orsay
Around standard conjectures for algebraic cycles
Abstract.
Given a (reasonable) topological space X, one can study its shape by defining a simple invariant attached to X: its cohomology. When moreover X is a complex algebraic variety (i.e. the set of zeroes of a finite collection of complex polynomials) one would like to understand the collection of its algebraic cycles, namely the formal linear combinations of its algebraic subvarieties, and their relations with the cohomology of X. In the 60's Grothendieck proposed a set of simple statements describing some naturally expected relations between algebraic cycles and cohomology: the standard conjectures. In characteristic zero they are implied by the Hodge conjecture. Voisin proved that the following conjecture is a consequence of the standard conjectures: Conjecture N (Voisin): Let X be an algebraic variety. If Z is an algebraic cycle in X whose cohomology class is supported on a closed subvariety Y, then Z is homologically equivalent to a cycle supported on Y. In the first part of this talk, I will present in an elementary way the notion of algebraic cycle, and the above conjectures. In the second part, I will explain a converse to Voisin's result: Theorem (O.) In characteristic 0, Voisin's conjecture N is equivalent to the standard conjectures.
Jörg
Wolf
Recent regularity results concerning the Navier-Stokes equations
Peter
Teichner
MPIM Bonn
Four-dimensional Manifolds revisited
Michael
Harrison
Penn State University
An h-principle for totally skew embeddings
Abstract.
An embedding f : M \to \R^n is called totally skew if for every pair of distinct points x,y \in M, the tangent spaces at f(x) and f(y) neither intersect nor contain parallel directions. Following Ghomi and Tabachnikov, we ask: Given a manifold M, what is the smallest dimension n(M) such that M admits a totally skew embedding into \R^n? The answer is known in only three cases: n(\R)=3, n(\R^2) = 6, and n(S^1)=4. In their investigation, Ghomi and Tabachnikov established a deep relationship between totally skew embeddings and the generalized vector field problem, as well as to immersions and embeddings of real projective spaces. We give a brief overview of their work and discuss our recent progress on the existence problem for totally skew embeddings, which is based on an h-principle technique due to Gromov and Eliashberg.
Veit
Wiechert
TU Berlin
On the boxicity of graphs with no K_t minor.
November 2016
Alexander
Pilz
Deciding monotonicity of complete topological graphs
Anna
von Pippich
TU Darmstadt
A Rohrlich-type formula for the hyperbolic 3-space
Abstract.
Jensens's formula is a well-known theorem of complex analysis which characterizes, for a given meromorphic function $f$, the value of the integral of $|\log(f(z))|$ along the unit circle in terms of the zeros and poles of $f$ inside this circle. An important theorem of Rohrlich generalizes Jensen's formula for modular functions $f$ with respect to the full modular group, and expresses the integral of $|\log(f(z))|$ over a fundamental domain in terms of special values of Dedekind's Delta function. In this talk, we report on a Rohrlich-type formula for the hyperbolic 3-space.
Timo
de Wolff
Texas A&M
A New Approach to Nonnegativity and Polynomial Optimization
Abstract.
Deciding nonnegativity of real polynomials is a fundamental problem in real algebraic geometry and polynomial optimization. Since this problem is NP-hard, one is interested in finding sufficient conditions (certificates) for nonnegativity, which are easier to check. Since the 19th century, the standard certificates for nonnegativity are sums of squares (SOS). In practice, SOS based semidefinite programming (SDP) is the standard method to solve polynomial optimization problems. In 2014, Iliman and I introduced an entirely new nonnegativity certificate based on sums of nonnegative circuit polynomials (SONC), which are independent of sums of squares. We successfully applied SONCs to global nonnegativity problems. Recently, Dressler, Iliman, and I proved a Positivstellensatz for SONCs, which provides a converging hierarchy of lower bounds for constrained polynomial optimization problems. These bounds can be computed efficiently via relative entropy programming. This result establishes SONCs as a promising alternative to SOS based SDP methods. In this talk I will give an overview about sums of nonnegative circuit polynomials, their relation to combinatorics and optimization, and outline my future research program.
Veit
Wiechert
On the boxicity of graphs with no K_t minor
Torsten
Muetze
TU Berlin
Trimming and gluing Gray codes.
Remke
Kloosterman
University of Padova
Monomial deformations of Delsarte Hypersurfaces and Arithmetic Mirror Symmetry
Abstract.
In a recent preprint Doran, Kelly, Salerno, Sperber, Voight and Whitcher study for five distinct quartic Delsarte surfaces a one-parameter monomial deformation. Using character sums they find a remarkable similarity between the zeta functions of general members of each of the families. In this talk we present another approach to prove these results. Moreover, we give a necessary and sufficient combinational criterion to check for a pair of monomial deformations of Delsarte hypersurfaces whether their zeta functions are essentially the same or not.
Jens
Griepentrog
Solution regularity of parabolic variational inequalities
Martin
Skutella
A 2.542-Approximation for Precedence Constrained Single Machine Scheduling with Release Dates and Total Weighted Completion Time Objective
Stefan
Felsner
Circle- and Pseudocirclearrangements
Rainer
Sinn
Geometry of Sums of Squares
Abstract.
I will present recent results on the geometry of sums of squares on projective algebraic sets. A central theme will be the ranks of extreme points of spectrahedra, which are affine linear slices of the cone of positive semidefinite symmetric matrices. These spectrahedra arise naturally in real algebraic geometry in the context of sums of squares. Of special interest will be projective varieties of small degree and subspace arrangements. We will see that the latter case translates into the positive semidefinite matrix completion problem.
Tamás
Mészáros
FU Berlin
Some combinatorial applications of Gröbner bases and standard monomials.
Erik
Visse
University of Leiden
A heuristic for a Manin-type conjecture for K3 surfaces
Abstract.
In the late 1980's Manin came up with a conjecture for the growth of rational points of bounded height of Fano varieties; he predicted what the number of rational points on a suitable open subset of any given such variety should be asymptotically. Since Manin's original paper, many specific cases have been studied giving rise to refinements, proofs, upper and lower bounds, counterexamples and proposed fixes; all still concerning Fano varieties. In my PhD project, I study the same problem for the "next case" in dimension 2: K3 surfaces. Recently I was able to compute heuristics for certain diagonal quartic surfaces that agree with some numerical experiments that were done by my supervisor Ronald van Luijk a few years ago. In the talk I will explain the techniques involved and some problems that need to be overcome in order to formulate a reasonable conjecture.
Antareep
Mandal
HU Berlin
What is a modular form?
Abstract.
We define one of the most important invariants for a lattice in the Euclidean space called its theta series and show that it belongs to an interesting class of functions called modular forms, whose symmetry and nice properties makes it possible to prove strong statements about them.
Maryna
Viazovska
HU Berlin
Solving packing problems by linear programming
Darius
Wuttke
Efficient algorithm for computing middle level Gray codes
Antareep
Mandal
HU Berlin
Uniform sup-norm bound for the average over an orthonormal basis of Siegel cusp forms
Abstract.
This talk updates our progress towards obtaining the optimal sup-norm bound for the average over an orthonormal basis of Siegel cusp forms by relating it to the discrete spectrum of the heat kernel of the Siegel-Maass Laplacian. In particular, we study Maass operators and generalize the relation between the cusp forms and the Maass forms in higher dimensions along with presenting an interesting trick to obtain the heat kernel corresponding to the Maass Laplacians by adapting the 'method of images' used to obtain the heat kernel corresponding to the Laplace-Beltrami operator.
October 2016
Günter
Rote
Selecting K points with the largest convex hull area
Abstract.
We are given a set S of n points in three dimensions and an integer K. We are looking for a subset of at most K points whose convex hull has the largest possible volume. We show that this problem is W[1]-hard, by reduction from the so-called Grid Tiling problem. This implies, under the standard complexity-theoretic assumption that W[1] is not equal to FPT, that the problem cannot be solved in f(K)n^C time, for any function f and any constant C. In other words, the problem is not fixed-parameter tractable. For the related problem of selecting K points that maximize the dominated volume in the nonnegative orthant, we could show NP-hardness by a reduction from independent set in planar graphs of maximum degree 3. The problems are motivated by the task of reducing a large set of options to a small "representative" set. These results are joint work with Kevin Buchin, Karl Bringmann, Sergio Cabello, and Michael Emmerich.
Tibor
Szabó
FU Berlin
Vertex Folkman Numbers.
Stanley
Schade
FU Berlin
What is mixed integer linear programming?
Abstract.
Defining mixed integer linear programming (MILP) is easy, but not very instructive. What we want to understand is how we can approach MILP problems in practice and why they are so relevant. To do so, we will make a quick tour through the field of combinatorial optimization. We will explore some of its problems, mathematical concepts and tools and also gain a few geometric insights on MILP.
Alexander
Martin
FAU Erlangen-Nürnberg
Combinatorial optimization problems with physical constraints
Piotr
Micek
FU Berlin
Weak coloring numbers and poset's dimension.
Linda
Kleist
Popular matchings
Veit
Wiechert
Coloring, sparseness, and girth
September 2016
Christian
Krattenthaler
Uni Wien
Robinson-Schensted-Knuth correspondence, jeu de taquin, and growth diagrams
Abstract.
I shall review the classical Robinson-Schensted correspondence - a classical bijection between permutations and pairs of standard Young tableaux of the same shape - and then explain that there is a modern, 'better' way the present the same bijection in terms of Sergey Fomin's growth diagrams. These ideas are then put into action by establishing a recent conjecture of Sophie Burrill on a certain bijective relation between standard tableaux and oscillating tableaux.
August 2016
Hendrik
Schrezenmaier
Maximizing the Sum of Radii of Disjoint Balls or Disks
Stefan
Felsner
Coloring Segments in Space
July 2016
Sebastian
Hensel
An L^
infty estimate for local suitable weak solutions of the Navier-Stokes equations
Markus
von der Heyde
Rechtecks-Duale mit vorgegebenen Flächen
Aleksei
Shcherbina
Dissipative 2D magnetic Zakharov system in bounded domain
Tamara
Fastovska
Global attractor for a.radially symmetric fluid-structure interaction model in circular cylinders
Moritz
Schmitt
On the Number of Facets of a Monohedral Tile
Abstract.
A family of proper convex bodies is a (convex) tiling if the bodies cover space completely and have disjoint interiors. The tiling is monohedral if the bodies are pairwise congruent. Every body in such a tiling has to be polytopal, and determining the least upper bound on the number of facets is a major open problem. We give an overview of known upper bounds for special classes of monohedral tilings, survey a few classical results, and present recent constructions of lower bounds.
Boris
Bukh
Carnegie Mellon University
One-sided epsilon-approximants.
Abstract.
Two common approximation notions in discrete geometry are ε-nets and ε-approximants. Of the two, ε-approximants are stronger. For the family of convex sets, small ε-nets exist while small ε-approximants unfortunately do not. In this talk, we introduce a new notion "one-sided ε-approximants", which is of intermediate strength, and prove that small one-sided ε-approximants do exist. The proof is based on a (modification of) the regularity lemma for words by Axaenovich--Person--Puzynina. Joint work with Gabriel Nivasch.
Martin
Westerholt-Raum
Chalmers University of Technology
Automorphic constituents of tensor products of Harish-Chandra modules
Abstract.
Products of real-analytic automorphic forms in general generate more than one automorphic representation. We study this phenomenon at the infinite place for scalar valued Siegel modular forms of genus 2. It turns out that automorphic constituents of the specific tensor products that we inspect are all holomorphic (limits) of discrete series. This has applications to Poincaré series and harmonic weak Siegel Maaß forms.
Philipp
Arnold
A system of waves in two media separated by a vibrating membrane
Albert
Haase
FU Berlin
What is sphere packing?
Abstract.
We will discuss what a (lattice) sphere packing is and how its density can be defined. Then, we will look at examples in dimensions 2, 3, and perhaps 4. We will also introduce the related kissing number problem. Finally, we will survey known results and look at a rather puzzling plot of the densities — by dimension — of the densest sphere packings known. If we have time, we can discuss the two lattices (E_8 and Leech lattice) that lead to the packings in dimensions 8 and 24 that were recently shown to have highest possible density.
Raman
Sanyal
Two double poset polytopes
Henry
Cohn
Mysteries of sphere packing in high dimensions
Raman
Sanyal
FU Berlin
Tight upper bounds on colorful simplicial depth and Minkowski sums
Abstract.
The colorful simplicial depth of a collection of d+1 finite sets of points in Euclidean d-space is the number of choices of a point from each set such that the origin is contained in their convex hull. This is an extension of the notion of 'simplicial depth' to the colorful situation. If the origin is contained in all color classes, then Barany's beautiful colored version of Carathéodory's theorem asserts that the colorful depth is at least 1. Deza, Huang, Stephen, Terlaky (2006) conjectured lower and upper bounds on the colorful depth and, after many partial results, the lower bound conjecture was confirmed by Sarrabezolles (2015). Recently, we managed to prove the conjectured upper bound. For that we cast the problem into one of combinatorial topology and the goal of the talk is to give a sketch of our proof. We also discovered an interesting connection between colorful configurations and Minkowski sums of polytopes. If time permits, I will give the idea and show how this resolves a conjecture of Burton (2003) in the theory of normal surfaces. This is joint work with Adiprasito, Brinkmann, Padrol, Patak, and Patakova.
Anton
Dochtermann
U Texas
Syzygies of generalized permutohedra, with combinatorial applications
Abstract.
The lattice points of a generalized permutohedra P, when viewed as a Minkowski sum/difference of simplices, naturally define a monomial ideal I_P. This class of ideals includes matroidal ideals and certain artinian ideals generalizing powers of the maximal ideal. We employ techniques from tropical geometry to show that if P is a positive sum of simplices, then any regular fine mixed subdivision of P supports a minimal resolution of I_P. This basic observation leads to applications in a variety of contexts, including ladder determinantal ideals, h-vectors of cographical matroids, and chip-firing on graphs. For the case of arbitrary P we obtain nonminimal resolutions in certain cases (eg for matroid polytopes). We'll survey some of these connections, with an emphasis on the combinatorial/geometric aspects. Joint work, some in progress, with Raman Sanyal and Alex Fink.
Anita
Liebenau
Monash University
Ramsey equivalence of $K_n$ and $K_n + K_{n−1}$.
Abstract.
Szab\’o, Zumstein, and Z\”urcher proved in 2010 that, for $n\geq 4$, the complete graph $K_n$ is Ramsey equivalent to $K_n+K_{n-2}$, the graph formed by taking the disjoint union of a copy of $K_n$ and a copy of $K_{n-2}$. In fact, they proved that one can add {\em many} disjoint copies of $K_{n-2}$ to $K_n$, and the resulting graph is still Ramsey equivalent to $K_n$. The situation is quite different for $K_{n-1}$: Fox, Grinshpun, Liebenau, Person, and Szab\’o proved in 2014 that $K_n$ is {\em not} Ramsey equivalent to the graph $K_n + 2K_{n-1}$, the graph containing a copy of $K_n$ and two copies of $K_{n-1}$. Nevertheless, it was conjectured in both papers that $K_n$ and $K_n + K_{n−1}$ are Ramsey equivalent for $n\geq 4$. We prove this conjecture. This is joint work with Thomas Bloom. .
Ana Maria
Botero
HU Berlin
Intersection theory of b-divisors on toric varieties
Abstract.
We introduce toric b-divisors on complete smooth toric varieties and a notion of integrability of such divisors. We show that under some positivity assumptions, toric b-divisors are integrable and that their degree is given as the volume of a convex set. Moreover, we show that the dimension of the space of global sections of a nef toric b-divisor corresponds to the number of lattice points in this convex set and we give a Hilbert-Samuel type formula for its asymptotic growth. This generalizes classical results for classical toric divisors on toric varieties. We further investigate the question of extending these results to arbitrary toroidal varieties. Examples in which such b-divisors naturally appear are invariant metrics on line bundles over toroidal compactifications of mixed Shimura varieties. Indeed, the singularity type which the metric acquires along the boundary can be encoded using toroidal b-divisors.
Viktoria
Filatova
Numerical modeling of dynamic inverse problems in acoustics by the BC - method
June 2016
Benny
Sudakov
ETH Zürich
Equiangular lines and spherical codes in Euclidean spaces.
Abstract.
A set of lines in R^d is called equiangular if the angles between any two of them are the same. The problem of estimating the size of the maximum family of equiangular lines has had a long history since late 1940's. A closely related notion is that of a spherical code, which is a collection C of unit vectors in Rd such that x \cdot y \in L for any distinct x,y in C and some set of real numbers L. Spherical codes have been extensively studied since their introduction in the 1970's by Delsarte, Goethals and Seidel. Despite a lot of attention in the last forty years, there are still many open interesting questions about equiangular lines and spherical codes. In this talk we report recent progress on some of them. Joint work with I. Balla, F. Drexler and P. Keevash.
Franz
Brandenburg
Visibility Graphs of Geometric Objects
Spencer
Backman
Hausdorff Center Bonn
Riemann-Roch theory for graph orientations and the Tutte polynomial
Abstract.
Chip-firing is a certain simple game played on the vertices of a graph. In 2007, Baker and Norine proved that by utilizing chip-firing, one may prove a Riemann-Roch formula for graphs analogous to the classical statement for algebraic curves. I will describe a certain equivalence relation on partial graph orientations which captures chip-firing and explain how this setup allows for a more conceptual combinatorial understanding of Baker and Norine's result. I will then outline how such considerations of partial graph orientations led myself, Sam Hopkins, and Lorenzo Traldi to a new 12 variable expansion of the Tutte polynomial of a graph. Time permitting, I will describe some applications of this new Tutte expansion in algebraic and geometric combinatorics.
Dániel
Korándi
ETH Zürich
Saturation in random graphs.
Abstract.
A graph H is K_s-saturated if it is a maximal K_s-free graph, i.e., H contains no clique on s vertices, but the addition of any missing edge creates one. The minimum number of edges in a K_s-saturated graph was determined over 50 years ago by Zykov and independently by Erdős, Hajnal and Moon. In this talk, we consider the random analog of this problem: minimizing the number of edges in a maximal K_s-free subgraph of the Erdős-Rényi random graph G(n,p). We give asymptotically tight estimates on this minimum, and also provide exact bounds for the related notion of weak saturation in random graphs. Joint work with Benny Sudakov.
Fabian
Völz
TU Darmstadt
Elliptic and hyperbolic Eisenstein series as theta lifts
Abstract.
Generalising the concept of classical non-holomorphic Eisenstein series associated to cusps, one can define elliptic Eisenstein series associated to points in the upper-half plane, and hyperbolic Eisenstein series associated to geodesics. In my talk I will show that averaged versions of these elliptic and hyperbolic Eisenstein series can be obtained as theta lifts of signature (2,1) of some weighted Poincaré series, which generalizes a classical result in the parabolic case. Moreover, I will show how to realize a distinguished elliptic Eisenstein series as a theta lift of signature (2,2). Finally, if time permits, I will propose some applications of these results.
Loreen
Gräber
Ein hyperbolisches Randwertproblem und die optimale Steuerung eines elektromagnetischen Feldes
Veit
Wiechert
Dominating Sets in Sparse Graphs
Kathlén
Kohn
TU Berlin
What is the eigenvector of a tensor?
Abstract.
This talk gives an introduction to tensors and their eigenvectors. It is especially addressed to people that are not algebraists. We will see that tensors and their eigenvectors appear naturally when using Lagrange-multipliers in optimization and when studying higher order Markov chains in stochastics.
Bernd
Sturmfels
Tensors and their eigenvectors
Arnau
Padrol
Tropical Catalan Subdivisions
Abstract.
We revisit the associahedral subdivision of the Pitman-Stanley polytope to provide geometric realizations of the v-Tamari lattice of Préville-Ratelle and Viennot as the dual of a triangulation of a polytope, as the dual of a mixed subdivision, and as the edge-graph of a polyhedral complex induced by a tropical hyperplane arrangement. The method generalizes to type B. This is joint work with Cesar Ceballos and Camilo Sarmiento.
Jürg
Kramer
HU Berlin
On the volume of the Siegel modular variety
Abstract.
In our talk we provide a short proof of Siegel's formula for the volume of the Siegel modular variety. The proof makes essential use of the Klingen Eisenstein series and is based on a geometric induction argument.
Linda
Kleist
Union-closed set conjecture
Bernd
Sturmfels
UC Berkeley/TU Berlin
Convexity in Tree Spaces
Abstract.
We study the geometry of metrics and convexity structures on the space of phylogenetic trees, here realized as the tropical linear space of all ultrametrics. The CAT(0)-metric of Billera-Holmes-Vogtman arises from the theory of orthant spaces. While its geodesics can be computed by the Owen-Provan algorithm, geodesic triangles are complicated and can have arbitrarily high dimension. Tropical convexity and the tropical metric are better behaved, as they exhibit properties that are desirable for geometric statistics.
Matthias
Eller
Photoakustische Tomographie als inverses Problem für die Wellengleichung
Hendrik
Schrezenmaier
How to morph planar graph drawings
Tobias
Friedl
FU Berlin
Two Twisted Poset Polytopes
Abstract.
In 1986, Stanley introduced two polytopes associated with a finite poset P: The order polytope O(P) and the chain polytope C(P). Many geometric properties of these polytopes can be described only in terms of the underlying poset P. Moreover, there exists a continuous, piecewise linear bijection between O(P) and C(P), yielding that the volume (and even the Ehrhart polynomial) of the two polytopes is the same. In this talk we will generalize these results to the case of twisted prisms over order and chain polytopes. This is joint work with T. Chappell and R. Sanyal.
Christian
Reiher
University of Hamburg
A Ramsey Class for Steiner Systems.
Abstract.
We construct a Ramsey class whose objects are Steiner systems. In contrast to the situation with general $r$-uniform hypergraphs, it turns out that simply putting linear orders on their sets of vertices is not enough for this purpose: we also have to strengthen the notion of subobjects used from ``induced subsystems'' to something we call ``strongly induced subsystems''. Moreover we study the Ramsey properties of other classes of Steiner systems obtained from this class by either forgetting the order or by working with the usual notion of subsystems. This leads to an interesting induced Ramsey theorem in which {\it designs} get coloured. This is joint work with Vindya Bhat, Jaroslav Ne\v{s}et\v{r}il, and Vojt\v{e}ch R\"{o}dl.
May 2016
Fabien
Pazuki
University of Copenhagen
Bad reduction of curves with CM jacobians
Abstract.
An abelian variety defined over a number field and having complex multiplication (CM) has potentially good reduction everywhere. If a curve of positive genus which is defined over a number field has good reduction at a given finite place, then so does its jacobian variety. However, the converse statement is false already in the genus 2 case, as can be seen in the entry $[I_0-I_0-m]$ in Namikawa and Ueno's classification table of fibres in pencils of curves of genus 2. In this joint work with Philipp Habegger, our main result states that this phenomenon prevails for certain families of curves. We prove the following result: Let F be a real quadratic number field. Up to isomorphisms there are only finitely many curves C of genus 2 defined over $\overline{\mathbb{Q}}$ with good reduction everywhere and such that the jacobian Jac(C) has CM by the maximal order of a quartic, cyclic, totally imaginary number field containing F. Hence such a curve will almost always have stable bad reduction at some prime whereas its jacobian has good reduction everywhere. A remark is that one can exhibit an infinite family of genus 2 curves with CM jacobian such that the endomorphism ring is the ring of algebraic integers in a cyclic extension of $\mathbb{Q}$ of degree 4 that contains $\mathbb{Q}(\sqrt{5})$
Jörg
Wolf
Über eine erweiterte Hole-Filling-Methode mit Anwendung auf das stationäre Hall System
Yuri
Manin
MPIM Bonn
Time between real and imaginary: Big Bang and modular curves
Albert
Haase
FU
Open questions related to Tverberg's Theorem
Abstract.
Frick's counterexamples to the topological Tverberg conjecture in 2015 settled the main open question regarding Tverberg’s theorem, namely that Tverberg’s theorem cannot be extended to the non-prime-power case. The constraint method developed by Blagojevic, Ziegler, and Frick shows how Tverberg’s theorem and essentially the pigeonhole principle directly imply and, in some cases, strengthen many of the results related to Tverberg’s theorem that were published since the 1980s. So what – if any – are the remaining open questions related to Tverberg’s theorem? In my talk I will touch on one of most prominent open questions and tell you what progress has been made. This is ongoing work with Blagojević, Engström, and Ziegler.
Hao
Chen
TU Eindhoven
Selectively Balancing Unit Vectors
Abstract.
A set of unit vectors in R^n is selectively balancing if there is a non-zero linear combination with coefficients -1, 0 or 1 whose norm is smaller than 1. We prove that O(n log n) unit vectors suffice to ensure a selectively balancing set, and this estimate is asymptotically best possible. This result finds application in dot product representation of cubes. This is a joint work with Blokhuis.
Barbara
Jung
HU Berlin
The arithmetic volume of the moduli stack A2
Abstract.
The arithmetic volume of the (compactified) moduli stack An/Z of principally polarized n-dimensional abelian varieties is given by the arithmetic self intersection number of the bundle of Siegel modular forms on An, metrized by the Petersson norm. A generalized intersection theory applicable for this case was developed by Burgos, Kramer and Kühn in 2005. It is conjectured that the above intersection number consists of a sum of special values of the logarithmic derivative of the zeta function. We will present a way to compute the volume of A2, using results from Kudla and Kühn, and discuss how to handle the boundary.
Jörg
Wolf
Über eine erweiterte Hole-Filling-Methode mit Anwendung auf das stationäre Hall System
Raphael
Steiner
Existenz und Konstruktion von Dreiecks- und Fünfeckskontaktdarstellungen
Maren
Ring
Rostock
A geometric local formula for Ehrhart coefficients
Abstract.
Let P be a lattice polytope in R^d and tP its dilation by a positive integer t. In dimension 2, Pick's Theorem (1899) leads to a connection between the number of lattice points in tP and its volume: #(lattice points in tP) = vol(P) t^2 + (B/2) t + 1, where B is the number of lattice points on the boundary of P. Equivalently, B can be seen as the sum of relative volumes of the edges of P. The higher dimensional generalization of this function is the Ehrhart polynomial. As in the 2-dimensional case, the coefficients of Ehrhart polynomials can be determined via so-called local formulas, by weighting the relative volumes of the faces of P. We develop a construction of these weights based on an elementary geometric idea.
Tuan
Tran
Czech Academy of Sciences
Structure and supersturation for intersecting families.
Abstract.
A k-uniform family of subsets of [n] is called intersecting if it does not contain a pair of disjoint sets. The celebrated Erdos-Ko-Rado Theorem from 1961 asserts that, provided n>2k, any such system has size at most {n-1 \choose k-1}. A natural question is how many disjoint pairs must appear in a set system of larger size. In this paper, we determine the minimum number of disjoint pairs in small k-uniform families for k=o(\sqrt{n}), thus extending a result of Das, Gan and Sudakov (2016). Our main tool is a removal lemma for families with few disjoint pairs. As another application of the removal lemma, we show that almost all k-uniform intersecting families on [n], with n>(2+o(1))k, are trivial, i.e. all k-sets share a common element. This is based on my joint work with Balogh, Das, Liu and Sharifzadeh
Julia
Kraus
Magische Eigenschaften von Graphen
Anton
Mellit
SISSA Trieste
Mixed Hodge structures of character varieties.
Abstract.
The conjecture of Hausel, Letellier and Villegas gives precise predictions for mixed Hodge polynomials of character varieties. In certain specializations this conjecture also computes Hurwitz numbers, Kac's polynomials of quiver varieties, and zeta functions of moduli spaces of Higgs bundles. I will formulate the conjecture, give some examples, and talk about my proof of polynomiality of the generating functions that arise there.
Frank
Neumann
Packing While Traveling: Mixed Integer Programming for a Class of Nonlinear Knapsack Problems
Fei
Ren
FU Berlin
What is a Chow Ring?
Abstract.
In algebraic geometry, the Chow ring (named after W. L. Chow by Chevalley (1958)) of a smooth algebraic variety over a field is an algebro-geometric analogue of the cohomology ring of a complex variety considered as a topological space. The elements of the Chow ring are formed out of actual subvarieties (so-called algebraic cycles), and the multiplicative structure is derived from the intersection of subvarieties. In this talk, we will define what is a Chow ring, introduce basic properties and see a few examples if time permits.
Stefan
Felsner
On the Erdős-Szekeres convex polygon problem
Enrico
Arbarello
Enumerative geometry, intersection theory and moduli spaces
Tibor
Szabó
FU Berlin
On the complexity of Ryser's Conjecture.
Abstract.
Every r-uniform hypergraph has a vertex cover of size r times its matching number. Ryser's Conjecture states that if the hypergraph is also r-partite, then for a cover it is enough to take (r-1)-times the matching number many vertices. For r=2 Ryser's Conjecture reduces to Konig's Theorem, for r=3 it was proved by Aharoni by topolgical methods, and for larger r it is wide open. In a recent joint work with Abu-Khazneh, Barat, and Pokrovskiy, we argue that Ryser's Conjecture, if true, is probably difficult.
Maryna
Viazovska
HU Berlin
Modular forms and sphere packing
Abstract.
In this talk we will report on our recent result on the sphere packing problem in dimensions 8 and 24. We will explain the linear programming method for sphere packing introduced by N. Elkies and H. Cohn. Also we will present the construction of certificate functions providing the optimal estimate for the sphere packing problem in dimensions 8 and 24.
Joachim
Naumann
Über das Thermistor-Problem
Katherine
Staden
University of Warwick
The Erdős-Rothschild problem on edge-colourings.
Abstract.
Let $\bm{k} = (k_1,\ldots,k_s)$ be an $s$-tuple of positive integers. Given a graph $G$, how many ways are there to colour the edges of $G$ with $s$ colours so that there is no $c$-coloured copy of the complete graph on $k_c$ vertices, for any $c=1,\ldots,s$? Write $F(G;\bm{k})$ for this quantity and let $F(n;\bm{k})$ be its maximum over all graphs $G$ on $n$ vertices. What is $F(n;\bm{k})$ and which graphs $G$ attain this maximum? This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it has only been solved for a very small number of non-trivial cases. In this talk I will survey the history of the problem, and will discuss some recent general results with Oleg Pikhurko (Warwick) and Zelealem Yilma (Carnegie Mellon Qatar).
Veit
Wiechert
Centered colorings and order dimension
Miriam
Schlöter
Quickest Transshipments & Submodular Function Minimization
Joachim
Naumann
Über das Thermistor-Problem
April 2016
Mara
Ungureanu
HU Berlin
What is symplectic geometry?
Abstract.
Symplectic geometry is an even dimensional geometry that lives on even dimensional spaces and measures two dimensional quantities rather than the one dimensional quantities like lengths and angles familiar from Riemannian geometry. Moreover, symplectic geometry displays an intriguing interplay between rigidity and flabbiness, which makes the question of constructing invariants that distinguish between between different symplectic structures especially interesting. In this talk we shall explore some of these aspects and motivate the definition of a certain symplectic invariant, namely the symplectic capacity.
Stefanie
Wendisch
q-Catalan Zahlen
Shiri
Artstein-Avidan
U Tel Aviv
Billiards in convex domains
Emanuele
Ventura
Aalto University
Real rank geometry of ternary forms
Abstract.
The problem of expressing a homogeneous polynomial as a sum of powers of linear forms is very classical and goes back to the work of Sylvester, Hilbert, and Scorza among others. The real rank of a homogeneous polynomial is the smallest number of linear real forms such that the polynomial admits such a representation. The space parametrizing all real decompositions of a polynomial as a minimal sum of powers is a semialgebraic set sitting inside the classical Varieties of Sums of Powers. For plane cubics, we find a connection with oriented matroids. This is a joint work with M. Michalek, H. Moon and B. Sturmfels.
Dhruv
Mubayi
UI Chicago
Recent advances using the stepping up technique.
Abstract.
I will give some details of recent constructions in hypergraph Ramsey theory that settle longstanding problems in the field. These are obtained by adding some new ingredients to the stepping up method of Erdos and Hajnal that first appeared in 1965. Most of this is joint work with Andrew Suk.
D.
Chae
On the Hall-MHD equations
Stefan
Felsner
A Geometric Approach to Acyclic Orientations
Christian
Stump
Algebraic combinatorics of reflection arrangements and generalizations
Abstract.
Reflection arrangements have a long history in algebraic combinatorics. Unfortunately, their friends, the Catalan and Shi arrangements, have so far resisted from being generalized beyond Weyl types, and indeed several authors showed that such arrangements cannot be as beautiful for the reflection group of type H4. I start with recalling a few fundamental discrete-geometric properties of (reflection) arrangements. I then discuss surprising numerical observations in the module of derivations of a complex reflection arrangement that indicate that we still miss the "correct" combinatorics to study Catalan and Shi arrangements for general reflection groups. This talk is based on results in an ongoing collaboration with Torsten Hoge and Gerhard Röhrle.
Felix
Fischer
Truthful Outcomes from Non-Truthful Position Auctions
Linda
Kleist
Decomposing graphs into trees
Philip
Brinkmann
FU Berlin
Enumeration of 3-Spheres
Abstract.
In 1906 Steinitz gave a complete characterisation of the set of f-vectors of 3-polytopes. Since then people are trying to find a characterisation for the 4-dimensional case. However, there is still no such description that is complete. There is also no guess for a description of the set of f-vectors of 3-spheres (which have the boundary complexes of 4-polytopes as special cases), nor did we know up to now whether the set of f-vectors of 3-spheres is the same or strictly larger than the one of 4-polytopes - although most spheres are not polytopal (i.e. can be realised as the boundary complex of a convex polytope). In this talk I will present and explain an algorithm to enumerate 3-spheres for a given f-vector (f_0,f_1,f_2,f_3). Furthermore, I will present some of the enumeration results that finally prove that the set of f-vectors of 3-spheres is strictly larger than the set of f-vectors of 4-polytopes. This is joint work with Günter M. Ziegler.
Sven
Jäger
Iterative Algorithms for Integrated Optimization Problems
March 2016
Jannik
Matuschke
Recent developments in robust network flows
Veit
Wiechert
On Graphs that have the Erdos-Posa Property
Lena
Krauss
De Bruijn Graphen mit Eulerkreisen als Ansatz für DNA Fragment Assembly
February 2016
Stefan
Felsner
Embedding Graphs on Wheel Point Sets.
Katharina
Hoffmann
k-lokal planare Graphen
Codruţ
Grosu
FU Berlin
A lower bound for the Towers of Hanoi game with more pegs.
Abstract.
The Towers of Hanoi puzzle is played with N disks and 3 pegs. The disks are initially placed on one peg in increasing order according to size, and the goal is to move all the disks on another peg, without ever placing a larger disk on a smaller one. The solution is simple and uses a recursive algorithm for moving the disks. A well-known generalization asks for the minimum number of moves when p pegs are available. This problem is much more difficult and only recently the case p = 4 was solved by Bousch. The purpose of my talk is to present a lower bound on the number of moves needed for p ≥ 5 pegs and N disks. This lower bound is asymptotically better than the previously best known when p is fixed and N tends to infinity.
Irem
Portakal
FU Berlin
What is a rational function?
Abstract.
In this talk, we will investigate functions on affine space over a field. In the projective case, we naturally come across the notion of rational functions. More concretely, when we change the field to the complex numbers, for the projective space, one gets an identification of the field of rational functions with the field of meromorphic functions, which is a consequence of Chow's Theorem. We will prove this for the projective line.
Miriam
Schröder
TU Berlin
What is the disjoint
paths problem?
Abstract.
Imagine you are an electrical engineer and responsible for the power supply of a town. By building a large network of transmission towers you connected the town to the nearest power plant. Now your boss asks you to prove to him that the power supply of the town is still ensured if k transmission towers fail. How can you convince your boss that the network you built is failsafe? Of course, you can simply switch out every possible k-subset of transmission towers and check whether the city is still on power supply after that. Unfortunately, you built a large network, so this approach would take a long time. A good solution in this situaion would be to use Menger's theorem which provides us with a fast to check certificate that ensures that the network is failsafe. In this talk I will introduce Menger's Theorem and the related disjoint paths problem in different variants.
Lex
Schrijver
Horst-Wagon Saal H1012
The partially disjoint paths problem
Sören
Berg
TU Berlin
Tensor-valued Ehrhart theory
Abstract.
For a lattice polytope we can define the so-called discrete moment tensor of a certain rank. In the particular cases where this rank is zero or one this yields the number of lattice points or the sum of all lattice points contained in the polytope, respectively. Based on the work by Khovanskii and Pukhlikov the discrete moment tensor of a dilate of a lattice polytope by an integral factor is a polynomial in that dilation factor. For the discrete moment tensor of rank zero this collapses to the well-known Ehrhart theory. In this talk we want to discuss a few classical results from Ehrhart theory which extend naturally for discrete moment tensors of higher rank and some which do not. Starting with simple observations on some coefficients of the underlying polynomials we will carry on with the h-star polynomials arising from the corresponding generating functions. For these h-star polynomials having tensor-valued coefficients we will elaborate basic properties and introduce a notion of positivity and monotonicity. This is ongoing joint work with Katharina Jochemko and Laura Silverstein from TU Vienna.
Tamás
Mészáros
FU Berlin
A conjecture on shattering-extremal set systems
Abstract.
We say that a set system F ⊆ 2^[n] shatters a given set s ⊆ [n] if F|s = {f ∩ s : f ∈ F} = 2^s . One related notion is the VC-dimension of a set system: the size of the largest set shattered by F. The Sauer inequality states that in general, a set system F shatters at least |F| sets. A set system is called shattering-extremal if it shatters exactly |F| sets. Such families have many interesting features. Here we present several approaches to study shattering-extremal set systems together with a conjecture about the eliminability of elements from extremal families.
Josué
Tonelli Cueto
HU Berlin
What is an origami
constructible number?
Abstract.
When the people of the Greek city of Delos asked the Delphic oracle how to stop a plague that Apollo sent to them, the oracle answered that they should duplicate the cubic altar of this god. Although the efforts of the great mathematicians of the day, none of them was able to double the cube through straightedge and compass (as the fashion of the day requested in geometry). As we know nowadays, this latter task is impossible; however, if we allow ourselves to use origami, the duplication of the cube and other classical geometric problems not solvable by straightedge and compass can be solved. In the talk, we will explore the constructions that the use of origami permits in classical geometry.
Hendrik
Schrezenmaier
Ein neuer Existenzbeweis für Kontaktdarstellungen mit homothetischen gleichseitigen Dreiecken.
Christoph
Spiegel
FU Berlin
Van der Waerden Games - Part II
Irina
Kmit
Exponential dichotomy for hyperbolic PDEs
January 2016
Stefan
Rüdrich
FU Berlin
What is a stochastic
process?
Abstract.
Stochastic processes are families of random variables whose trajectories may differ with each realization, unlike deterministic processes, yet allow for analysis and simulation of dynamical systems, conserving some non-probabilistic quantities. This talk will be a rough introduction to some key concepts of stochastic processes, e.g. random walks, Markovianity and metastable sets. The latter typically correspond to almost independent substructures of the state space, such as functional subunits of a cellular network or stable conformations of molecules.
Christof
Schütte
ZIB
Model reduction for stochastic processes exhibiting multiple scales
Jürgen
Richter-Gebert
TU München
The Quest for Nice (Bracket) Formulas
Abstract.
The talk deals with the search for nice symmetric formulas for nice symmetric problems in Invariant theory. After a general introduction that highlights some formulas for geometric primitive operations. Two specific (more advanced) scenarios are addressed; both of them are related to the theory of algebraic curves in the plane. The first one deals with a problem related to the Cayley Bacharach Theorem: Calculate the ninth point of intersection of a pair of cubics if eight intersections are given. Surprisingly this point is uniquely determined and (as a consequence of the First Fundamental Theorem of Invariant Theory) should have a representation as a bracket expression free of polynomial roots. Two such formulas are presented. The second problem concerns the quest for a nice, short and symmetric bracket formula that that expresses the fact that 10 points in the projective plane lie on a common cubic. A nice formula and a short formula will be presented. Unfortunately so far we do not know about a nice and short formula.
Christoph
Spiegel
FU Berlin
Van der Waerden Games
Veit
Wiechert
Improved bounds for the dimension of posets whose cover graphs have bounded treewidth.
Irina
Kmit
Exponential dichotomy for hyperbolic PDEs
Tom
Trotter
Posets and Dimension
Giulia
Codenotti
FU Berlin
Kruskal-Katona and Macauley theorems in a relative setting
Abstract.
A relative simplicial complex is a family of sets which is the set-theoretic difference D\G of a simplicial complex D and a subcomplex G of D. Relative simplicial complexes were introduced by Reisner and Stanley, and in a recent paper of Adiprasito and Sanyal, properties of relative simplicial complexes play a key role in solving the upper bound problem for Minkowski sums of polytopes, which asks for an upper bound on the number of k-faces of a Minkowski sum of polytopes. This has raised interest in better understanding which properties of simplicial complexes can be "relativized". One fundamental result about simplicial complexes is the Kruskal-Katona theorem, which characterizes f-vectors of simplicial complexes. I will show a simple extension of the Kruskal-Katona theorem to relative simplicial complexes, and an analogue extension for Macauley theorem [which can be interpreted as an analogue of the Kruskal-Katona theorem for multicomplexes]. This is joint work with Raman Sanyal.
Claudius
Heyer
HU Berlin
What is a $p-$adic
number?
Abstract.
$p-$adic numbers are an integral part of algebraic number theory. In this talk we will define the field $Q_p$ of $p-$adic numbers and observe some differences and similarities between $Q_p$ and the field of real numbers. Furthermore, we will state Hensel's Lemma and deduce some immediate, yet interesting, properties of $Q_p$.
Philippe
Michel
EPFL
Applied l-adic Cohomology
Torsten
Muetze
TU Berlin
Bipartite Kneser graphs are Hamiltonian
Abstract.
For integers k>=1 and n>=2k+1, the bipartite Kneser graph H(n,k) is defined as the graph that has as vertices all k-element and all (n-k)-element subsets of {1,2,...,n}, with an edge between any two vertices (=sets) where one is a subset of the other. It has long been conjectured that all bipartite Kneser graphs have a Hamilton cycle. The special case of this conjecture concerning the Hamiltonicity of the graph H(2k+1,k) became known as the 'middle levels conjecture' or 'revolving door conjecture', and has attracted particular attention over the last 30 years. One of the motivations for tackling these problems is an even more general conjecture due to Lovasz, which asserts that in fact every connected vertex-transitive graph (as e.g. H(n,k)) has a Hamilton cycle (apart from five exceptional graphs). In this talk I present a simple and short induction proof that all bipartite Kneser graphs H(n,k) have a Hamilton cycle (assuming that H(2k+1,k) has one). This is joint work with Pascal Su (ETH Zurich).
Lutz
Recke
1d hyperbolic systems and integration along characteristics
Linda
Kleist
Coloring Graphs on Surfaces.
Ivan
Izmestiev
U Fribourg
Connecting balanced triangulations by cross-flips
Abstract.
A d-dimensional simplicial complex is called balanced, if its vertices can be properly colored in d+1 colors. In a joint work with Isabella Novik and Steve Klee we introduced a notion of cross-flips: certain moves that transform a balanced simplicial complex into another balanced simplicial complex. These moves form a natural balanced analog of bistellar flips (also known as Pachner moves). We established the following analog of Pachner's theorem: two balanced simplicial complexes that represent closed combinatorial manifolds are PL homeomorphic if and only if they can be connected by a sequence of cross-flips.
December 2015
Tillmann
Miltzow
An Approximation Algorithm and Parameterized Hardness of the Art Gallery Problem
Christoph
Spiegel
FU Berlin
What is discrete Fourier analysis?
Abstract.
Discrete Fourier analysis can be a powerful tool when studying the additive structure of sets. Sets whose characteristic functions have very small Fourier coefficients act like pseudo-random sets. On the other hand well structured sets (such as arithmetic progressions) have characteristic functions with a large Fourier coefficient. This dichotomy plays an integral role in many proofs in additive combinatorics from Roth?s Theorem and Gower?s proof of Szemer�di?s Theorem up to the celebrated Green-Tao Theorem. We will introduce the discrete Fourier transform of (balanced) characteristic functions of sets as well some basic properties, inequalities and exercises. No prior knowledge of combinatorics or number theory is necessary.
Julia
Wolf
U Bristol
Quadratic Fourier Analysis
Felix
Seibert
Square Dissections and Transversal Structures.
Matthias
Henze
FU
On lattice points in strictly convex sets and Helly-type numbers in integer programming
Abstract.
Doignon proved a discrete version of Helly's theorem claiming that a finite family of convex sets in R^n intersects in an integral point if every subfamily of size at most 2^n does so. Motivated by applications in integer programming, Aliev et al. recently obtained a quantitative version of this result, which guarantees that a finite family of convex sets intersects in k integral points whenever every subfamily of size at most c_n(k) does so. The best current upper bound on the minimal such constant c_n(k) grows linearly with the parameter k. Based on a connection to the number of boundary integral points in strictly convex sets, we show that the asymptotic behavior of c_n(k) is sublinear in dimension two and we determine the exact value of c_n(k) for k at most four.
Julia
Wolf
University of Bristol
Counting monochromatic structures in finite abelian groups
Abstract.
In this talk we aim to determine the minimum number of monochromatic additive configurations in any 2-colouring of a finite abelian group (such as Z/pZ for p a prime). The techniques used to address this question, which include additive combinatorics and quadratic Fourier analysis, originate in quantitative approaches to Szemeredi’s theorem. Perhaps surprisingly, some of our results in the arithmetic setting have implications for a graph-theoretic problem that has been open since the 1960s.
Maryna
Viazovska
HU Berlin
An application of the theory of automorphic forms to discrete geometry
Abstract.
In this talk we will give an overview of classical and recent results on energy optimization problems in discrete geometry. We will focus on the interplay of the theory of automorphic forms and Fourier analysis and their applications to discrete geometry.
Martin
Groß
Dealing with Big Data - An Introduction to Streaming Algorithms
K.
Kang
Green tensor of the steady Stokes system in the half space and asymptotics of stationary Navier-Stokes flows
Franz
Brandenburg
Semi-bar 1-visibility graphs.
Shagnik
Das
FU Berlin
The Erdős-Rothschild problem for intersecting families
Abstract.
The Erdős-Rothschild problem is a twist on the typical extremal problem, asking for the structure with the maximum number of r-colourings without a monochromatic forbidden substructure. While originally phrased in the context of Turán theory, it can extend any extremal problem. Recently, there has been a great deal of interest in the Erdős-Rothschild problem for intersecting families, particularly for families of sets or vector spaces. We will present a general framework that allows us to extend these previous results to much larger, and, in some cases, optimal, choices of parameters of the families in question. This proof also applies to new settings, such as intersecting families of permutations. Joint work with Dennis Clemens and Tu(r)an Tran.
Daniel
Loughran
Leibniz Universität Hannover
The Hasse norm principle for abelian extensions
Abstract.
A classical theorem of Hasse states that, for a cyclic extension of number fields L/K, an element of K is a norm from L if and only if it becomes a norm over all completions of K. In this talk, we study the extent to which this "Hasse norm principle" holds for other abelian extensions. Namely, the distribution of abelian extensions of bounded discriminant that fail the Hasse norm principle. This is joint work with Christopher Frei and Rachel Newton.
November 2015
Emo
Welzl
ETH Zurich
Crossing-Free Perfect Matchings, etc., on Wheel Point Sets
Udo
Hoffmann
Polygon obstacle representations of graphs.
Sarah
Brodsky
The D4 cluster algebra and tropical positivity
Abstract.
We show that the number of combinatorial types of clusters of type $D_4$ modulo reflection-rotation is exactly equal to the number of combinatorial types of tropical planes in $\mathbb{TP}^5$. This follows from a result of Sturmfels and Speyer which classifies these tropical planes into seven combinatorial classes using a detailed study of the tropical Grassmannian $\Gr(3,6)$. Speyer and Williams show that the positive part $\Gr^+(3,6)$ of this tropical Grassmannian is combinatorially equivalent to a small coarsening of the cluster fan of type~$D_4$. We provide a structural bijection between the rays of $\Gr^+(3,6)$ and the almost positive roots of type $D_4$ which makes this connection more precise. This bijection allows us to use the pseudotriangulations model of the cluster algebra of type $D_4$ to describe the equivalence of "positive" tropical planes in $\mathbb{TP}^5$, giving a combinatorial model which characterizes the combinatorial types of tropical planes using automorphisms of pseudotriangulations of the octogon.
József
Balogh
University of Illinois
On Some Recent Applications of the Container Method
Abstract.
I will talk about some recent applications of the container method. There are two types of applications, I present examples for both; one where it is unexpected that the method would work, the other where it is expected, but current versions of container lemmas are not applicable. Partly it is joint work with Jozsef Solymosi and Adam Wagner.
David
Holmes
University of Leiden
Néron models over bases of higher dimension
Abstract.
Néron models for 1-parameter families of abelian varieties were defined and constructed by Néron in the 1960’s, and provide a ‘best possible’ model for the degenerating family. For a degenerating family of abelian varieties over a base scheme of dimension greater than 1, it is much less clear what the ‘best possible' model for the family would be. If one naively extends Néron’s original definition to this setting then these objects fail to exist, even if we allow blowups or alterations of the base space of the family - more precisely, we give a combinatorial characterisation of exactly when such Néron models of jacobians exist. In the case of the jacobian of the universal curve we will describe the minimal base-change required in order that a Néron model exist, giving a possible answer to the shape of the ‘best possible degeneration’.
Andreas
Feldmann
On the Equivalence of the Bidirected and Hypergraphic Relaxations for Steiner Tree
Jörg
Wolf
Methoden zur Behandlung der Navier-Stokes-Gleichungen
Björn
Kapelle
Kontakt- und Schnittdarstellungen planarer Graphen.
Fatemeh
Mohammadi
Applications of Combinatorial Commutative Algebra in system reliability theory, and Signature analysis for Erdös-Rényi model
Abstract.
I will talk about the application of syzygy tool from commutative algebra in two concrete examples arisen in system reliability theory. Let $G$ be a connected graph whose vertices are reliable but each edge $e$ may fail with the probability $p_e$. There is a canonical ideal associated to $G$ whose Hilbert series encodes the reliability of the system which is the probability that $G$ is connected. I will talk about the connections between matroid theory and system reliability theory applying some tools from convex geometry and commutative algebra.
Olaf
Parczyk
Goethe Universität Frankfurt
Universality in random and sparse hypergraphs
Abstract.
Finding spanning subgraphs is a well studied problem in random graph theory, in the case of hypergraphs less is known and it is natural to study the corresponding spanning problems for random hypergraphs. We study universality, i.e. when does an r-uniform hypergraph contain any hypergraph on n vertices and with maximum vertex degree bounded by $\Delta$? For $\mathcal{H}^{(r)}(n,p)$ we show that this holds for $p=\omega( (\ln n/n)^{ 1/ \Delta } )$ a.a.s. Furthermore we derive from explicit constructions of universal graphs due to Alon, Capalbo constructions of universal hypergraphs of size almost matching the lower bound $\Omega (n^{r-r/ \Delta})$. This is joint work with Samuel Hetterich and Yury Person.
Anna
von Pippich
TU Darmstadt
On the wave representation of Eisenstein series
Abstract.
Let Γ ⊂ PSL2 (ℝ) be a Fuchsian subgroup of the first kind and let X = Γ\H be the associated finite volume hyperbolic Riemann surface. Eisenstein series attached to parabolic subgroups of Γ play a fundamental role in the theory of automorphic forms on X. Analoguously, one can consider Eisenstein series associated to hyperbolic or elliptic subgroups of Γ. In this talk, we present a unified approach to the construction of these Eisenstein series in terms of the wave kernel. This is joint work with Jay Jorgenson and Lejla Smajlovi'.
Jörg
Wolf
Methoden zur Behandlung der Navier-Stokes-Gleichungen
Yann
Disser
The Online Matrix-Vector Multiplication Conjecture
Jan
Hofmann
FU Berlin
What is reciprocity?
Abstract.
Pick's theorem allows us to compute the area of any lattice polygon just by counting integer points. We will sketch a proof of Pick's theorem. It is now natural to ask if there are analogues in higher dimensions. This leads to the introduction of the Ehrhart polynomial and Ehrhart reciprocity.
Christian
Haase
FU Berlin
Hodge Numbers & Lattice Points
Stefan
Felsner
Drawings of K_{r,n}.
Raman
Sanyal
FU
Extension complexity of Hypersimplices -- fun and other feelings
Abstract.
A (n,k)-hypersimplex is the convex hull of all 0/1-vectors of length n with exactly k ones. This is a very explicit class of polytopes that appears in many contexts including algebraic geometry and geometric combinatorics. The extension complexity of a polytope P is the minimal number of facets of a polytope Q that projects onto P. This simple-to-state invariant has its origins in combinatorial optimization and is extremely difficult to compute in general. In this talk I will tell you about the extension complexity of hypersimplices and combinatorial hypersimplices. For hypersimplices, this is a lot of fun and involves geometry, combinatorics, and computers. For combinatorial hypersimplices, the other feelings kick in. This is based on joint work with Francesco Grande and Arnau Padrol.
Pedro
Berrizbeitia
Universidad Simón Bolívar
On additive Properties of Multiplicative Groups on finite Index in Fields
Abstract.
Let m > 1 be an integer. Except for finitely many primes p, the equation x^m + y^m = t mod p has integer solutions x and y for all integer t. The problem is a classical one in Number Theory. In fact, if N_{p,t} is the number of solutions (x mod p, y mod p) of the equation, then |N_{p,t} - p| = O( p^{1/2} ). The analog problem for infinite or more general fields is the following: Let G be a multiplicative group of finite index in F*, the multiplicative group of units of the field F, what does G + G look like, in particular, when is G + G = F? The first publication on the subject appeard in 1989, when Leep and Shapiro, gave a positive answer when the index of G in F* is 3. That is, they proved that G + G = F, except for a small and short list of exceptional fields. They conjectured that the result was true in the case of index 5, for any infinite field F. They mentioned that they had not been succesfull even for F = Q, the rational numbers. In the talk we will prove that if G is of finite index in Q*, then G − G = Q, that is, every rational number is writen as the difference of two elements of G. The proof is a very nice application of the Van der Waerden Theorem on artithmetic progressions, that we will state without proof. More general results in Ramsey Theory allow to prove the same result of any infinite field, and even for some other more general rings. We will brieffly describe these more general results and consequences. We will also discuss the behaviour of G + G, of G + G + G, etc. In fact, we will scketch the solution published in 2011 of a conjecture posed in 1992 by Bergelson and Shapiro on an additive question and give Florian’s theorem on the general behaviour of G_n + G_n, for a specific sequence of subgroups of index n ∈ Q*.
Matthias
Eller
On boundary value problems for hyperbolic systems
Kathlén
Kohn
Voronoi Cells of Lattices w.r.t. Arbitrary Norms
Abstract.
Micciancio and Voulgaris gave in 2010 'A Deterministic Single Exponential Time Algorithm for Most Lattice Problems based on Voronoi Cell Computations'. Among others, their algorithm solves the prominent Shortest/Closest Vector Problem, but it works solely for the Euclidean norm. One open problem listed by Micciancio and Voulgaris is the extension of this algorithm to other p-Norms. We show that a direct extension with deterministic single exponential time is not possible, for the following reason: Micciancio and Voulgaris represent the Voronoi cell of a lattice (w.r.t. the Euclidean norm) by a finite set of lattice vectors, called the Voronoi-relevant vectors. In particular, they use that the number of the Voronoi-relevant vectors is upper bounded by a bound solely depending on the lattice dimension. We show that such a bound does not even exist for the 3-norm.
Ping
Hu
University of Warwick
Short induced cycles in graphs
Abstract.
Let C(n) denote the maximum number of induced copies of 5-cycles in graphs on n vertices. For n large enough, we show that C(n)=abcde + C(a)+C(b)+C(c)+C(d)+C(e), where a+b+c+d+e = n and a,b,c,d,e are as equal as possible. Moreover, for n being a power of 5, we show that the unique graph on n vertices maximizing the number of induced 5-cycles is an iterated blow-up of a 5-cycle. The proof uses flag algebra computations and stability methods. Joint work with Balogh, Lidický and Pfender.
Anilatmaja
Aryasomayajula
University of Hyderabad
Heat kernels, Bergman kernels, and estimates of cusp forms
Abstract.
In this talk, we describe a geometric approach to study estimates of cusp forms. The approach relies on the micro-local analysis of the heat kernel and the Bergman kernel. Using which we can derive qualitative estimates of cusp forms of integral weight or half-integral weight associated to arbitrary Fuchsian subgroups and groups commensurable with the Hilbert modular group.
Matthias
Eller
On boundary value problems for hyperbolic systems
October 2015
Filippo
Nuccio
Université St. Étienne
What is a period?
Abstract.
Not available.
Linda
Kleist
Drawing planar graphs with prescribed face areas.
Pierre
Cartier
U Paris-Diderot and IHÉS
Towards a Galois theory of transcendental numbers
Moritz
Schmitt
Freie Universität Berlin
On the number of space groups
Abstract.
A space group is a discrete group of isometries with bounded fundamental domain. Prominent examples are the 17 wallpaper groups and 219 crystallographic groups. Bieberbach proved that in each dimension there are only finitely many non-isomorphic space groups, and much later Buser gave the first upper bound. We will show how to improve Buser's bound considerably and get close to known lower bounds.
Christoph
Spiegel
FU Berlin
Threshold functions for systems of equations in random sets
Abstract.
We present a unified framework to deal with threshold functions for the existence of certain combinatorial structures in random sets. The structures will be given by certain linear systems of equations M · x = 0 and we will use the binomial random set model where each element is chosen independently with the same probability. This covers the study of several fundamental combinatorial families such as k-arithmetic progressions, k-sum-free sets, B_h [g] sequences and Hilbert cubes of dimension k. Furthermore, our results extend previous ones about B_h [2] sequences by Godbole et al. We show that there exists a threshold function for the property "A^m contains a non-trivial solution of M · x = 0" where A is a random set. This threshold function depends on a parameter maximized over all subsystems, a notion previously introduced by Rödl and Rucinski. The talk will contain a formal definition of trivial solutions for any combinatorial structure, extending a previous definition by Ruzsa. Furthermore, we will study the behavior of the distribution of the number of non-trivial solutions in the threshold scale. We show that it converges to a Poisson distribution whose parameter depends on the volumes of certain convex polytopes arising from the linear system under study as well as the symmetry inherent in the structures, which we formally define and characterize. Joint work with Juanjo Rué and Ana Zumalacárregui.
Davide
Veniani
Leibniz Universität Hannover
Counting lines on quartic surfaces: New techniques and results
Abstract.
In the last years, two independent research teams (the one based in Ankara, Turkey, the other in Hannover, Germany) have tackled the problem of counting lines on smooth quartic surfaces; the former aimed at a complete classification using computer algebra system GAP, the latter strove for more geometrical insight. The synergy between the two methods has fostered new ideas towards three goals: (1) finding a proof of the fact that the maximal number of lines is 64 which does not involve the flecnodal divisor; (2) proving the uniqueness of the surface with 64 lines with a geometrical approach; (3) adapting the methods to the K3 quartic case. I will report about the state of the art.
Veit
Wiechert
Graphs with locally bounded treewidth.
Michał
Lasoń
http://www.mi.fu-berlin.de/math/groups/discgeom/members/Lason.html
On the toric ideal of a matroid and related combinatorial problems
Abstract.
When an ideal is defined only by combinatorial means, one expects to have a combinatorial description of its set of generators. An attempt to achieve this description often leads to surprisingly deep combinatorial questions. White's conjecture is an example. It asserts that the toric ideal associated to a matroid is generated by quadratic binomials corresponding to symmetric exchanges. In the combinatorial language it means that if two multisets of bases of a matroid have equal union (as a multiset), then one can pass between them by a sequence of symmetric exchanges. White's conjecture resisted numerous attempts since its formulation in 1980. We will discuss its relations with other open problems concerning matroids.
Tibor
Szabó
FU Berlin
Half-random Maker-Breaker games
Abstract.
We study Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In both such scenarios, that is when Maker plays randomly and when Breaker plays randomly, we determine the sharp threshold bias of classical graph games, such as connectivity, Hamiltonicity, and minimum degree-k. The traditional, deterministic version of these games, with two optimal players playing, are known to obey the so-called probabilistic intuition. That is, the threshold bias of these games is asymptotically equal to the threshold bias of their random counterpart, where players just take edges uniformly at random. We find, that despite this remarkable agreement of the results of the deterministic and the random games, playing randomly against an optimal opponent is not a good idea: the threshold bias becomes significantly higher in favor of the clever player. An important qualitative aspect of the probabilistic intuition nevertheless carries through: for Maker to occupy a connected graph or a Hamilton cycle, the bottleneck is still the ability to achieve that there is no isolated vertex in his graph. The talk represents joint work with Jonas Groschwitz.
Patrick
Da Silva
HU Berlin
What is representation theory?
Abstract.
This What Is seminar is meant to be a rough introduction to representation theory to be able to follow Geordie Williamson's talk. We will sketch the basic concepts : definition of a representation, Maschke's theorem, Schur's Lemma, character theory and if we have some time perhaps the link between representations and \(k[G]-\)modules. Since Geordie will mention modular representation theory I will give an example in characteristic p, where Maschke's theorem fails.
Geordie
Williamson
MPI Bonn
Mathematics in light of representation theory
Udo
Hoffmann
The slope number of graphs.
Alexander
Engström
Aalto University
Self-polar resolutions
Abstract.
Polytopes with strong duality properties are introduced. They are interesting for constructing explicit projective resolutions of ideals with a strong duality on the syzygies. I will provide some evidence for that this is a fairly general possibility, and discuss results towards that with Linusson on Stanley-Reisner ideals of cyclic polytopes.
Stefan
Felsner
Strongly monotone drawings of planar graphs.
Torsten
Mütze
Bipartite Kneser graphs are Hamiltonian.
September 2015
Felix
Seibert
An algorithm for square contact representations.
Linda
Kleist
Points in the plane and depth.
Hannah
Schäfer Sjöberg
The combinatorial parameters (f_0, f_03) of 4-dimensional polytopes
Abstract.
For dimensions d ≥ 4, the complete set of all f-vectors of d-polytopes has not been characterized. For dimension 4, the projections of f-vectors onto 2 of the 4 coordinates have been determined by Grünbaum (1967), Barnette–Reay (1973) and Barnette (1974). We expand these results to the flag vectors of 4-polytopes. We will characterize the projection of the set of all flag vectors of 4-polytopes to the two coordinates f_0 and f_03.
Veit
Wiechert
The dimension of planar posets.
August 2015
Udo
Hoffmann
Illumination of polygons with vertex lights.
Stefan
Felsner
Arrangements of Pseudolines on Dual Arrangements
Linda
Kleist
Counting annular non-crossing matchings.
July 2015
Dániel
Korándi
ETH Zürich
A random triadic process
Abstract.
Given a random 3-uniform hypergraph H=H(n,p) on n vertices where each triple independently appears with probability p, consider the following graph process. We start with the star G_0 on the same vertex set, containing all the edges incident to some vertex v_0, and repeatedly add an edge xy if there is a vertex z such that xz and zy are already in the graph and xzy∈H. We say that the process propagates if it reaches the complete graph before it terminates. In this talk we show that the threshold probability for propagation is p=1/2√n using the differential equation method. As an application, we obtain the upper bound p=1/2√n on the threshold probability that a random 2-dimensional simplicial complex is simply connected. Joint work with Yuval Peled and Benny Sudakov.
Veit
Wiechert
Queue layouts of graphs with bounded treewidth.
Clément
Requilé
FU Berlin
Triangles in random cubic planar graphs
Abstract.
This talk will be about the exact counting of triangles in cubic planar graphs, and some asymptotic consequences. The starting point is the study, done in [BKLMcD] by means of generating functions, of the asymptotic number of labeled cubic planar graphs with a fixed number of vertices. Our approach, following theirs, is based on connectivity decomposition and generating functions which reduce the problem to a map enumeration question. We show how to adapt this decompositions in order to encode triangles. At the end of the talk we will explain how to use these equations to get limiting laws for the number of triangles in cubic planar graphs, as well as the asymptotic number of triangle-free planar graphs on a given number of vertices. Work in progress with Juanjo Rué. [BKLMcD] M. Bodirsky, M. Kang, M. Löffler and C. McDiarmid. Random Cubic Planar Graphs. Random Structures and Algorithms, 30 (2007) 78-94.
Ariyan
Javanpeykar
Uni Mainz
Good reduction of complete intersections
Abstract.
In 1983, Faltings proved the Shafarevich conjecture: for a finite set of finite places of a number field K and an integer g>1, the set of isomorphism classes of curves of genus g over K with good reduction outside S is finite. In this talk we shall consider analogues of the Shafarevich conjecture for complete intersections. This is joint work with Daniel Loughran.
Manh Tuan
Tran
FU Berlin
What is Ramsey theory?
Abstract.
Ramsey theory refers to a branch of mathematics whose underlying philosophy is captured by the statement that "Every large system contains a large well-organized subsystem." There are examples of such statements in many areas, including geometry, number theory and analysis. In this talk, we shall discuss some examples and focus on Roth's theorem, which guarantees that every set of integers with positive density contains a 3-term arithmetic progression. No prior knowledge of combinatorics and number theory is assumed.
Benjamin
Sudakov
ETH Zürich
Induced matchings, arithmetic progressions and communication
Udo
Hoffmann
Slope minimization of segment intersection graphs.
Jozsef
Balogh
University of Illinois at Urbana-Champaign
On the applications of counting independent sets in hypergraphs
Abstract.
Recently, Balogh-Morris-Samotij and Saxton-Thomason developed a method of counting independent sets in hypergraphs. I will survey the field, and explain several applications. This includes counting maximal triangle-free graphs, counting -free graphs, estimate volume of metric spaces, etc. These results are partly joint with Liu, Morris, Petrickova, Samotij, Sharifzadeh, and Wagner.
Stefan
Felsner
Drawings of K_{2,n}.
Boris
Bukh
Carnegie Mellon University
Common subsequences in words and permutations
Abstract.
Every sufficiently large collection of objects always contains two similar objects. For example, a large collection of words contains a pair of words with a long common subsequence. How long? In this talk, we look at several versions of this problem. We will see how this problem leads to some algebraic constructions, Fourier-like arguments, and an innocently-looking problem about monotone functions. The talk is based on the joint works with Lidong Zhou, Jie Ma and Venkatesan Guruswami. The work began 3 years ago, on a visit to FU Berlin.
June 2015
Antareep
Mandal
HU Berlin
Uniform sup-norm bound for the average over an orthonormal basis of Siegel cusp forms
Abstract.
The average over an orthonormal basis of cusp forms on a given Siegel modular curve can be viewed as the lower part of the discrete spectrum, corresponding to the eigenvalue 0, of the heat kernel associated to the Siegel-Maaß Laplacian. Therefore, one can attempt to use long time asymptotics of this heat kernel to derive an optimum sup-norm bound for the average over an orthonormal basis of cusp forms uniformly in a finite degree cover of the given Siegel modular curve. This method has been proven to work in the one-dimensional case of classical modular forms. In this talk, we present our progress towards a generalization of this method to the higher dimensional case of Siegel modular forms.
John
Walsh
U British Columbi
Some remarks on stochastic partial differential equations
Jonathan
Spreer
University of Queensland
Parameterised complexity theory for topological problems
Abstract.
Many of the most important algorithmic problems in computational geometry are known to be NP-complete or even have unknown complexity. On the other hand, these difficult problems can often be solved efficiently with the use of heuristics. I will explain how the framework of parameterised complexity can be used to explain this inconsistent behaviour and present a number of examples and recent results.
Christopher
Kusch
FU Berlin
Strong Ramsey Games: Drawing on an infinite board
Abstract.
We will consider a strong Ramsey-type game played on the edges of the complete -uniform hypergraph with infinitely many vertices. Two players, first player and second player, try to claim edges to build a copy of a fixed graph . First player starts and whoever builds a copy of first wins. It is known that if they play on finitely many vertices, player one can guarantee to win, if the number of vertices is large enough. We will then prove that this is false on infinitely many vertices and provide sufficient conditions a hypergraph needs to satisfy in order for second player to guarantee a draw as well as a -uniform example of such a hypergraph. This is joint work with Dan Hefetz, Lothar Narins, Alexey Pokrovskiy, Clément Requilé and Amir Sarid.
Slawomir
Rams
Jagiellonian University Krakow, z.Zt. Leibniz Universität Hannover
On Enriques surfaces with four cusps
Abstract.
One can show that maximal number of A2-configurations on an Enriques surface is four. In my talk I will classify all Enriques surfaces with four A2-configurations. In particular I will show that they form two families in the moduli of Enriques surfaces and I will construct open Enriques surfaces with fundamental groups (Z/3Z)^2 × Z/2Z and Z/6Z, completing the picture of the A2-case and answering a question put by Keum and Zhang. This is joint work with M. Schuett/LUH Hannover.
Asilya
Suleymanova
HU Berlin
What is the heat kernel?
Abstract.
First we will discuss the notion of a kernel of an operator and the heat operator itself. On the Euclidean space the scalar heat kernel is given by the exact formula . For an arbitrary Riemannian manifold it is usually impossible to find an exact expression for the heat kernel. However for many problems approximate solution suffices. For example, I will show how the local Atyiah-Singer index theorem can be proven using heat kernel approach.
Waldo
Gálvez
Improved Online Algorithms for the Machine Covering Problem with Bounded Migration
Steve
Chaplick
Partial Bar Visibility Representation Extension.
Emmanuel
Tsukerman
Circumcenter of mass
Abstract.
I shall define and study a variant of the center of mass of a polygon, called the circumcenter of mass. The circumcenter of mass is an affine combination of the circumcenters of the triangles in a non-degenerate triangulation of a polygon, weighted by their areas, and is independent of the triangulation. It satisfies an analog of the Archimedes Lemma, similarly to the center of mass of the polygonal lamina, and hence gives rise to an isometry-covariant valuation. The line connecting the circumcenter and the centroid of a triangle is called the Euler line. Taking affine combinations of the circumcenter of mass and the center of mass, one obtains an Euler line of a polygon. The construction of the circumcenter of mass extends to simplicial polytopes and to the spherical and hyperbolic geometries.
Florian
Pfender
UC Denver
Semidefinite programming in extremal graph theory and Ramsey theory
Abstract.
Ten years ago, Razborov developed the theory of flag algebras. In this theory, many extremal problems on densities of small substructures in large structures can easily be expressed and studied. One of the most successful methods inside this theory is the so called "plain flag algebra method." In this method, semidefinite programming is used to optimally combine a large set of true equalities and inequalities to bound densities in the extremal object. Many new results in extremal graph theory have been proved this way. In general, the method is often successful if the extremal example is a simple blow up of a small graph --- a very common outcome in this field. Another common outcome is an iterated blow up of a small graph. In this situation, the plain flag algebra method alone rarely succeeds. In this talk I will present a way how to extend the plain method to deal with a number of questions of this flavor. In the last part of the talk I will show a surprising trick how to use the plain method to improve the upper bounds on some small Ramsey numbers. This is joint work with Bernard Lidický.
Valentina
Di Proietto
FU Berlin
On the homotopy exact sequence for the log algebraic fundamental group
Abstract.
There is a strong link between the fundamental group of a variety and the linear differential equations we can define on it. The definition of the fundamental group given in terms of homotopy classes of loops does not generalize easily to algebraic varieties defined over an arbitrary field. But exploiting this link we can give another definition that makes sense in very general contexts: it is called the algebraic fundamental group. We prove the homotopy exact sequence for the algebraic fundamental group for a fibration with singularities with normal crossing and we explain how this gives a monodromy action. This is a joint work with Atsushi Shiho.
Peter
Patzt
FU Berlin
What is group homology?
Abstract.
In this talk we recall a few notions and definitions from homological algebra. Then we give a quick-and-dirty definition of group homology. The talk finishes with Shapiro's Lemma, which is an easy implication from the definitions.
Nathalie
Wahl
U Copenhagen
Stabilizing symmetries
Linda
Kleist
Rainbow Connectivity
Anton
Dochtermann
Warmth and edge spaces of graphs
Abstract.
In recent years two novel approaches for finding lower bounds on the chromatic number of a graph have been introduced. One involves studying the topological connectivity of the 'edge space' of a graph, dating back to Lovasz's celebrated proof of the Kneser conjecture. The other is motivated by constructions in statistical physics and involves the notion of long range action of random branching walks and the 'warmth' of a graph, as introduced by Brightwell and Winkler. We seek to relate these two constructions, and in particular we provide evidence for the conjecture that the warmth of a graph G is always less than three plus the connectivity of its edge space. We succeed in establishing the first nontrivial case of the conjecture, and calculate the warmth of a family of graphs with relevant edge space topology. We also demonstrate a connection between the warmth of a graph and the collection of complete bipartite subgraphs that it contains, providing an analogue for a similar result in the context of edge spaces. This is joint work with Ragnar Freij.
Samuel
Fiorini
ULB
No Small Linear Program Approximates Vertex Cover within a Factor 2-epsilon
Abstract.
The vertex cover problem is one of the most important and intensively studied combinatorial optimization problems. Khot and Regev (2003) proved that the problem is NP-hard to approximate within a factor 2-epsilon, assuming the Unique Games Conjecture (UGC). This is tight because the problem has an easy 2-approximation algorithm. Without resorting to the UGC, the best inapproximability result for the problem is due to Dinur and Safra (2002): vertex cover is NP-hard to approximate within a factor 1.3606. We prove the following unconditional result about linear programming (LP) relaxations of the problem: every LP relaxation that approximates vertex cover within a factor of 2-epsilon has super-polynomially many inequalities. As a direct consequence of our methods, we also establish that LP relaxations that approximate the independent set problem within any constant factor have super-polynomially many inequalities. This is joint work with Abbas Bazzi, Sebastian Pokutta and Ola Svensson.
Jürg
Kramer
HU Berlin
Degeneration of the hyperbolic heat kernel
Abstract.
In our talk we will investigate the degeneration of the hyperbolic heat kernel and its trace at the cusps of modular curves. This degeneration behavior should be similar to the degeneration of the arithmetic self-intersection number of the corresponding line bundle equipped with a metric that is logarithmically singular at the cusps.
Adam
Nielsen
ZIB
What is a transfer operator?
Abstract.
The transfer operator is a very important tool in applied mathematics, originating from measure theory. Its beauty and simplicity is challenged by a horrible wikipedia article which will be rectified during the talk with a clean and plain introduction of transfer operators.
Miriam
Schlöter
Mechanism Design for Crowdsourcing: An Optimal 1–1/e Competitive Budget-Feasible Mechanism for Large Markets
Udo
Hoffmann
Visibility graphs of pseudo-polygons.
Benny
Sudakov
ETH Zürich
Grid Ramsey problem and related questions
Abstract.
The Hales-Jewett theorem is one of the pillars of Ramsey theory, from which many other results follow. A celebrated result of Shelah says that Hales-Jewett numbers are primitive recursive. A key tool used in his proof, known as cube lemma, has become famous in its own right. In its simplest form, it says that if we color the edges of the Cartesian product in colors then, for sufficiently large, there is a rectangle with both pairs of opposite edges receiving the same color. Hoping to improve Selah’s result, Graham, Rothschild and Spencer asked more than 20 years ago whether cube lemma holds with , which is polynomial in . We show that this is not possible by providing a superpolynomial lower bound in . We also discuss a number of related questions, among them a solution of the problem of Erdos and Gyarfas on generalized Ramsey numbers. Joint work with Conlon, Fox and Lee.
Bram
Petri
Fribourg
The systole of a random surface
Abstract.
In this talk a random surface will be a surface constructed by randomly gluing together an even number of triangles that carry a fixed metric. The model lends itself particularly well to studying the geometry of typical high genus hyperbolic surfaces. For example, it turns out that the expected value of the length of the shortest non-contractible curve, the systole, of such a surface converges to a constant. In this talk I will explain what goes into the proof of this fact and how this relates to the theory of random regular graphs and random elements in the symmetric group.
Ana
Zumalacárregui
University of New South Wales
Additive bases for intervals
Abstract.
A set is said to be an additive basis for the interval if every element in the interval can be represented as sum of two elements of the set. A natural question arises in this context: how small could such a basis be? In this talk we will address this question and study the minimal cardinality of a Basis for (that is, the number of representations of every element in is at least ). Even though it is clear that such quantity is of order , an asymptotic for is not known to exist (even in the simplest case ). We will study this quantity and show that both the and the tend to the same constant as grows. The strategy follows the lines of [CRV'10] where the case of -Sidon sets for intervals was studied (that is the number of representations is at most ) and approaches the problem by considering successive constructions of sets whose support is restricted. A key point in the argument is to exploit good constructions -based on ideas from Ruzsa [R'90]- of sets on finite groups whose representation function is closed to be constant. Joint work with Javier Cilleruelo and Carlos Vinuesa. [CRV'10] J. Cilleruelo, I. Ruzsa, and C. Vinuesa. Generalized Sidon sets. Adv. Math., 225(5):2786-2807, 2010. [R'90] I. Z. Ruzsa. A just basis. Monatsh. Math., 109(2):145-151, 1990.
Andriy
Bondarenko
Norwegian University of Science and Technology
On Helson's conjecture
Abstract.
[see here]
May 2015
Kolja
Knauer
Drawing graphs with vertices and edges in convex position
Sylwia
Antoniuk
Adam Mickiewicz University
On the connectivity Waiter-Client game
Abstract.
We consider a variation of the connectivity Waiter-Client game played on an -vertex graph which consists of disjoint spanning trees. In this game in each round Waiter offers Client edges of which have not yet been offered. Client chooses one edge and the remaining edges are discarded. The aim of Waiter is to force Client to build a connected graph. If this happens Waiter wins. Otherwise Client is the winner. It is known that for and for and even , Waiter can always force Client to build a connected graph. Bednarska-Bzdega et al. [BBHKL2014] asked if this is true for the remaining values of . We give and answer to this question, namely we show that for most of the Waiter does not always have a Winning strategy. This is a joint work with Codrut Grosu and Lothar Narins. [BBHKL2014] M. Bednarska-Bzdega, D. Hefetz, M. Krivelevich, T. Łuczak, Manipulative waiters with probabilistic intuition
Wouter
Zomervrucht
Universität Leiden
Bhargava's cube law and cohomology
Abstract.
In his Disquisitiones Arithmeticae, Gauss described a composition law on (equivalence classes of) integral binary quadratic forms of fixed discriminant D. The resulting group is the class group Cl(S), where S is the quadratic algebra of discriminant D. More recently, Bhargava explained Gauss composition as a consequence of a composition law on (equivalence classes of) 2x2x2-cubes of integers. Here one obtains the group Cl(S) x Cl(S). Bhargava's proof is arithmetic. We show how to obtain Bhargava's cube law instead from geometry, with the class groups arising as cohomology. This is work in progress.
Cédric
Villani
UPMC/CNRS
Of particles, stars, and eternity
Hao
Chen
Dressing code for the sphere - scribability problems of polytopes
Abstract.
We will study various scribability problems, including the classical one studied by Steinitz and Schulte, the weak one proposed by Grünbaum-Shephard and Schulte, and the new (i, j)-scribability which requires faces to cut or avoid the sphere instead of being tangent. We obtain new results for each of these problems. In particular, classical scribabilities are fully answered for stacked and cyclic polytopes; the loose end left by Schulte on weak scribability is tied up. As for the new (i, j)-scribability, we find counter examples in most cases, but leave two cases open. This is a joint work with A. Padrol.
Shagnik
Das
FU Berlin
Comparable pairs in set families
Abstract.
Given a family of subsets of , we say two sets are comparable if or . Sperner’s celebrated theorem gives the size of the largest family without any comparable pairs. This result was later generalised by Kleitman, who gave the minimum number of comparable pairs appearing in families of a given size. In this talk we shall consider a complimentary problem posed by Erdös and Daykin and Frankl in the early ‘80s. They asked for the maximum number of comparable pairs that can appear in a family of subsets of , a quantity we denote by . We will first resolve an old conjecture of Alon and Frankl, showing that when . Time (and interest) permitting, we shall then discuss some further improvements to bounds on for various ranges of the parameters. This is joint work with Noga Alon, Roman Glebov and Benny Sudakov.
Piotr
Micek
Sparsity and dimension.
Yingkun
Li
TU Darmstadt
Special values of Green function at twisted big CM points
Abstract.
A Green function on an arithmetic variety is a function with logarithmic singularity along an algebraic divisor. Their values at CM points play an important role in the theory of arithmetic intersection. In the case of Hilbert modular surface, one could use Poincare series to explicitly construct the Green function with log singularity along Hirzebruch-Zagier divisors. Its values averaging over Galois orbits of a big CM point are rational numbers and have interesting factorizations. In this talk, we will recall these notions and use harmonic Maass forms of weight one to give a modular interpretation of the values of these Green function at twisted big CM points.
Tuan Anh
Nguyen
TU Berlin
What is a supercritical percolation cluster?
Abstract.
With a lot of pictures my talk will explain you what a supercritical percolation cluster is and give you an example of two random walks among random conductances on this cluster: i) the constant speed and ii) the variable speed random walk, which are typical examples of discrete and continuous time Markov chains. (They may not be exactly the objects in Nina Gantert's talk, but they will give you some feeling about this field.) Very easy examples will show one of the main difficulties when studying the macroscopic properties of the random walks. That is, the trap problem, which can be roughly understood as follows. While the constant speed random walk stays a very long time in very high conductances, the variable speed one cannot easily get out of very low conductances. To get rid of this difficulty, one needs some assumptions on the law of the random conductances, which is introduced at the end of the talk.
Veit
Wiechert
Classes of graphs with bounded expansion: examples and characterizations.
Nina
Gantert
Random media and percolation
Torsten
Hoge
Leibniz Universität Hannover
The freeness of ideal subarrangements of Weyl arrangements
Abstract.
The talk is based on joint work with Takuro Abe, Mohamed Barakat, Michael Cuntz and Hiroaki Terao on Weyl arrangements. A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. Our proof of the main theorem heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula.
Alberto
Marchetti-Spaccamela
Global EDF Scheduling of Systems of Conditional Sporadic DAG Tasks
April 2015
Bas
Edixhoven
Universität Leiden
Gästeseminar 'Arithmetische Geometrie' der FU Berlin
Karim
Adiprasito
Alexander
Richter
Lower bounds on the sizes of integer programs without additional variables
Pierre
Lairez
TU Berlin
What is the complexity of matrix multiplication?
Abstract.
Studies in computer algebra led to numerous efficient algorithms to solve many kinds of difficult problems (e.g., large integer multiplication). Complexity theory focuses on problems for which an efficient solution is not known and tries to grasp the intrinsic difficulty of problems. This involves both searching for lower bounds and upper bounds. Matrix multiplication is one of these hard problems. The obvious algorithm needs $n^3$ scalar multiplications to multiply two n by n matrices, and it is far from optimal. Simple methods and sophisticated tools allow to lower this bound, but it seems likely that we are still not close to optimal.
Kolja
Junginger
1:d-Graphen: Eine Verallgemeinerung planarer Triangulierungen.
Peter
Bürgisser
TU Berlin
Geometry, invariants, and the elusive search for complexity lower bounds
Tobias
Friedl
Zero Sets of Polynomials Invariant under Finite Reflection Groups
Abstract.
The degree principle states that the non-emptiness of the zero set of a symmetric polynomial of degree at most d can be tested on a set of dimension d, independent of the dimension of the ambient space (Timofte, 2003, Riener, 2012). In this talk we generalize this to the case of arbitrary reflection groups. We show that the non-emptiness of the zero set of an invariant polynomial can be tested on the union of certain flats in the corresponding reflection arrangement. This is joint work with Cordian Riener and Raman Sanyal.
Stefan
Felsner
Hook-formulae
Steve
Chaplick
Intersection Graphs of Non-Crossing Paths.
Ewan
Davies
London School of Economics
Counting in hypergraphs via regularity inheritance
Abstract.
We develop a theory of regularity inheritance in 3-uniform hypergraphs. As a simple consequence we deduce a slight strengthening of a counting lemma of Frankl and Rödl. We believe that the approach is sufficiently flexible and general to permit extensions of our results in the direction of a hypergraph blow-up lemma.
Marvin
Künnemann
Towards Understanding the Smoothed Approximation Performance of the 2-OPT heuristic
March 2015
Lin
Chen
Polynomiality for Bin Packing with a Constant Number of Item Types (part II)
Linda
Kleist
Chromatic Art Gallery Problem
Lin
Chen
Polynomiality for Bin Packing with a Constant Number of Item Types (part I)
Imre
Leader
University of Cambridge
Tiling with Arbitrary Tiles
Abstract.
A `tile' is a finite subset T of Zn. It may or may not be possible to partition Zn into copies of T. However, Chalcraft conjectured that every T does tile Zd for some d. In this talk, we will discuss some examples, and also a proof of the conjecture, recently obtained in joint work with Vytautas Gruslys and Ta Sheng Tan.
Alexey
Pokrovskiy
FU Berlin
Rainbow matchings and rainbow connectedness
Roman
Rischke
On the power of sampling in stochastic optimization
Udo
Hoffmann
Catching light rays with mirrors
Bruno
Benedetti
Two results on collapsibility
Abstract.
We discuss higher dimensional versions of two basic results: (1) Every tree is planar, (2) Every tree has at least two leaves. If time permits, we show a few applications. This is joint work with Karim Adiprasito and Frank Lutz (arxiv:1403.5217, arxiv:1404.4239).
Lluis
Vena
Charles University Prague
Deducing an arithmetic removal lemma from the removal lemma for hypergraphs and its applications
Abstract.
The (hyper)graph removal lemma says that if a large (hyper)graph K does not have many copies of a given (hyper)graph H, then K can be made free of copies of H by deleting a small number of edges. An arithmetic removal lemma says the following. Given a group G and some subset Xi of G, if a linear system Ax = 0 does not have many solutions with xi in Xi, then we can obtain new sets Xi' where the linear system Ax = 0 does not have any solutions if the variables xi take values in Xi'. The sets Xi' have been obtained from Xi by removing few of its elements. One of the applications of the arithmetic removal lemmas is a more direct proof of Szemerédi's Theorem, regarding finding arbitrarily long arithmetic progressions on the integers (or in other groups), and its multidimensional counterpart, regarding finding k-dimensional simplicies in Zk. In this talk we show a deduction of an arithmetic removal lemma using the combinatorial one.
Sebastian
Schenker
A strongly polynomial time algorithm for multicriteria global minimum cuts (part II)
Veit
Wiechert
A new proof for the Aztec diamond theorem.
February 2015
Piotr
Micek
Computing dimension of bounded width posets
Alexey
Pokrovskiy
FU Berlin
Open problems in Ramsey Theory (pt 2)
Abstract.
A selection of fascinating open problems in all areas of Ramsey Theory will be presented. These problems were collected at the AIM Graph Ramsey Theory workshop, San Jose, January 2015. If anyone has a problem they would like to present (not necessarily in Ramsey theory), feel free to bring it and present it.
Stefan
Felsner
Crossing Numbers and Rotation Systems
Tibor
Szabó
FU Berlin
and
Alexey
Pokrovskiy
FU Berlin
Open problems in Ramsey Theory
Sebastian
Schenker
A strongly polynomial time algorithm for multicriteria global minimum cuts (part I)
Thomas
von Larcher
FU Berlin
What is a scale in motion?
Nieke
Aerts
Power line route optimization.
Philip
Brinkmann
Flag Vector Spaces of Polytopes, Spheres and Eulerian Lattices
Abstract.
This talk deals with open questions concerning the flag vector spaces of polytopes, strongly regular spheres and Eulerian lattices. The latter two are natural generalizations of polytopes, since polytope boundaries are strongly regular spheres, and the combinatorial structure behind forms Eulerian lattices. The open questions discussed concern characteristics and description of the sets of flag-vectors that belong to these objects. Emphasis lies here on the distinguishing of the sets of flag-vectors of the three classes of objects. For dimension $d=4$ this relates to the parameters fatness and complexity, which are introduced and studied.
Christopher
Pütz
FU Berlin
An exposition of the Green-Tao theorem
Abstract.
Szemerédi's theorem states that any subset A of the positive integers with positive natural density limsupN→∞ |A ∩ [N]|/N contains arbitrary long arithmetic progressions. Allowing sets to have natural density zero the assertion remains true under certain conditions. This result is called the relative Szemerédi theorem. One is able to construct a function that corresponds to a superset of almost all prime numbers, for which the relative Szemerédi theorem holds and provides arbitrary long arithmetic progressions in the prime numbers. This result is also known as the Green-Tao theorem and was first proven by Ben Green and Terence Tao. Lately there was a paper of Conlon, Fox and Zhao who gathered simplifications of the original proof that have been made in recent years and introduced a new key step that further simplifies the proof. In this talk I will give a sketch of the proof of the relative Szemerédi theorem for the special case of 3-term arithmetic progressions as well as an outline of how to construct such a function associated with a superset of almost all prime numbers. I will include the proofs of some well-chosen intermediate results.
Jesko
Hüttenhain
TU Berlin
What is (algebraic) complexity theory?
Abstract.
Complexity theory is generally the study of algorithms, and the notion of an algorithm is mathematically not among the most accessible. In many cases however, we want to solve problems with an inherent mathematical structure, like multiplication of matrices. In algebraic complexity theory, we only look at very special classes of algorithms, those which have algebraic descriptions and interpretations. This way, stronger mathematical tools can be employed to answer computational questions. We give a short introduction to some of these algebraic models.
Joseph M.
Landsberg
Texas A&M U
What can geometry tell us about computer science?
Steve
Chaplick
Drawing the Complete Graph with a Planar Subgraph
Henrik
Bachmann
Universität Hamburg
Multiple zeta values and multiple Eisenstein series
Abstract.
In the first half of the talk I will introduce multiple zeta values and discuss their algebraic structure. Multiple zeta values can be seen as a multiple version of the Riemann zeta values appearing in different areas of mathematics and theoretical physics. The product of these real numbers can be expressed in two different ways, the so called stuffle and shuffle product, which yields a large family of linear relations. The second part of the talk is dedicated to multiple Eisenstein series which can be seen as a multiple version of the classical Eisenstein series for the full modular group. By definition the multiple Eisenstein series functions also fulfill the stuffle product. I will explain a recent result which solves the problem of getting also the shuffle product for these functions.
Cecília
Salgado
Universidade Federal do Rio de Janeiro, z.Zt. Leibniz Universität Hannover
Classification of elliptic fibrations on certain K3 surfaces
Abstract.
Let X be an algebraic K3 surface endowed with a non-symplectic involution. We classify all elliptic fibrations on X under some hypothesis on the non-symplectic involution. The idea behind it involves transferring the classification problem to a 'simpler' surface from the geometric point of view. This is work in progress with Alice Garbagnati (Milano).
January 2015
Udo
Hoffmann
Obstacle Representations of Graphs
Sarah
Brodsky
Moduli of Tropical Plane Curves
Abstract.
Tropical curves have been studied under two perspectives; the first perspective defines a tropical curve in terms of the tropical semifield \(\mathbb{T}=(\mathbb{R}\cup \{-\infty\}, \max, +)\), and the second perspective defines a tropical curve as a metric graph with a particular weight function on its vertices. Joint work with Michael Joswig, Ralph Morrison, and Bernd Sturmfels, we study which metric graphs of genus \(g\) can be realized as smooth, plane tropical curves of genus \(g\) with the motivation of understanding where these two perspectives meet. Using Polymake, TOPCOM, and other computational tools, we conduct our study by constructing a map taking smooth, plane tropical curves of genus \(g\) into the moduli space of metric graphs of genus \(g\) and studying the image of this map. In particular, we focus on the cases when \(g=2,3,4,5\). In this talk, we will introduce tropical geometry, discuss the motivation for this study, our methodology, and our results.
Rafael
von Känel
MPI Bonn
Discriminants and small points of cyclic covers
Abstract.
Let K be a number field. We first consider a generalization of Szpiro's discriminant conjecture to arbitrary smooth, projective and geometrically connected curves X/K of positive genus. Then we present an unconditional exponential version of this conjecture for cyclic covers of the projective line, and we discuss a related work (jointly with A. Javanpeykar) in which we established Szpiro's small points conjecture for cyclic covers. We also plan to explain the proofs. They combine the theory of logarithmic forms with Arakelov theory for arithmetic surfaces.
Amir
Džambić
Kiel
The cohomology of the smallest Hurwitz ball quotient
Abstract.
Recently, M. Stover showed that there exists the unique compact arithmetic 2-dimensional ball quotient of smallest volume. Its smooth Galois coverings, called Hurwitz ball quotients, thus have the maximal automorphism group among the arithmetic ball quotients with the same Euler number. We study the smallest Hurwitz ball quotient and use the knowledge of the automorphisms and the fundamental group to determine its Picard number and the Albanese variety and study some other of its cohomological properties (joint work with Xavier Roulleau).
Torsten
Mütze
Combinatorial Gray codes and the Chung-Feller theorem
Shagnik
Das
FU Berlin
What is the probabilistic method?
Abstract.
Has your mathematics become mundane? Are you suffering through a mid-Masters crisis? If so (or even if not), come along and witness the wonder of the mysterious probabilistic method at work. Guaranteed* to reignite your passion for mathematics. In this talk, we shall see some random number of applications of the probabilistic method in various fields. No prior knowledge** of combinatorics is assumed, and only very basic probability will be used. * Or your money back. **Indeed, for maximum dramatic effect, it is hoped no such knowledge exists.
Angelika
Steger
ETH Zürich
The determinism of randomness and its use in combinatorics
Matthias
Henze
Congruence Arguments in the Geometry of Numbers and a General Discrete Minkowksi-type Theorem
Abstract.
One of the most fruitful results from Minkowski's geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It says that the volume of every o-symmetric n-dimensional convex body whose only interior lattice point is the origin is bounded from above by the volume of the orthogonal n-cube of edge length two. Minkowski also obtained a discrete analog by identifying the n-cube as a maximizer of the number of lattice points in the boundary of such convex bodies. Whereas the volume inequality has been generalized to any number of interior lattice points already by van der Corput in the 1930s, a corresponding result for the discrete case remained to be proven. Using congruence arguments for lattice points and an inequality in additive combinatorics, we determine a best possible relation of this kind. In the talk, we will moreover highlight the usefulness of considering congruences on lattice points in the geometry of numbers. This is joint work with Bernardo González Merino.
Dennis
Clemens
FU Berlin
Reiher's Clique Density Theorem
Abstract.
One of the most classic results in Extremal Graph Theory is Turán's Theorem, which guarantees the existence of cliques of given order r in a graph G, provided that its edge density d exceeds a certain threshold. Naturally, one may wonder how many such cliques must necessarily be contained in G when the edge density d becomes larger than this threshold. Lovász and Simonovits [1] conjectured their number to be at least Fr(d)nr + O(nr-2), where F describes a certain piecewise concave function. After some partial results in the last decades, the conjecture was proven by Christian Reiher [2] recently, by using weighted graphs and Lagrange multipliers which help to transfer this discrete problem into a continuous setting. In the talk we will see a construction showing that the above bound is tight asymptotically and then we will see an overview on the proof by Christian Reiher. [1] L. Lovász and M. Simonovits: On the number of complete subgraphs in a graph II, Studies in pure mathematics, pages 459-495, Birkhäuser, 1983. [2] C. Reiher: The Clique Density Theorem, Hamburger Beiträge zur Mathematik 462, pages 1-25, 2012.
Codruţ
Grosu
FU Berlin
On transversals avoiding complete subgraphs
Abstract.
Let G be a multipartite graph with each block of size b and maximum degree d. It was shown by Haxell that b ≥ 2d is a sufficient (and best possible) condition for G to contain an independent transversal. Szabó and Tardos studied a generalization of this problem, where one requires the transversal to be Ks-free instead of independent. They conjectured that the optimal lower bound for b is of the order of d/s. The purpose of this talk is to present an symptotic proof of this conjecture, due to Harris and Srinivasan. The proof uses a new version of the Local Lemma, which I will try to explain.
Kay
Rülling
FU Berlin
Vanishing of the higher direct images of the structure sheaf
Abstract.
Let f: X---> Y be a birational and projective morphism between excellent and regular schemes. Then the higher direct images of the structure sheaf of X under f, R^i f_* O_X, vanish for all positive integers i. In case X and Y are smooth schemes over a field of characteristic zero, this vanishing was proved by Hironaka as a corollary of his proof of the existence of resolutions of singularities. In case X and Y are smooth over a field of positive characteristic the statement was proved by Chatzistamatiou-Rülling in 2011. In this talk I will explain the proof in the general case. This is joint work with Andre Chatzistamatiou.
Evdokia
Nikolova
The Burden of Risk Aversion in Selfish Routing
Jan Hendrik
Bruinier
TU Darmstadt
Classes of Heegner divisors in generalized Jacobians
Abstract.
In parallel to the Gross-Kohnen-Zagier theorem, Zagier proved that the traces of the values of the j-function at CM points are the coefficients of a weakly holomorphic modular form of weight 3/2. Later this result was generalized in different directions and also put in the context of the theta correspondence. We recall these results and report on some newer aspects, which arise from considering classes of Heegner divisors in generalized Jacobians. This is joint work with Y. Li.
Isaac
Mabillard
Institute of Science and Technology Austria
What is a van Kampen obstruction cocycle?
Abstract.
The Kuratowski theorem provides a nice criterion for graph planarity, ie, to decide whether a simplicial $1$-complex can be embedded into $\mathbb{R}^2$. A natural generalization of the problem is to find a criterion to decide whether a simplicial $n$-complex $K$ can be embedded into $\mathbb{R}^{2n}$. This is what the van Kampen obstruction cocycle gives us. By using standard tricks in PL topology, one can show that $K$ is embeddable if and only if (the class of) its cocycle is zero. This is (maybe?) surprising because embeddability is a geometric question, whereas a cocycle is an algebraic object, but it still carries enough information to solve the geometric problem.
Linda
Kleist
Long Paths in Line Arrangements
Leif
Naundorf
FU Berlin
Generalized Schur-Horn orbitopes and zonoids
Abstract.
In this talk I will introduce a special class of convex bodies, the so-called orbitopes. We will take a closer look at a special class of orbitopes, namely the Schur-Horn orbitopes and their generalizations which can be seen as continuous generalizations of permutahedra. We will introduce the notion of a zonoid which is the continuous generalization of a zonotope. Since some permutahedra are zonotopes it is a natural question to ask if there are Schur-Horn orbitopes which are zonoids. We will see that this is in general not the case.
Juanjo
Rué
FU Berlin
Arithmetic removal lemmas and independent sets in hypergraphs
Abstract.
In the last years, several authors have studied sparse and random analogues of a wide variety of results in extremal combinatorics. Very recently, due to the work of Balogh, Morris, and Samotij, and the work of Saxton and Thomasson on the structure of independent sets on hypergraphs, several of these questions have been addressed in a new framework by using the so-called containers in hypergraphs. In this talk I will present how to use this technology together with arithmetic removal lemmas due to Serra, Vena and Kral in the context of arithmetic combinatorics. We will show how to get sparse (and random) analogues of well-known additive combinatorial results even in the non-abelian situation. This talk is based on a work (still in progress!) joint with Oriol Serra and Lluís Vena.
Maryna
Viazovska
HU Berlin
Strongly regular graphs from the geometrical point of view
Abstract.
A strongly regular graph with parameters (v, k, l, m) is a k-regular graph in which every pair of adjacent vertices has l common neighbors and every pair of non-adjacent vertices has m common neighbors. In this talk we will give an overview of the theory of these graphs. Also we will report on new non-existance results for strongly regular graphs.
Jürg
Kramer
On the analytic continuation of the heat kernel
Abstract.
In our talk we will present an approach of how to analytically continue the heat kernel associated to the Laplacian of quotient spaces of the hyperbolic plane associated to Fuchsian subgroups of the first kind of PSL(2,R).
Wang-Q
Lim
TU Berlin
What is sparse and redundantrepresentation modelingfor image processing?
Abstract.
Recently, sparse and redundant representation modeling has received extensive attention and shown to be quite effective in signal and image processing. This framework allows for the successful reconstruction of the true image from severely corrupted or undersampled data. In this talk, we will introduce an image reconstruction scheme based on this framework and show the effectiveness of such a scheme in various applications.
Joachim
Weickert
Image Compression with Differential Equations
Piotr
Micek
Posets and minors.
Victor M.
Buchstaber
Steklov Mathematical Institute, Moscow
Combinatorial 2-truncated cubes and applications
Abstract.
The talk is based on the results of Victor Buchstaber, Vadim Volodin, Mikhail Gorsky and Nikolai Erochovets. In the first part of the talk we will discuss the family of combinatorial polytopes that can be obtained from a cube by sequence of truncations of codimension 2 faces (2 - truncated cubes). Every 2-truncated cube is flag simple polytope. These polytopes have remarkable properties and this family contains classes of polytopes playing important roles in different areas of mathematics. The second part will be devoted to the results of the flag simple polytopes, which turns the operation of 2-truncation of the remarkable simple flag polytopes (dodecahedron, fullerenes and their multidimensional analogues) that are not 2-truncated cubes.
Christopher
Kusch
FU Berlin
Random Algebraic Constructions
Abstract.
The aim of this talk is to introduce Bukh's random algebraic construction of large graphs not containing a given bipartite subgraph. If time permits, we discuss this method in the context of graphs with few paths of prescribed length between any two vertices as done by Conlon.
Danylo
Radchenko
MPIM Bonn
Higher cross-ratios and functional equations for polylogarithms
Abstract.
A well-known conjecture of Zagier states that the value of a Dedekind zeta function of a number field at an integer m>1 can be expressed in terms of the m-th polylogarithm function. This conjecture remains widely open for all m>3. A general strategy for proving it was outlined by Goncharov, and the main ingredient involves constructing certain higher-dimensional generalizations of the classical cross-ratio. In this talk I will give a general definition of higher cross-ratios, show how they can be used to construct interesting functional equations for polylogarithms, and report on the recent progress towards proving Zagier's conjecture in case m=4.
December 2014
Jean-Philippe
Labbé
Lower bound for the discrepancy of triangulations of the square with an odd number of triangles
Abstract.
Paul Monsky proved in 1970 that it is impossible to dissect a square using an odd number of triangles all of which having the same area. Guenter Ziegler posed the problem to study the discrepancy of such triangulations, that is: as the number of triangles increases, study how close can dissections be to the ideal situation where all triangles would have equal area. In this talk, we will prove a doubly exponential lower bound using real algebraic geometry results bounding the distance between zeros of a multivariate polynomial. Further, we will describe a family of dissections which could achieve a exponentially decreasing discrepancy supported by experimental results.
Dimitrios
Thilikos
University of Athens
Erdős–Pósa property for minor and topological minor models
Abstract.
A class of graphs C satisfies the Erdős–Pósa property if there exists a function f such that, for every integer k and every graph G, either G contains k vertex-disjoint subgraphs each isomorphic to a graph in C, or there is a subset S of V(G) of at most f(k) vertices such that G ∖ S has no subgraph in C. Erdős and Pósa (1965) proved that the set of all cycles satisfies this property for all graphs with f(k) = O(k log k). Given a connected graph H, let M(H) be the class of graphs that contain H as a minor. Robertson and Seymour (1986) proved that M(H) satisfies the Erdős-Pósa property if and only if H is planar. When H is planar, finding the smallest possible function fM(H) has been an active area of research in the last years. In this talk we will survey some recent combinatorial and algorithmic results in this direction, and we will particularly focus on other variants of the Erdős–Pósa property such as the case where the k subgraphs have to be edge-disjoint or the case where M(H) is the class of graphs that contain a graph H as a topological minor. On going work with Dimitris Chatzidimitriou, Jean-Florent Raymond, Ignasi Sau.
Arne
Smeets
KU Leuven/Orsay
On the insufficiency of the étale Brauer-Manin obstruction
Abstract.
Since Poonen's construction of a variety X defined over a number field k for which X(k) is empty and the étale Brauer-Manin set X(\mathbf{A}_k)^\text{Br,ét} is not, several other examples of smooth, projective varieties have been found for which the étale Brauer-Manin obstruction does not explain the failure of the Hasse principle. All known examples are constructed using "Poonen's trick", i.e. they have the distinctive feature of being fibrations over a higher genus curve; in particular, their Albanese variety is non-trivial. In this talk, we construct examples for which the Albanese variety is trivial. The new geometric ingredient in our construction is the appearance of Beauville surfaces. Assuming the abc conjecture and using geometric work of Campana on orbifolds, we also prove the existence of an example which is simply connected.
Jonathan
Youett
FU Berlin
What is a Hamiltonian system?
Abstract.
Hamiltonian systems provide an elegant way to describe the evolution equations of certain mechanical systems. The corresponding flow mappings can be shown to be symplectic and usually possess conserved quantities like momentum or energy. In this talk we will introduce the basic concepts, first for Euclidean space and then extend them to finite dimensional manifolds.
Veit
Wiechert
Dimension and Standard Examples
Alberto
Abbondandolo
Ruhr U Bochum
A promenade on the two-sphere
Emily
McCullough
On the Unimodality of h^*-vectors for Lattice Parallelepipeds and Zonotopes
Abstract.
It is a famous open question whether the Ehrhart h^*-vector of integrally closed lattice polytopes is unimodal. This question was answered in the affirmative for lattice parallelepipeds by Schepers and Van Langenhoven. In a joint project with Matt Beck and Katharina Jochemko, we give a geometric proof of this argument and a combinatorial interpretation to the entries of the h^*-vectors in terms of permutation descent statistics on the symmetric group. Via a refinement of these descent statistics, the result is extended to include half-open parallelepipeds (closed parallelepipeds with any j adjacent facets removed). From this, together with results from Shephard, and Köppe and Verdoolaege, we show that the h^*-vector for lattice zonotopes is also unimodal. In this talk I introduce the main actors in our story, namely, permutation descent statistics (in a familiar and a refined variety), half-open unit cubes, lattice parallelepipeds and unimodular simplices. I will discuss the interplay of the geometry and the combinatorics and I will review our main results and motivate their proofs with illustrative examples.
Piotr
Micek
Jagiellonian University/TU Berlin
Posets and minors
Abstract.
Posets of height 2 may have arbitrarily large dimension. But posets of bounded height with somehow restricted cover graphs do have bounded dimension. (If you are about to give up this talk. Let me just mention that I will try to make it a kind of introductory into posets dimension theory) In 1977, Trotter and Moore proved that a poset has dimension at most 3 whenever its cover graph is a forest. In a series of papers it is proved that when a poset has bounded height and its cover graph is planar (2014), or has bounded treewidth (2015+), or most generally excludes a fixed graph as a topological minor (2015+), then the poset has bounded dimension. We conjecture that a restriction on bounded height, forbidding a long chain in a poset, may be relaxed to exclusion of two long incomparable chains. We support our feeling providing three theorems: (1) Every poset without two long incomparable chains whose cover graph has bounded pathwidth has bounded dimension; (2) For every poset without two long incomparable chains whose cover graph excludes a fixed graph as a topological minor, all standard examples in the poset are small; (3) Every interval order, i.e. (2+2)-free poset, whose cover graph excludes a fixed graph as a topological minor has bounded dimension.
Jan-Philipp
Kappmeier
Threesomes, Degenerates, and Love Triangles
Florian
Frick
TU Berlin
What is the ham sandwich theorem?
Abstract.
Some problems in discrete geometry can be solved using topological methods by exploiting inherent symmetries. We will discuss some examples of this and focus on the Ham sandwich theorem, which guarantees that $d$ finite measures with continuous density in Euclidean $d$-space can be simultaneously cut in half by an affine hyperplane.
Christoph
Standke
The list chromatic index of 1-factorable graphs.
Alexander
Hopp
Topologische Bucheinbettungen planarer Graphen.
Mónica
Blanco
Classification of lattice 3-polytopes with 5 or 6 lattice points
Abstract.
We undertake the complete classification of lattice 3-polytopes with n lattice points, modulo unimodular equivalence, for n=5 and n=6. We first argue that for each n there is only a finite number of (equivalence classes) of polytopes of lattice width larger than one and exactly n lattice points. Polytopes of width one are easy to classify, an infinitely many, so we concentrate on an exhaustive classification of those of width larger than one. For n=4, all empty tetrahedra have width one (classical result by White, Howe). For n=5 we show that there are exactly 9 different polytopes of width 2, and none of larger width. For n=6, we show that there are 74 classes of width 2, 2 of width 3, and none of larger width. We give explicit coordinates for representatives of each class, together with other invariants such as their oriented matroid and volume vector. Our motivation comes partly from the concept of distinct pair sum (or dps) polytopes, which are known to have at most 8 lattice points in dimension 3. Among the 9 + 74 + 2 classes mentioned above, exactly 9 + 44 + 1 are dps.
Cathryn
Supko
FU Berlin
Martin
Bright
Universität Leiden
The Brauer-Manin obstruction and reduction mod p
Abstract.
The Brauer-Manin obstruction plays an important role in the study of rational points on varieties. Indeed, for rational varieties (such as cubic surfaces) it is conjectured that this obstruction determines whether or not a variety has a rational point. I will give a brief introduction to the Brauer-Manin obstruction and then look at recent results relating it to the geometry of the variety at primes of bad reduction.
Ágnes
Cseh
An improved approximation algorithm for the stable marriage problem with one-sided ties
November 2014
Arnau
Padrol Sureda
FU Berlin
What is a (complex) hyperplane arrangement?
Abstract.
Every arrangement of real linear hyperplanes induces a decomposition of the sphere into cells. We will study the combinatorics of such a decomposition (explained by an oriented matroid) and define an analogous notion for complex hyperplane arrangements.
Stefan
Felsner
Independent and hitting sets of some classes of rectangle intersection graphs.
Anders
Björner
Random walks on complex hyperplane arrangements and self-organising libraries
Anders
Björner
KTH Stockholm
A q-analogue of the FKG inequality and some applications
Abstract.
The FKG inequality, due to Fortuin, Kasteleyn and Ginibre (1971), is a correlation inequality stemming from work in statistical mechanics. In this talk I will discuss some uses of the FKG inequality in extremal combinatorics, and I will present a polynomial generalization. The polynomial FKG inequality has applications to f-vectors of joins of simplicial complexes, to Betti numbers of intersection of certain Schubert varieties, and to power series weighted by Young tableaux. The talk will be quite elementary. No previous familiarity with these topics will be assumed.
Clément
Requilé
FU Berlin
A parameterized algorithm for the diameter improvement problem for plane graphs
Abstract.
A problem mentioned by Chung in 1987: given a plane graph and an integer d, is it possible to reduce the diameter of the graph to at most d by adding non-edges, while conserving planarity? Fellows and Dejter have shown in an unpublished paper in 1993 that the oriented case was NP-Complete and that both the oriented and undirected cases were FPT (FIxed Parameter Tractable). But their proof is mostly based on the Robertson and Seymour theorem and is therefore non-constructive as of yet. This type of result is in essence theoretical, but it gives us good hopes of finding efficient FPT algorithms. We will focus on a natural variant of this problem in the undirected case, where the number of edges that can be added per face is upper-bounded by an integer k. We will then give an FPT algorithm (parameterized by d and k) for this variant, build on dynamic programming. This talk is based on joint work with Dimitrios Thilikos (LIRMM, France/University of Athens).
Peter
Bruin
Universität Leiden
Optimal bounds for the difference between the Weil height and the Néron-Tate height for elliptic curves over Qbar
Abstract.
Consider an elliptic curve E over Qbar given by a Weierstraß equation with algebraically integral coefficients. For the purpose of computing Mordell-Weil groups, one would like to bound the difference between the two standard height functions on E(Qbar) (the Weil height and the Néron-Tate height) as sharply as possible. I will describe an algorithm that computes the infimum and the supremum of the difference between these height functions to any desired precision. The main source of difficulties are the Archimedean places; it turns out that these can be treated using the classical Weierstraß elliptic functions.
Kevin
Schewior
Faster Maximum-Flow Computation via Electrical Flows
Steve
Chaplick
Overlap Representations of Planar Graphs
Thomas
Stollin
FU Berlin
The Brunn-Minkowski Inequality for Zonotopes
Abstract.
The Brunn-Minkowski inequality is the most prominent example of a geometric volume inequality of convex geometry with far reaching implications. This inequality together with several other related inequalities like Minkowski’s First and Second Mixed-Volume inequalities and the Aleksandrov-Fenchel inequality are of particular interest for the field of Discrete Geometry. Several proofs arose out of the theory of polytopes exploiting relations of discrete structures and the continuous measures of volume and mixed volume. A particularly interesting class of polytopes in Discrete Geometry are zonotopes which are finite sums of segments. On the one hand, zonotopes feature strong combinatorial properties such as their relation to hyperplane arrangements that allow for relating the volume of the zonotope to volumes of projections and subzonotopes via the deletion and contraction. On the other hand, the interpretation of a zonotope as an affine image of a possibly higher-dimensional cube offers an association with rectangular matrices. This relation can be used to compute volumes and mixed volumes of zonotopes by applying means of linear algebra. Thus, zonotopes and their subclass of parallelotopes are a link between geometry, combinatorics, and linear algebra providing rich structure and easy computation of volumes. Hence, they are a natural choice for investigating details concerning Minkowski’s geometric volume inequalities. In this talk I will present results and observations obtained by following this approach in my master’s thesis.
Tuan
Tran
FU Berlin
A removal lemma for Kneser graphs
Abstract.
Given natural numbers n, k with 2 ≤ k < n/2. We write ([n] choose k) for the family of all subsets of size k of [n]. The Kneser graph K(n, k) is the graph whose vertex set is ([n] choose k ) where two sets A, B ∈ ([n] choose k ) are adjacent if and only if A ∩ B = ∅. The celebrated theorem of Erdős-Ko-Rado from 1961 says that every independent set in K(n, k) has size at most (n - 1 choose k - 1), and the only independent sets of this size are stars. We prove a removal lemma for these graphs. Loosely speaking, it tells us that every subfamily of ([n] choose k) of size roughly (n - 1 choose k - 1) with few disjoint pairs must be very close to a star. Armed with this lemma, we establish a sparse version of the Erdős-Ko-Rado theorem. This settles a question of Bollobás, Narayanan and Raigorodskii (2014). The talk represents joint work with Shagnik Das.
Benjamin
Göbel
Universität Hamburg
Arithmetic local coordinates and applications to arithmetic self-intersection numbers
Abstract.
In order to calculate the arithmetic self-intersection number of an arithmetic prime divisor on an arithmetic surface, we need to move the prime divisor by the divisor of a rational function. Since there is no canonical choice for the rational function, we may ask whether there is an analytic shadow of the prime divisor that replaces the geometric intersection number at the finite places on the arithmetic surface by an analytic datum on the induced complex manifold. This leads to the definition of an arithmetic local coordinate. In this talk we show that the arithmetic self-intersection number of an arithmetic divisor can be written as a limit formula using an arithmetic local coordinate. We also apply this idea to the intersection theory of H. Gillet and C. Soule and to the generalized intersection theory of J. I. Burgos Gil, J. Kramer and U. Kühn.
Fidaa
Abed
MPII
Optimal Coordination Mechanisms for Multi-Job Scheduling Games
Sebastian
Meinert
FU Berlin
What is a $\operatorname{CAT}(0)$ space?
Abstract.
A $\operatorname{CAT}(0)$ space is a metric space in which geodesic triangles are at most as thick as in Euclidean space. We will make this notion precise and explain some basic features of $\operatorname{CAT}(0)$ spaces, like contractibility.
Udo
Hoffmann
Point visibility graphs and the existential theory of the reals.
Stephanie B.
Alexander
UIUC
Alexandrov Geometry
Matthias
Beck
Tibor
Szabó
FU Berlin
On the minimum degree of minimal Ramsey graphs
Abstract.
A graph G is r-Ramsey for H if for any r-coloring of its edges there is a monochromatic copy of H. G is called r-Ramsey minimal for H if it is r-Ramsey for H, but no subgraph of it is r-Ramsey for H. We study the smallest minimum degree an r-Ramsey-minimal graph for the clique Kk can have. This function was introduced and determined for two colors by Burr, Erdős and Lovász in 1975. We extend their theorem for r colors and connect the function to the so-called Erdős-Rogers function, which was introduced and studied more than a decade earlier. This represents joint work with Jacob Fox, Andrey Grinshpun, Anita Liebenau, and Yury Person.
Olaf
Parczyk
FU Berlin
Relative entropy and Sidorenko’s conjecture
Abstract.
Following the paper by Balázs Szegedy we introduce the family of graphs H that admit a probability distribution of copies in an arbitrary graph G obtained by iterating conditionally independent couplings starting from the uniform distribution on edges. We investigate the famous conjecture by Erdős-Simonovits and Sidorenko inside this family. As a tool we use an inclusion exclusion type formula for relative entropy. The method gives a unified treatment for most known cases of the conjecture and it implies various new results as well.
Barbara
Jung
HU Berlin
An integral around the toroidal boundary of A2
Abstract.
To obtain the arithmetic degree of the Hodge bundle on a (toroidal) compactification of the Siegel modular variety A2 of degree two, an integral over the regularized star product of Green objects, corresponding to Siegel modular forms, has to be computed. By definition of this product and Stokes’ Theorem, an integral over an ε-tube around the boundary appears. As the integrand degenerates near the boundary, it is not clear that this converges. We will give an explicit description of the geometric situation and show that the integral is in fact going to zero for ε approaching zero.
Max
Klimm
The Complexity of the parity argument and other inefficient proofs of existence
October 2014
Kolja
Knauer
Orienting Triangulations.
Martin
Henk
TU Berlin
Geometry of Numbers: Convex Bodies and Lattice Points
Niels
Lindner
HU Berlin
Bertini theorems for simplicial toric varieties over finite fields
Abstract.
The classical Bertini theorems on generic smoothness and irreducibility do not hold for varieties over finite fields. However, by the work of Poonen and Charles, it is still possible to give a "probability" for smoothness or geometric irreducibility in certain linear systems on subvarieties of projective space over a finite field. Both versions extend to the context of projective simplicial toric varieties. As an application, one finds that "almost all" hypersurfaces in a nice simplicial toric variety are geometrically irreducible and have a finite singular locus, which is small compared to the degree.
Martin
Groß
A strongly polynomial algorithm for generalized flow maximization
Daniele
Agostini
HU Berlin
What is a syzygy?
Abstract.
In astronomy, a "syzygy" is an alignment of three celestial bodies. The term was imported into mathematics by Sylvester to denote a linear relation between the generators of a module. Nowadays, syzygies and, more generally, free resolutions are powerful tools to relate the algebra and geometry of a projective variety. In this talk I will introduce these concepts and present some concrete examples.
Jonatan
Krolikowski
Refined counting of linear extensions of posets.
Benjamin
Rahman
Würfelkonstaktdarstellungen von Graphen
Jannik
Matuschke
A short introduction to extended formulations
Volker
Remmert
U Wuppertal
Visual representations of the mathematical sciences in early modern Europe
Yann
Disser
The Power of a Pebble: Exploring and Mapping Directed Graphs
Martin
Skutella
Integer multi-commodity flows and the cut condition
September 2014
Torsten
Ueckerdt
On-line coloring between two lines.
August 2014
Udo
Hoffmann
Representations of point visibility graphs.
Tillmann
Miltzow
Token Swapping
Veit
Wiechert
Large dimensional posets with planar cover graphs
Linda
Kleist
On the complexity of recognizing unit proper contact graphs of unit regular polygons.
July 2014
Steve
Chaplick
Clique Separators in Neighbourhood Subtree Intersection Graphs
Ulrich
Reitebuch
FU Berlin
What is a gyroid?
Abstract.
The gyroid is a minimal surface which belongs to the family of the Schwarz P and Schwarz D surfaces. We will first have a short introduction to minimal surfaces and their symmetries and then see some minimal surfaces in visualizations and 3D models.
Myfanwy
Evans
U Erlangen
Geometry... in a material world...
Pablo
Soberón
U Michigan
Tverberg’s theorem and the Birkhoff polytope
Abstract.
Tverberg’s theorem describes the number of points needed in R^d to guarantee the existence of a partition of them into k sets whose convex hulls intersect. This result has led to many variations and extensions, among them what we call the “colorful versions”. In this talk we will discuss how some of these versions can be mixed together, where some conditions involve points in the Birkhoff polytope instead of usual convexity relations. No prior knowledge of the subject is required.
David
Ouwehand
HU Berlin
Local rigid cohomology of weighted homogeneous singularities
Abstract.
In 2005 Abbott, Kedlaya and Roe have given a method to compute the rigid cohomology of a smooth hypersurface. The goal of this talk is to explain how this method can be applied to the computation of the local rigid cohomology of weighted homogeneous singularities.
Jonas
Krone
FU Berlin
and
Zoi
Tokoutsi
FU Berlin
What is time-of-death modeling?
Abstract.
In forensics, there is a natural interest in determining the exact time of death of murder victims. Thus far this was done by measuring the victim's body temperature and extrapolating the time of death based on a rather crude physical model. Recently, there have been attempts to come up with more realistic models that take into account all the different body properties (height, weight, clothing, etc.) as well as environmental factors (outside temperature, wind, heat sources, etc.). Our goal is to develop a physically accurate, three-dimensional heat flow simulation for the body that can be adjusted to a multitude of scenarios, and estimate the time of death by a curve-fitting scheme for the heat curve.
Nieke
Aerts
Touching Triangle Representations of Biconnected Outerplanar Graphs
Alice
Händel
FU Berlin
Partielle Transversale in Lateinischen Quadraten
Abstract.
Ein lateinisches Quadrat der Ordnung n ist ein n x n Feld bestehend aus n2 Zellen, welche mit n verschiedenen Symbolen gefüllt sind, wobei in jeder Zeile und jeder Spalte jedes Symbol genau einmal vorkommen darf. Eine Menge aus n Zellen, genau eine aus jeder Zeile und jeder Spalte, bestehend aus k verschiedenen Symbolen, heißt partielles Transversal der Länge k. Im Rahmen dieses Seminarvortrages wird der Beweis von Shor und Hatami, dass jedes lateinische Quadrat der Ordnung n ein partielles Transversal der Länge mindestens n - 11,053 log2 n besitzt, bewiesen.
Hiệp
Hàn
Emory University
Maximum number of colourings without monochromatic Schur triples
Abstract.
We investigate subsets of finite abelian groups which maximize the number of r-colourings without monochromatic Schur triples, i.e. without triples of the form (a, b, c) such that a + b = c. For r = 2, 3 and abelian groups G of order |G| = 2 mod 3 we show that the maximum is achieved only by largest sum-free sets. For abelian groups of even order and r = 4 the maximum is achieved by the union of two largest sum-free sets, whenever they exist. Joint work with Andrea Jimenez.
Giovanni
de Gaetano
HU Berlin
A metric degeneration approach to a special case of the arithmetic Riemann - Roch theorem
Abstract.
In 2010 G. Freixas proved an arithmetic Riemann – Roch theorem for arbitrary powers of the bundle of cusp forms on modular curves equipped with log-singular metrics; his approach heavily relied on the properties of the moduli space of pointed Riemann surfaces. The goal of the talk is to introduce a different method to prove this theorem, namely the metric degeneration process, which does not require the existence of such a moduli space. Partial results and open problems of such a program will be widely discussed.
Lara
Skuppin
TU Berlin
What is a Riemannian manifold?
Abstract.
A Riemannian manifold is a smooth manifold together with a Riemannian metric giving rise to notions of geometric quantities such as length, angle, distance, volume and curvature. After discussing surfaces in $\R^3$, I will give the definition of a Riemannian manifold and some examples, state the Nash embedding theorem, and explain a few properties.
Christian
Bär
U Potsdam
Positive curvature in 4 dimensions
Ivan
Izmestiev
FU Berlin
Fitting the barycenters by a projective transformation
Abstract.
For two subsets A and B of a euclidean space, does there exist a projective transformation f such that the barycenters of f(A) and f(B) coincide? In the following situations f exists and is unique (up to an affine transformation): 1) A is the vertex set of a convex polytope, B a single point inside the polytope (related to polarity wrt a higher order algebraic curve); 2) A is a convex body, B a single point inside (this case was previously known and is related to the Blaschke-Santalo theorem on minimizing the volume of the polar dual); 3) A is the unit sphere, B a finite set of points on it (this is related to the uniqueness of edge-circumscribed realizations of 3-polytopes). If A and B are convex bodies, one sitting inside the other, then a projective transformation exists but is not always unique. It is unique, provided that B is "small enough" wrt A in the projective sense. A suitable transformation corresponds to a critical point of a certain functional, related to the moment of inertia of the classical mechanics.
Ana Maria
Botero
HU Berlin
The singularities of the invariant metric on the line bundle of Jacobi forms on the universal elliptic curve
Abstract.
A theorem by Mumford implies that an automorphic line bundle on a pure open Shimura variety, equipped with an invariant smooth metric, can be uniquely extended as a line bundle on a toroidal compactification of the variety acquiring only logarithmic singularities. This result is the key point of being able to compute arithmetic intersection numbers from these line bundles. It is then natural to ask, whether this result extends to mixed Shimura varieties. In this talk, we examine the case of the sheaf of Jacobi forms on the universal elliptic curve. We see that Mumford's theorem cannot be applied here since a new kind of singularities appear. However, we show that a natural extension in this case is a so called b-divisor. This extension is meaningful because it satisfies Chern-Weil theory and a Hilbert-Samuel type formula. This is work by J. I. Burgos, J. Kramer and U. Kühn.
June 2014
Lucas
Braune
IMPA Rio de Janeiro
What is the Atiyah-Singer Index Theorem?
Abstract.
The Atiyah-Singer index theorem is a general result that gives an integral formula for the index of an elliptic operator on a compact manifold. It has as immediate corollaries, fundamental theorems in different areas of geometry — theorems whose statements have seemingly nothing to do with an index. The main examples are the Chern-Gauss-Bonnet theorem, the Hirzebruch signature formula, and the Riemann-Roch-Hirzebruch theorem. My purpose for this talk is to show each of these theorems as a solution to an index problem and, with this as the motivation, to explain the statement of a version of the Atiyah-Singer index theorem.
Piotr
Micek
Pathwidth and nonrepetitive colorings of graphs
Alexey
Pokrovskiy
FU Berlin
Linkage structures in tournaments (part II)
Abstract.
Thomassen conjectured that there is a function f(k) such that every strongly f(k)-connected tournament contains k edge-disjoint Hamiltonian cycles. This conjecture was recently proved by Kuhn, Lapinskas, Osthus, and Patel who showed that f(k) < O(k2(log k)2) and conjectured that there was a constant C such that f(k) < Ck2. We'll discuss a proof of this conjecture, giving an overview of the Kuhn, Lapinskas, Osthus, and Patel proof and modifications which are needed to obtain the quadratic bound on f(k).
Slawomir
Cynk
Jagiellonian University Krakow, z.Zt. Universität Mainz
Picard-Fuchs operators for one parameter families of Calabi-Yau threefolds
Abstract.
I will present some methods of computation of Picard-Fuchs operators for one parameter families of Calabi-Yau threefolds. The motivation comes from the Mirror Symmetry and an attempt to classify Calabi-Yau threefolds with the Picard number equal 1. I will discuss examples with special properties: families without points of maximal unipotent monodromy (MUM) and families with several MUM points. This is joint research with D. van Straten (Mainz).
Giulia
Battiston
FU Berlin
What is the moduli spaces of curves?
Abstract.
Moduli spaces are constructed in order to parametrize specific classes of objects. We will explicitly construct moduli spaces for simple parametrization problems, and then introduce the moduli space of curves of genus g and discuss some of their basic geometric properties.
Rahul
Pandharipande
ETH Zurich
Cohomology of the moduli space of curves
Stefan
Felsner
Abstract Ham-Sandwich Cuts and Selection in Arrangements
Serkan
Hosten
San Francisco State University
The degree of the central curve in quadratic programming
Abstract.
For convex optimization problems, such as linear, quadratic, or semidefinite programming, a class of interior point algorithms track the so-called central path to an optimal solution. The central curve, the Zariski closure of the central path, is an algebraic curve and it has been recently studied by De Loera, Sturmfels, and Vinzant in the linear case. In particular, the degree of the central curve for linear programming has been computed, and this has implications for the complexity of the interior point algorithms. We tackle the next case, the degree of the central curve for quadratic programming. After a reduction to the 'diagonal' case, we conjecture a formula and present a strong case for the conjecture. Also, in the diagonal case, we construct a Groebner basis for the ideal defining the central curve.
Alexey
Pokrovskiy
FU Berlin
Linkage structures in tournaments
Abstract.
A (directed) graph is k-linked if for two sets of vertices {x1, ..., xk} and {y1, ..., yk}, there are vertex disjoint paths {P1, ..., Pk}, such that Pi goes from xi to yi. A theorem of Bollobas and Thomason says that every 22k-connected (undirected) graph is k-linked. It is desirable to obtain analogues for directed graphs as well. Although Thomassen showed that the Bollobas-Thomason Theorem does not hold for general directed graphs, for tournaments he showed that there is a function f(k) such that every strongly f(k)-connected digraph is k-linked. The bound on f(k) was reduced to O(k log k) by Kuhn, Lapinskas, Osthus, and Patel, who also conjectured that a linear bound should hold. We'll talk about a proof of this conjecture. The proof uses linkage structures in tournaments which were recently introduced by Kuhn, Lapinskas, Osthus, and Patel in order to prove a conjecture of Thomassen about edge disjoint Hamiltonian cycles in tournaments. As a consequence of our results we will also show that there is a constant C such that every Ck2 connected tournament contains k edge-disjoint Hamiltonian cycles, solving another conjecture of Kuhn, Lapinskas, Osthus, and Patel.
Miguel Grados
Fukuda
HU Berlin
Zeta functions of ray ideal classes and applications to Arakelov geometry
Abstract.
Pursuing asymptotic formulas for the self-intersection number of the relative dualizing sheaf of a modular curve, one encounters zeta functions associated to congruence subgroups. In this talk, we identify such zeta functions and those attached to ray ideal classes of real quadratic fields. The purpose of this identification is to obtain a residue formula at s=1 for the former class of zeta functions. The special case of $\\(Gamma(N)\\)$ is worked out in detail. If time allows, an idelic interpretation of this approach will be presented.
Leszek F.
Demkowicz
Finite elements for Maxwell equations
Karola
Meszaros
Cornell
h-polynomials of triangulations of flow polytopes
Abstract.
The h-polynomial of a simplicial complex is a way of encoding the number of faces of each dimension. I will introduce a multivariate generalization of the h-polynomials of unimodular triangulations of flow polytopes. The inspiration for this generalization lies in the subdivision algebra of flow polytopes, whose relations prescribe a way of subdividing flow polytopes. The multivariate generalization of the h-polynomials can be used to prove certain nonnegativity properties in the subdivision and related algebras.
Tamás
Mészáros
Central European University
Shattering extremal set systems
Abstract.
We say that a set system F ⊆ 2[n] shatters a given set S ⊆ [n] if 2S = {F ∩ S : F ∈ F}. One related notion is the VC-dimension of a set system: the size of the largest set shattered by F. The Sauer inequality states that in general, a set system F shatters at least |F| sets. A set system is called shattering-extremal if it shatters exactly |F| sets. Here we present two methods, one algebraic and one graph theoretical, to study shattering-extremal set systems. When considering a set system as a set of characteristic vectors, one can define the corresponding vanishing ideal of polynomials. The standard monomials and Gröbner bases of these ideals can be used to characterize extremal set systems and to obtain many interesting results in connection with these combinatorial objects. This algebraic characterization leads to an efficient method for testing extremality. The inclusion graph of a set system F ⊆ 2[n] is the labelled graph GF whose vertices are the elements of F, and there is an edge going from G to F with label j iff F = G ∪ {j}. GF is just the Hasse diagram of F labelled in a natural way. This graph theoretical point of view enables us to study the problem in a more familiar environment. We managed to characterize extremal set systems of VC-dimension at most 2 in terms of their inclusion graphs.
Michael
Joos
TU Berlin
What is an isohedral tiling?
Abstract.
A tiling is a tessellation of the plane and an isohedral tiling is a tiling in which all tiles are related by a symmetry of the tiling. One can categorize these tilings into 93 types. The definition leads to a compact symbolic encoding. We will focus on some computational aspects of isohedral tilings and also have a look at an implementation.
Sarah
Spoenemann
Kombinatorische Eigenschaften von bichromatischen Punktmengen und Arrangements.
Craig S.
Kaplan
U Waterloo
Metamorphoses and deformations of tilings
Timo
de Wolff
Sarbrücken
Nonnegative Polynomials and Sums of Squares Supported on Circuits
Abstract.
Nonnegativity of real polynomials is a key question in real algebraic geometry with crucial importance in polynomial optimization. In this talk we completely characterize sections of the cones of nonnegative polynomials and sums of squares with ‘polynomials supported on circuits’ -- a genuine class of sparse polynomials. In particular, nonnegativity is characterized by an invariant, which can be immediately derived from the initial polynomial using a new norm based relaxation strategy. Based on these results, we obtain an entire new class of nonnegativity certificates independent from sums of squares certificates. On the optimization side these results significantly extend known geometric programming approaches for the computation of lower bounds. These results generalize earlier works by Fidalgo, Ghasemi, Kovacec, Marshall and Reznick. The talk is based on joint work with Sadik Iliman.
Lander
Ramos
Universitat Politècnica de Catalunya
Random planar graphs with minimum degree two
Abstract.
We illustrate how to use the symbolic method to determine asymptotic parameters of a combinatorial class. In particular, we compute the asymptotic exponential growth of the class of planar graphs with minimum degree at least two, as well as the expected number of edges of a random graph from this class.
Siddarth
Sankaran
Universität Bonn
Towards an arithmetic Siegel-Weil formula
Abstract.
The Siegel-Weil formula, first discovered by Siegel in 1951, is a classical theorem that equates the integral of a theta function with a special value of an Eisenstein series. A beautiful series of papers by Kudla and Millson from the late 80's casts this theorem a geometric light, in terms of certain 'special cycles' on Hermitian symmetric domains and their quotients. Since then, evidence has emerged for a deeper arithmetic significance of this geometric interpretation; in many cases, there are deep and surprising connections between integral models of these cycles on the one hand, and derivatives of Eisenstein series on the other. In this talk, I will introduce this circle of ideas, which has come to be known as Kudla's programme, and in particular focus on recent developments in the context of unitary Shimura varieties.
May 2014
Udo
Hoffmann
(Pseudo) Vertex-Edge Visibility Graphs in Polygons.
Zavosh
Amir-Khosravi
HU Berlin
Double coset spaces for the compact unitary groups
Abstract.
I will speak about B. Gross's construction of a definite Shimura curve, its special points and height pairing, and the Gross-Zagier-type formula to which this leads. I will then describe an ongoing project to carry out a similar construction for the compact unitary group U(n), replacing special points by certain special cycles, and defining a corresponding notion of heights. One expects to obtain a generating series that's an automorphic form on U(r,r) with coefficients in Chow groups.
Miguel
Grados
HU Berlin
What is a partial zeta function?
Abstract.
Zeta functions are ubiquitous in number theory and usually its residue at s=1 contains relevant information of the problem in question. This talk is two-fold. First, we introduce two variants of the Riemann zeta function: one that appears in practice, and the other that is good for theoretical purposes. Second, we show an identity between these two variants and its consequences.
Irina
Mustaţă
Intersection graphs of unit squares and partial grid graphs
Alina
Marian
Northeastern U
Volumes of moduli spaces – will take place within the framework of the MAF Intensive Course @ HU
Melanie
Koser
TU Berlin
On a variational model for low-volume fraction microstructures in martensites - Properties of minimizers
Mirjam
Plietzsch
Durchmesser von höheren symmetrischen Shift-Graphen
Remke Nanne
Kloosterman
HU Berlin
The Hilbert function of the singular locus of hypersurfaces
Abstract.
Given a zero dimensional scheme $X$ in $P^n$, it is hard to determine if $X$ occurs as the singular locus of a degree d hypersurface. In this talk we use some results on the topology of singular hypersurfaces to obtain restrictions on the Hilbert function of the singular locus of a degree d hypersurface with isolated singularities. This result has several corollaries: It enables us to determine the Mordell-Weil rank of several isotrivial fibrations of abelian varieties. Moreover, it enables us to give constructions of Severi-Enriques varieties with dimension bigger than expected. For the final application let $p_1, ... p_t$ be points in $P^n$ and $ $m an integer. We give a non-trivial lower bound for the degree of a hypersurface in $P^n$ with $m$-fold points at the $p_i$ and no further singularities.
Veit
Wiechert
Balanced Pairs in Sets of Orderings with many initial Elements
Martin
Hairer
U Warwick
Taming infinities
Olaf
Parczyk
FU Berlin
On the logarithmic calculus and Sidorenko's conjecture
Abstract.
Sidorenko's conjecture states that if H is a bipartite graph then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. It is known to be true for various families of graphs, including trees, even cycles and bipartite graphs with one vertex complete to the other side. We take a look at the paper by Li and Szegedy where they establish the logarithmic calculus. This is their abstract: We study a type of calculus for proving inequalities between subgraph densities which is based on Jensen's inequality for the logarithmic function. As a demonstration of the method we verify the conjecture of Erdős-Simonovits and Sidorenko for new families of graphs. In particular we give a short analytic proof for a result by Conlon, Fox and Sudakov. Using this, we prove the forcing conjecture for bipartite graphs in which one vertex is complete to the other side.
Yuri
Bilu
Université de Bordeaux
Mini Course: The Subspace Theorem in Diophantine Analysis Part II
Abstract.
I will speak on the work of Corvaja, Zannier and others, on applying the Subspace Theorem to integral points on curves and surfaces.
Adam
Streck
FU Berlin
What is a random graph?
Abstract.
Unsurprisingly, the random graph is a graph that has been constructed in a random manner. However, the exact nature of this construction is of interest as it has been shown that many real-world networks share the structural properties of particular sorts of randomly constructed networks. We will focus on the two most common approaches, Erdős-Rényi and Barabási-Albert models, and show some well known emergent patterns of these approaches. We will also refresh some of the graph-related notions that will be discussed in the following talk.
Mario
Arioli
A gentle introduction to complex and quantum graphs and their applications
Ragnar
Freij
Polytopes from graph homomorphisms
Lothar
Narins
FU Berlin
Tau and the Connectedness of Line Graphs
Abstract.
The topological connectedness of the independence complex is a useful and important tool that has lead to such things as the proof of Ryser's Conjecture for 3-partite 3-graphs. In this talk, I will present a lower-bound on the connectedness of the line graph of a hypergraph in terms of the hypergraph's vertex-cover number (tau). The bound is tight for general hypergraphs, but there are potentially improvements to be made in the case of r-partite r-graphs. I will sketch the proof of a conjectured lower-bound improvement for 3-partite 3-graphs for tau at most 12.
Yuri
Bilu
Université de Bordeaux
Mini Course: The Subspace Theorem in Diophantine Analysis Part I
Abstract.
I will explain the statement of the Subspace Theorem of Schmidt and Schlickewei and will show some of its applications. In particular, I will prove the Adamcszewki-Bugeaud theorem on transcendence of automatic numbers, and the theorem of Corvaja-Zannier-Levin-Autissier on the non-density of integral points on algebraic surfaces.
Deryk
Osthus
University of Birmingham
Proof of Two Conjectures of Thomassen on Tournaments
Abstract.
We prove the following two conjectures of Thomassen on highly connected tournaments: (i) For every k, there is an f(k) so that every strongly f(k)-connected tournament contains k edge-disjoint Hamilton cycles (joint work with Kühn, Lapinskas and Patel). (ii) For every k, there is an f(k) so that every strongly f(k)-connected tournament has a vertex partition A, B for which both A and B induce a strongly k-connected tournament (joint with Kühn and Townsend). Our proofs introduce the concept of `robust dominating structures', which will hopefully have further applications. I will also discuss related open problems on cycle factors and linkedness in tournaments.
Stefan
Felsner
Generating and counting perfect matchings in bipartite regular graphs
April 2014
Juanjo
Rué
FU Berlin
Applications of Tutte's tree decomposition in the enumeration of bipartite graph families
Abstract.
We adapt the grammar introduced by Chapuy, Fusy, Kang and Shoilekova to study bipartite graph families which are defined by their 3-connected components. More precisely, in this talk I will explain how to get the counting formulas for bipartite series-parallel graphs (and more generally of the Ising model over this family of graphs), as well as asymptotic estimates for the number of such graphs with a fixed size. This talk is based in a work in progress joint with Kerstin Weller.
Ariane
Papke
FU Berlin
What is the Navier-Stokes equation?
Abstract.
The motion of a fluid can be described by the Navier-Stokes equations. We discuss the forces acting on a fluid element and explore some applications.
Yannik
Stein
The Complexity Class 'Polynomial-Time Local Search' (PLS)
Eduard
Feireisl
Czech Academy of Sciences
Interaction of scales in mathematical fluid dynamics
Christophe
Hohlweg
UQAM
Tonghai
Yang
University of Wisconsin, zzt. MPI Bonn
The story of $j$ and generalizations
Abstract.
The classical modular invariant $j$ has a long history. For example, it was long known that $j(\frac{D +\sqrt D}2)$ is an algebraic integer generating the Hilbert class field of the imaginary quadratic field $Q(\sqrt D)$. As early as 1920's, Berwick observed that the norm of the difference of some singular moduli, like $j(\frac{-163+\sqrt{-163}}2) -j(i)$, although very big, has a very small and interesting prime factorization. In 1985, Gross and Zagier confirmed this guess and gave a beautiful factorization form in general. In 1990s, Borcherds, in attempting to prove the celebrated Moonshine conjecture, discovered and proved a beautiful and surprising product formula for the modular function $j(z_1) -j(z_2)$. His idea of 2nd proof, the regularized theta lifting, can be used to prove Gross-Zagier's formula easily. Moreoever, the method is very soft and can be extended to give direct link between the central derivative of some L-series and the height pairing on some Shimura varieties of orthogonal/unitary type. In this talk, I will mainly focus on the proof of Gross-Zagier formula using regularized theta lifting (less notation and general concepts) after some background review. In the last 20-30 minutes, I will explain the extension.
Camilo
Sarmiento
Magdeburg
Dyck path triangulations and extendability
Abstract.
In this talk, we introduce the Dyck path triangulation of the cartesian product of two simplices: its maximal simplices are given by Dyck paths along with their orbit under a cyclic action. The construction also naturally produces triangulations of the product of two simplices consisting of rational Dyck paths. Our study of the Dyck path triangulation is motivated by an extendability problem for certain kinds of partial triangulations of the product of two simplices. We present a complete solution to this extendability problem and, with an explicit construction of non-extendable partial triangulations, we prove that our characterization of extendability is optimal. Time permitting, we will briefly mention interesting interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory.
Steve
Chaplick
On (K_{1,3},even-hole)-free graphs.
Linda
Kleist
On Area-Universal Contact Representations with Orthogonal Polygons.
Tibor
Szabó
FU Berlin
What is Ramsey-equivalent to a clique?
Abstract.
A graph G is Ramsey for H if every two-colouring of the edges of G contains a monochromatic copy of H. Two graphs H and H' are Ramsey-equivalent if every graph G is Ramsey for H if and only if it is Ramsey for H'. In the talk we discuss the problem of determining which graphs are Ramsey-equivalent to the complete graph Kk. In particular we prove that the only connected graph which is Ramsey-equivalent to Kk is itself. This gives a negative answer to a question of Zumstein, Zürcher, and the speaker. For the proof we determine the smallest possible minimum degree that a minimal Ramsey graph for Kk ⋅ K2 (the graph containing the clique with a pendant edge) can have. This represents joint work with Jacob Fox, Andrey Grinshpun, Anita Liebenau, and Yury Person.
Nieke
Aerts
Grid-Path Contact Representations.
March 2014
Simeon
Ball
Universitat Politècnica de Catalunya
On the Turán number of complete bipartite graphs
Udo
Hoffmann
Slopes of bipartite segment intersection graphs.
Katinka
Becker
Counting triangulations and related structures.
Piotr
Micek
Lower bounds for on-line graph colorings
Irina
Mustaţă
Opaque forests
Codruţ
Grosu
FU Berlin
An algebraic approach to Turán densities
February 2014
Stefan
Felsner
Bookembeddings and topological bookembeddings
Alexey
Pokrovskiy
FU Berlin
Reading Group: Independent Sets in Hypergraphs Part 5 - The KŁR Conjecture
Alexey
Pokrovskiy
FU Berlin
Reading Group: Independent Sets in Hypergraphs Part 4 - Applications
Ananyo
Dan
HU Berlin
What is the prime number theorem?
Veit
Wiechert
On the Dimension of Posets that do not have a large Standard Example as a subposet.
Johannes
Rauh
MPI Leipzig
Polytopes from Subgraph Statistics
Codruţ
Grosu
FU Berlin
Reading Group: Independent Sets in Hypergraphs Part 3 - The Proof Continued
Anna
von Pippich
TU Darmstadt
An arithmetic Riemann-Roch theorem for weighted pointed curves
Abstract.
In this talk, we report on work in progress with G. Freixas generalizing the arithmetic Riemann-Roch theorem for pointed stable curves to the case where the metric is allowed to have conical singularities at the marked points. We will first outline the main ideas of the proof and then focus on some analytical ingredients, e.g. the explicit computation of the regularized determinant for hyperbolic cusps and cones.
Steve
Chaplick
Graph Traversals and Restricted Families of Graphs
Tim
Dokchitser
University of Bristol
L-functions of curves
Abstract.
L-functions of elliptic curves have been studied a lot and their local invariants (local factors, conductors, Tate module, root numbers etc.) are well-understood, both theoretically and computationally. For curves of higher genus the situation is more complicated, and I will report on a joint work in progress with Vladimir Dokchitser that attempts to develop the corresponding theory and classification. Basically, there are two approaches to understand these L-functions, one using regular models and one using semistable models. I will explain what they are and what they can achieve, focussing in particular on hyperelliptic curves over the rationals.
January 2014
Yizhi
Sun
TU Berlin
What is a Carleson operator?
Abstract.
We recall some fundamental facts about Fourier series and different types of convergence. Then we will see how the Carleson operator and its boundedness work in the proof of Lusin's conjecture for $L^2$ functions.
Hannes
Pollehn
TU Berlin
What is an Ehrhard Polynomial?
Abstract.
A lattice polygon is a shape in the plane bounded by a sequence of line segments $\overline{v_1 v_2},\ldots,\overline{v_{k-1} v_k}$, where the vertices $v_i$ have integral coordinates. Georg Alexander Pick described a relation between the area of the polygon and the number of lattice points contained in the polygon by a formula which is now known as Pick's Theorem. Ehrhart later generalized this result to higher dimensions, which is now an important tool in the field of Geometry of Numbers.
Dennis
Clemens
FU Berlin
and
Lothar
Narins
FU Berlin
Reading Group: Independent Sets in Hypergraphs Part 2 - The Proof
Lilian
Pierce
Bringing the Carleson operator out of Flatland
Jean-Philippe
Labbé
Counting singletons and related structures
Miloš
Stojaković
University of Novi Sad
Threshold for Maker-Breaker H-game
Abstract.
We will look at the Maker-Breaker H-game played on the edge set of the random graph Gn,p. In this game two players, Maker and Breaker, alternately claim unclaimed edges of Gn,p, until all the edges are claimed. Maker wins if he claims all the edges of a copy of a fixed graph H; Breaker wins otherwise. It turns out that, with the exception of trees and triangles, the threshold for this game is given by the threshold of the corresponding Ramsey property of Gn,p with respect to the graph H. This is joint work with Rajko Nenadov and Angelika Steger.
Julia
Ehrenmüller
TU Hamburg-Harburg
On the intersection of longest paths
Abstract.
A longest path in a graph has the property that there exists no other path in the graph that is strictly longer. In 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. However, the answer to Gallai's question is positive for several well-known classes of graphs, e.g. trees, outerplanar graphs, split graphs, and 2-trees. In this talk we discuss these results and present a proof that all series-parallel graphs have a vertex that is common to all of its longest paths. We also comment on how this vertex can be found in quadratic time. Joint work with Cristina G. Fernandes and Carl G. Heise.
James
Stankewicz
University of Copenhagen
Palindromic Properties and Descent Obstructions
Abstract.
For most curves that you might think of, it is possible to find a twist which has a rational point. For the first time we exhibit an infinite family of curves over the rational numbers for which this explicitly does not apply. That is to say that we find Shimura curves C whose lack of rational points is palindromic or preserved by twists. Using this family of curves, we find a related set of twists of Shimura curves which all violate the Hasse Principle. This violation is explicitly given by a descent obstruction.
Victor
Falgas-Ravry
Umeå University
On the connectivity of k-nearest-neighbours random geometric graphs
Abstract.
Many real-life networks have an inherent geometry, which should be reflected in the random graph models used to analyse their behaviour. A typical example is that of radio networks: radio devices may find it easier to exchange information if they lie close to each other than if they lie far apart. This and other examples have motivated research into specifically geometric models of random graphs. In this talk I will present one such model, the k-nearest-neighbours random geometric graph model, and discuss its connectivity properties. I will present in particular a sharp threshold result, which is joint with Mark Walters (QMUL), as well as some more recent work of mine on subcritical components.
Linda
Kleist
On unit distance graphs.
Hao
Chen
FU Berlin
Juanjo
Rué
FU Berlin
Reading Group: Independent Sets in Hypergraphs Part 1 - Introduction
Faniry
Razafindrazaka
FU Berlin
What is a 4D object?
Abstract.
As human beings, we are only able to see 2D images. The extra dimension comes from our ability to understand shadows as depth information. While 3D objects have 2D shadows, 4D objects have 3D shadows. In this talk, we are going to explore, first, the 4D variant of the platonic solids together with their visualization and second, we will construct some surfaces embedded in 4-dimensional space.
Sylwia
Antoniuk
Adam Mickiewicz University
Random triangular groups
Jeff
Weeks
Visualizing Four Dimensions
Nieke
Aerts
Triangle contact representation of a planar graph and its dual.
Raman
Sanyal
FU Berlin
Upper Bound Problems and relative Stanley-Reisner theory
Abstract.
The Upper Bound Theorem (UBT) states that among all simplicial d-spheres on n vertices precisely the neighborly spheres maximize the f-vector componentwise. McMullen established the UBT for convex polytopes taking a very geometric approach via shellings. Stanley's generalization to simplicial spheres elegantly combines ideas from combinatorics, commutative algebra, and combinatorial topology. Together with Karim Adiprasito (IHES), we extend these ideas into an algebraic framework for treating relative simplicial complexes. This allows us to resolve upper bound problems of various types including a tight Upper Bound Theorem for Minkowski sums. In this (slightly informal) talk I will try to give a hands-on introduction to this 'relative Stanley-Reisner theory' and its applications.
Frank
Gounelas
HU Berlin
Cohomology of SRC varieties in positive characteristic
Abstract.
This talk will offer two different perspectives on the fact that a separably rationally connected variety in characteristic p has H^1(X, O_X)=0. Over C, this result follows from Hodge theory. The first proof comes from a result of Biswas-dos Santos on triviality of vector bundles on SRC varieties (following from deep recent results of Langer in char p), whereas the second is cohomological (etale and crystalline) in nature. I will introduce some of the necessary background theory and try to include full proofs.
Peter
Patzt
FU Berlin
What is the representation theory of finite groups?
Abstract.
A representation of a group is a linear group action. The representations of finite groups are well understood and their study led to new results formulated without representations. Also linear group actions occur naturally and representations theory yields more inside to the underlying vector spaces. We give an introduction to the representation theory of finite groups and some examples.
Piotr
Micek
On the number of edges in families of pseudo-discs
Benson
Farb
U Chicago
Braids, homology, permutations and polynomials – ZIB lecture hall
December 2013
Udo
Hoffmann
Slopes of k-dir graphs.
Jean-Philippe
Labbé
FU Berlin
Christian
Reiher
University of Hamburg
Counting odd cycles in locally dense graphs
Abstract.
Roughly speaking we prove that if a graph on n vertices has the property that each set of vertices whose size is linear in n spans at least as many edges as it spans in a random graph of density d, then the graph contains at least as many (odd) cycles of fixed length as the random graph does. With "at least" being replaced by "(almost) exactly" this is well known, but the present version seems to require a different approach. The result addresses a question of Y. Kohayakawa, B. Nagle, V. Rödl, and M. Schacht.
Bernhard
Schmitt
Nanyang Technological University, Singapore
Unique Sums and Differences in Finite Abelian Groups
Abstract.
Let A and B be subsets of a finite abelian group G. We say that A + B contains a unique sum if there exists g ∈ G such that g = a + b for exactly one pair (a, b) ∈ A × B. Unique differences are defined analogously. The main goal in the investigation of these phenomena is to find sufficient conditions for the existence of unique sums and differences. Such results have a variety of applications, for instance, in the context of cyclotomic integers of small modulus, field extensions, spectral properties of subsets of Fp, balanced sets, and circulant weighing matrices. I will present a new method to study unique sums and differences, which mainly relies on the Smith Normal Form of the corresponding linear relations. This yields substantial improvements upon previously known results. For instance, we obtain the following. Let G be a finite abelian group and let p be the smallest prime divisor of the order of G. If A and B are subsets of G with ∜(12)|A|+|B|-2 < p, then A + B contains a unique sum.
Wojciech
Gajda
Adam Mickiewicz University, Poznań
Independence of \\(ell\\)-adic representations
Abstract.
We discuss certain arithmetical properties of Galois representations attached to etale cohomology of algebraic varieties and schemes defined over finitely generated fields of any characteristic. The talk will contain a report on recent joint work with Gebhard Boeckle and Sebastian Petersen.
Juan Camilo
Orduz
HU Berlin
What is the Burger's equation and heat equations?
Abstract.
In this short talk we will describe two concrete examples of partial differential equations (PDEs): Burger's and the heat equation. We will mention some useful techniques to solve them: method of characteristics and Fourier theory.
Felix
Otto
MPI Leipzig
Playing with Partial Differential Equations
Irina
Mustaţă
Graceful labelings of graphs
Thilo
Rörig
TU Berlin
Kathleen
Johst
FU Berlin
The k-color Ramsey number of a graph with m edges
Abstract.
The k-color Ramsey number rk(H) of a graph H is the smallest integer N, such that in each k-coloring of the edges of the complete graph KN on N vertices there is a monochromatic copy of H. In my talk I show an upper bound rk(H) ≤ k2√(km)(1 + o(1)) for bipartite graphs H with m edges and no isolated vertices and k ≥ 2. Further, for a general graph H with m edges and no isolated vertices I discuss an upper bound rk(H) ≤ k3km2/3, for k ≥ 3.
Aprameyo
Pal
Uni Heidelberg, zzt. HU Berlin
Non-commutative Iwasawa theory and its applications
Abstract.
In Arithmetic Geometry one of the main themes has always been to understand the interplay between analytic invariants and algebraic invariants. One of the most famous example of this interplay is Birch and Swinnerton-Dyer conjecture. Iwasawa theory is one of the important tools which sheds some light on this issue. It provides a crucial link between the characteristic ideal of the Selmer groups (which are defined algebraically) and p-adic L-functions (which are defined analytically). In this talk, I will explain how non-commutative Iwasawa theory fits in the bigger picture. I will explain some results in function field and number field case. If time permits, I will sketch some proofs.
Eva
Martínez
FU Berlin
What is a Grassmannian?
Abstract.
Although the definition of a Grassmannian can be understood by anyone who has taken a course on linear algebra, it actually has a very rich and subtle structure and has become a very important tool in many fields. In this talk we will introduce the definition of a Grassmannian and some of its basic properties in a very elementary way with the aim to convince the audience about its significance.
Bernd
Sturmfels
Berkeley/Bonn
Quartic Spectrahedra
Abstract.
Spectrahedra are the feasible regions in semidefinite programming. This includes convex polyhedra, the feasible regions in linear programming. We present a tiny first step towards the classification of spectrahedra of a given degree and dimension, by focusing on the 24-dimensional family of all quartic spectrahedra in 3-space. These come in 20 generic types, according to the location of their nodal singularities. This lecture is based on joint work with John Christian Ottem, Kristian Ranestad, and Cynthia Vinzant. It intertwines convex geometry with classical algebraic geometry, and offers many colorful pictures.
Veit
Wiechert
On the Dimension of Posets that have a Cover Graph of bounded Treewidth
Peter
Bürgisser
TU Berlin
Intrinsic volumes and their applications in convex optimization
Abstract.
To a closed convex cone C \\subseteq R^d, one can assign the probability distribution v_0(C),...,v_d(C) of its (spherical) intrinsic volumes. These invariants arise naturally in spherical integral geometry. They are of relevance for understanding algorithms in convex optimization, e.g., for the probabilistic analysis of condition numbers. Numerous fascinating open questions involving spherical intrinsic volumes call for a solution: we conjecture that the sequence v_0(C),...,v_d(C) is always log-concave (a property euclidean inner volumes are known to satisfy). In the talk, I will report about a recent insight due to Amelunxen, Lotz, McCoy, and Tropp. They proved that the distribution of intrinsic volumes concentrates sharply around its mean \delta(C) := \sum_{k=0}^d k v_k(C), for which they coined the name "statistical dimension of C". This has important implications. The kinematic formula, a crowning achievement in integral geometry, implies that a random linear subspace of codimension m hits C nontrivially with the probability hit_C(m) = 2 ( v_{m+1}(C) + v_{m+3}(C) + ...). The concentration result for intrinsic volumes implies that for m >= \delta(C) + \lambda, the hitting probability hit_C(m) decays exponentially with \lambda, while for m <= \delta(C) - \lambda, hit_C(m) is exponentially close to 1. This threshold result enables a uniform treatment of phase transitions in convex optimization, as needed for compressed sensing.
November 2013
Matthias
Liero
WIAS Berlin
What is a Wasserstein distance?
Abstract.
Gradient flows are an important subclass of evolution equations. A gradient-flow structure can be exploited to obtain additional information about the evolution, such as existence and the stability of solutions. Moreover, gradient flows can provide additional physical and analytical insight, such as the maximum dissipation of entropy and energy, or the geometric structure induced by the dissipation distance. A special subclass is formed by gradient flows with respect to the Wasserstein distance. This class was first identified in the seminal work by Jordan, Kinderlehrer, and Otto in the late nineties. In my talk I will introduce the Wasserstein distance and discuss its relation to diffusion equations.
Kolja
Knauer
Connected Covering Numbers.
Martin
Burger
U Münster
Patterns and Minimal Models
Alexey
Garber
Moscow State University
Parallelohedra and the Voronoi Conjecture
Abstract.
A parallelohedron is a d-dimensional polytope which can tile the d-dimensional Euclidean space with translation copies. In 1909 Voronoi conjectured that every parallelohedron is an affine image of a Dirichlet-Voronoi polytope for some lattice. Since Voronoi stated his conjecture there were several results for different families of parallelohedra (G.Voronoi himself, O.Zhitomirskii, R.Erdahl, A.Ordine) but the conjecture remains unproved in the general case. In this talk we will discuss some of the mentioned results and the way they were achieved, also we will sketch the proof of the Voronoi conjecture in a new special case. Furthermore we will discuss some other problems related to parallelohedra theory and the Voronoi conjecture. This is joint work with A.Gavrilyuk and A.Magazinov.
Alexey
Pokrovskiy
FU Berlin
Nonnegative sums in a set of numbers
José
Burgos
ICMAT, Madrid
Equidistribution of small points on toric varieties
Abstract.
As the culmination of work of many mathematicians, Yuan has obtained a very general equidistribution result for small points on arithmetic varieties. Roughly speaking Yuan's theorem states that given a "very" small generic sequence of points with respect to a positive hermitian line bundle, the associated sequence of measures converges weakly to the measure associated to the hermitian line bundle. Here very small means that the height of the points converges to the lower bound of the essential minimum given by Zhang's inequalities. The existence of a very small generic sequence is a strong condition on the arithmetic variety because it implies that the essential minimum attains its lower bound. We will say that a sequence is small if the height of the points converges to the essential minimum. By definition every arithmetic variety contains small generic sequences. We show that for toric line bundles on toric arithmetic varieties Yuan's theorem can be split in two parts: a) Given a small generic sequence of points, with respect to a positive hermitian line bundle, the associated sequence of measures converges weakly to a measure. b) If the sequence is very small, the limit measure agrees with the measure associated to the hermitian line bundle.
Christoph
von Tycowicz
FU Berlin
What is the spacetime constraints paradigm?
Abstract.
Creating motions of objects or characters that are physically plausible and follow an animator's intent is a key task in computer animation. The spacetime constraints paradigm is a valuable approach to this problem. It computes optimal physical trajectories that are solutions of a variational spacetime problem. Such techniques calculate acting forces that minimize an objective functional while guaranteeing that the resulting motion satisfies prescribed spacetime constraints, e.g., interpolates a set of keyframes. Resulting forces are optimally distributed over the whole animation and show effects like squash-and-stretch, timing, or anticipation that are desired in animation. In this talk, we give an introduction to spacetime constraints and — if time permits — sketch a recent approach to interactive spacetime optimization for deformable objects.
Keita
Kunikawa
Tohoku University
What is mean curvature flow and its self-shrinkers?
Abstract.
This talk is a short introduction to mean curvature flow (MCF). MCF is one of the geometric flows (PDE on manifolds). Start with a generic initial surface having singularities. After rescaling near a singularity, we can see a self-shrinker of the flow. They are special solutions of MCF and they do not change their shape under the flow (up to scaling). Classifying self-shrinkers is an important topic of research. In general, however, it is impossible to do so without some additional conditions. I will explain a method using the Gauss map to approach this problem.
Christoph
Hansknecht
Methoden zur Berechnung von Tabellenkartogrammen
Stefan
Felsner
Basics of Multitriangulations.
Alexander
Maksimenko
k-Neighborly polytopes in combinatorial optimization
Abstract.
Definition: k-neighborly polytope is a convex polytope in which every set of k or fewer vertices forms a face. E.g., every pair of vertices of a 2-neighborly polytope is connected by an edge. It is known today that as a rule, polytopes of NP-complete problems have a 2-neighborly faces with large (superpolynomial) number of vertices. It is known also that 2-neighborly polytopes are more common than any other among random polytopes. These and some other facts force to look more carefully at the properties of such polytopes. We will try to shed light on this phenomenon and answer the main question: could the k-neighborliness be good complexity measure of a polytope.
Piotr
Micek
Jagiellonian University
Nonrepetitve sequences, colorings and entropy compression method
Abstract.
A sequence is nonrepetitive if it does not contain two adjacent identical blocks. The remarkable construction of Thue asserts that 3 symbols are enough to build an arbitrarily long nonrepetitive sequence. It is still not settled whether the following extension holds: for every sequence of 3-element sets L1, ..., Ln there exists a nonrepetitive sequence s1, ..., sn with si ∈ Li. We propose a new non-constructive way to build long nonrepetitive sequences and provide an elementary proof that sets of size 4 suffice confirming the best known bound. The simple double counting (entropy compression method) in the heart of the argument is inspired by the recent algorithmic proof of the Lovász local lemma due to Moser and Tardos. We will discuss recent results concerning nonrepetitive colorings of graphs with a special emphasis on applications of entropy compression method. Joint work with Jarosław Grytczuk and Jakub Kozik.
Remke Nanne
Kloosterman
HU Berlin
Alexander polynomials of curves and Mordell-Weil ranks of Abelian Varieties
Abstract.
Let C={f(z_0,z_1,z_2)=0} be a plane curve with ADE singularities. Let m be a divisor of the degree of f and let H be the hyperelliptic curve [ y^2=x^m+f(s,t,1).] defined over $\mathbb{C}(s,t)$. In this talk we explain how one can determine the Mordell-Weil rank of the Jacobian of H. For this we use the Alexander polynomial of C. This extends a result by Cogolludo-Augustin and Libgober for the case of where C is a curve with ordinary cusps. In the second part of the talk we sketch how the work of David Ouwehand can be used to obtain a similar result for Jacobians of curves over $\mathbb{F}_q(s,t)$.
Wayne
Lam
TU Berlin
What is the conformal geometry of surfaces?
Abstract.
A brief introduction to conformal geometry and its relation to complex analysis is shown. Some basic results, like the Riemann mapping theorem, are emphasized for their relations to the discrete theory. The goal is to give the audience a taste of discrete differential geometry.
Boris
Springborn
TU Berlin
Discrete conformal maps
Piotr
Micek
Making Octants Colorful
Frederik
Garbe
FU Berlin
On Graphs with Excess 2
Abstract.
The Moore bound m(d, k) = 1 + d Σi=1,...,k-1(d - 1)i is a lower bound for the number of vertices of a graph by given girth g = 2k + 1 and minimal degree d. Hoffmann and Singleton, Bannai and Ito, Damerell showed that graphs with d > 2 tight to this bound can only exist for girth 5 and degree 3, 7, 57. The difference to the Moore bound by given girth is called the excess of a graph. In the case of girth 5 Brown showed that there are no graphs with excess 1 and Bannai and Ito showed that for g ≥ 7 there are also no graphs with excess 1. We generalize the result of Kovács that, under special conditions for the degree d, there are no graphs with excess 2 and girth 5 and prove the new result that an excess-2-graph with odd degree and girth 9 cannot exist. In this proof we discover a link to certain elliptic curves. Furthermore we prove the non-existence of graphs with excess 2 for higher girth and special valences under certain congruence conditions.
Giovanni
De Gaetano
HU Berlin
Towards an arithmetic Riemann Roch theorem for non-compact modular curves
Abstract.
The goal of the talk is to survey the construction and the explicit computation of the Quillen metric on the determinant of cohomology of powers of the Hodge bundle on a modular curve, this corresponds to the definition and the computation of the left hand side of an analogue of Gillet and Soulè's arithmetic Riemann Roch theorem for smooth arithmetic surfaces. In specific we want to describe how the case of the first power of the Hodge bundle is problematic in this situation. If time permits we would like to discuss the right hand side of the formula in the already solved cases, and specify what our task in this context would be.
Deniz
Genlik
TU Berlin
What is the relation between Laplace transform and Z-transform?
Abstract.
The Laplace transform plays an important role in the analysis of continuous linear time invariant systems and the Z-transform does the same for Discrete Linear Shift Invariant Systems. In this talk, these systems will be briefly characterized and these transformations will be related to the characterizations. The relation between the Laplace transform and the Z-Transform will then be introduced, which is important for the analysis of mixed-type CLTI and DLSI Systems.
Hao
Chen
FU Berlin
Boyd--Maxwell ball packings
Abstract.
In the recent study of infinite root systems, fractal patterns of ball packings were observed while visualizing roots in affine space. We show that the observed fractals are exactly the ball packings generated by inversions described by Boyd and Maxwell, generalizing Apollonian ball packings. We describe the tangency graph of these packings in terms of Coxeter complexes, and list all Coxeter groups that generate Boyd--Maxwell packings. This is joint work with J-P Labbe.
Peter
Keevash
University of Oxford
Dynamic concentration of the triangle-free process
Abstract.
The triangle-free process begins with an empty graph on n vertices and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. We determine the asymptotic number of edges in the maximal triangle-free graph at which the triangle-free process terminates. We also bound the independence number of this graph, which gives an improved lower bound on Ramsey numbers: we show R(3, t) > (1/4 - o(1))t2/log t, which is within a 4 + o(1) factor of the best known upper bound. Furthermore, we determine which bounded size subgraphs are likely to appear in the maximal triangle-free graph produced by the triangle-free process: they are precisely those triangle-free graphs with maximal average density at most 2.
Codrut
Grosu
FU Berlin
What is a graph limit?
Abstract.
The goal of this talk is to define the notion of graph limit. I will present the relevant definitions and state the Lovász-Szegedy theorem characterizing graph limits. I will also compute a graph limit in a special case as an example, and as time permits, consider some generalizations of graph limits to digraphs, and different metric distances on graphs and graphons.
László
Lovász
Eötvös Loránd University Budapest
and
Katalin
Vesztergombi
Eötvös Loránd University Budapest
Limits of graphs and nondeterministic property testing
Michael
Kaufmann
Slanted Orthogonal Drawings.
October 2013
László
Lovász
Eötvös Loránd University
Extremal graphs and graph limits
Abstract.
Growing sequences of dense graphs have a limit object in terms of a symmetric measurable 2-variable function. A typical use of this fact in graph theory is the following: we want to prove a result, say an inequality between subgraph densities. We look at a sequence of counterexamples, and consider their limit. Often this allows clean formulations and arguments that would be awkward or impossible in the finite setting. This setting also allows us to pose and in some cases answer general questions about extremal graph theory: which inequalities between subgraph densities are valid, and what is the possible structure of extremal graphs.
Ana Maria
Botero
HU Berlin
Spherical varieties
Abstract.
First, we will introduce the notion of spherical varieties and discuss important subclasses (horospherical, toroidal) and many examples of them. We present their description by so-called colored fans and, finally, we show how the Tits fibration can be used to understand spherical varieties as T-varieties. Thus, colored fans turn into p-divisors. The latter is recent work by Klaus Altmann, Valentina Kiritchenko and Lars Petersen.
Uğur
Doğan
HU Berlin
What is a hyperreal number?
Abstract.
The set of hyperreal numbers is a field which contains $\mathbb R$ (real numbers) and some "infinitesimal" and "infinite" numbers. It will be constructed using model theory, so some basic facts and theorems in model theory will also be mentioned, such as Łoś's theorem and compactness of first order logic (and some applications if time permits).
Daniel
Heldt
A (q-analogon)^2 for the number of bounded Motzkin paths.
Tuan
Tran
FU Berlin
Subgraphs of dense multipartite graphs
Abstract.
Let H be a non-bipartite graph. Pfender conjectured that for large ℓ depending on H, every balanced ℓ-partite graph whose parts have pairwise edge density greater than (χ(H) - 2)/(χ(H) - 1) contains a copy of H. He verified the conjecture for cliques. In this work, we show that the same premise implies the existence of much larger graphs. As a consequence, we confirm the conjecture for a family of graphs. Finally, we disprove the conjecture in general by constructing several families of counterexamples. This is a joint work with Lothar Narins.
Jürg
Kramer
HU Berlin
Effective bounds for Faltings's delta function
Abstract.
In his seminal paper on arithmetic surfaces Faltings introduced a new invariant associated to compact Riemann surfaces. For a given compact Riemann surface X of genus g, this invariant is roughly given as minus the logarithm of the distance of the point in the moduli space of genus g curves determined by X to its boundary. In our talk, we will first give a formula for Faltings's delta function for compact Riemann surfaces of genus g>1 in purely hyperbolic terms. This formula will then enable us to deduce effective bounds for Faltings's delta function in terms of the smallest non-zero eigenvalue and the shortest closed geodesic of X. If time permits, we will also address a question of Parshin related to bounding the height of rational points on curves defined over number fields.
Louis
Theran
FU Berlin
What is combinatorial rigidity?
Abstract.
A bar-joint framework is a structure made of fixed-length bars connected by joints with full rotational degrees of freedom. The allowed continuous motions preserve the lengths and connectivity of the bars. If all the allowed motions come from Euclidean isometries, the the framework is rigid and otherwise it is flexible. Combinatorial rigidity theory is concerned with how much geometric information about a framework (e.g., if it is rigid or what its motions are like) from just the graph that has as its edges the bars. In this talk, I'll introduce frameworks in more detail and discuss some basic results and techniques in the area.
Ileana
Streinu
Smith College
Degrees of Freedom
Nathan
Williams
Minnesota
Cataland
Abstract.
I will talk about two combinatorial miracles relating purely poset-theoretic objects with purely Coxeter-theoretic objects. The first miracle is that there are the same number of linear extensions of the root poset as reduced words of the longest element (occasionally), while the second is that there are the same number of order ideals in the root poset as certain group elements (usually). I will conjecturally place these miracles on remarkably similar footing and examine the generality at which we should expect such statements to be true.
Björn
Kapelle
Anwendungen der Entladungsmethode
Manuel
Hensel
Layer Aware Codes
Juanjo
Rué
FU Berlin
On the probability of planarity of a random graph near the critical point
Abstract.
Consider the uniform random graph G(n, M) with n vertices and M edges. Erdős and Rényi (1960) conjectured that the limit lim P[G(n, n/2) is planar] exists and is a constant strictly between 0 and 1. Łuczak, Pittel and Wierman (1994) proved this conjecture and Janson, Łuczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this work we determine the exact probability of a random graph being planar near the critical point M = n/2. More precisely, for each u, we find an exact analytic expression for p(u) = lim P[G(n, n/2(1 + u n-1/3) is planar] In particular, we obtain p(0) is approximately equal to 0.99780. Additionally, we are also capable to extend these results to classes of graphs closed under taking minors. This is a joint work with Marc Noy (UPC-Barcelona) and Vlady Ravelomanana (LIAFA-Paris).
Linda
Kleist
Plane Cubic Graphs and the Air-Pressure Method
Stefan
Felsner
Evaluating Determinants Combinatorially
September 2013
Udo
Hoffmann
Extending partial representations of circle graphs
Matthias
Köppe
UC Davis
Intermediate sums on polyhedra and real Ehrhart theory
Abstract.
We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449–1466]. For a given polytope P with facets parallel to rational hyperplanes and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope P parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step polynomials. These formulas are naturally valid for REAL (not just integer) dilations and thus provide a direct approach to real Ehrhart theory. The algorithms are polynomial time in fixed dimension. Following A. Barvinok (2006), the intermediate sums also provide an efficient algorithm to compute, for a fixed number k, the highest k Ehrhart coefficients in polynomial time if P is a simplex of varying dimension. The talk is based on joint papers with Velleda Baldoni, Nicole Berline, Jesus De Loera, and Michele Vergne.
Laura
Gellert
Gradsequenzen von gerichteten Graphen.
Nieke
Aerts
Henneberg Steps for Triangle Representations
Arnau
Padrol
FU Berlin
A universality theorem for projectively unique polytopes and a conjecture of Shephard
Abstract.
In this talk I will present the following universality theorem for projectively unique polytopes: every polytope described by algebraic coordinates is the face of a projectively unique polytope. This result can be used to construct a combinatorial type of polytope that is not realizable as a subpolytope of any stacked polytope. This disproves a classical conjecture in polytope theory, first formulated by Shephard in the seventies. This is joint work with Karim Adiprasito.
August 2013
Katharina
Jochemko
An order-theoretic generalization of Polya's enumeration theorem
Irina
Mustaţă
Untangling two systems of non-crossing curves
Michael
Dobbins
POSTECH
Convexly independent subfamilies of convex bodies
Abstract.
A theorem of Erdős and Szekeres says that for any $n$ a sufficiently large family of point in the plane in general position will have $n$ points in convex position. In this talk I will show how this generalizes to families of convex bodies in the plane, provided that the number of common supporting tangents of each pair of bodies is bounded and every subfamily of size 5 is in convex position. This confirms a conjecture of Pach and Tóth. This is joint work with Andreas Holmsen, Alfredo Hubard.
July 2013
Daniel
Heldt
For some $\alpha$ the face flip markov chain on the set of $alpha$-orientations is not rapidly mixing on planar triangulations of degree \leq 7
Trung
Ngo Viet
Bhattacharya function of complete monomial ideals in two variables
Abstract.
Let R be a polynomial ring in d variables over a field k. Let I,J be two M-primary homogeneous ideals, where M is the maximal homogeneous ideal. Then one can consider the length function dim_k(R/I^mJ^n) for nonnegative integers m, n. Bhattacharya proved that this function is a polynomial P(m,n) of for m, n large enough. In general, it is very difficult to compute the Bhattacharya polynomial. If I, J are monomials, the computation of the Bhattacharya function can be reduced to a problem of counting integral points of Minkowski sum of polytopes. In this talk I will show how to compute the Bhattacharya function in the case I,J are complete monomial ideals in two variables. Our results show that the coefficients of the Bhattacharya polynomial can be explicitly expressed in terms of the vertices of the Newton polyhedra of I and J. This is a report on a joint work with Hong Ngoc Binh (Hanoi, Vietnam).
Daniel C.
Cohen
On the topology of matrix conguration spaces
Abstract.
We discuss some topological aspects of matrix configuration spaces, certain generalizations of the classical configuration space of n distinct ordered points in the plane. This is part of work in progress with Benson Farb.
Veit
Wiechert
Unit Interval Graphs and Balanced Pairs
Carsten
Gräser
FU Berlin
What is a variational inequality?
Abstract.
Solutions of optimization problems can be characterized nicely by variational equations if their associated functional is smooth. For nonsmooth functionals, variational inequalities provide a useful generalization while only requiring that part of the functional is differentiable. After discussing the relation of minimization problems to variational inequalities, I will show how we can use the latter to extract some important features of such problems.
Barbara
Wohlmuth
TU Munich
Numerical analysis for computational science
Nieke
Aerts
Decompositions of optimal 1-planar graphs
Adrian
Vasiu
U Binghamton
Generalized Serre-Tate Ordinary Theory
Abstract.
In the late 60's, Serre and Tate developed an ordinary theory for abelian varieties over a perfect field of positive characteristic. The ordinary locus corresponds to the generic, dense, open stratum of the Newton polygon stratification of the moduli spaces of principally polarized abelian varieties of a fixed dimension. In this talk, we report on a generalized Serre-Tate ordinary theory that involves Shimura-ordinariness and Uni-ordinariness, in both abstract and geometric contexts. In particular, geometric applications to the study of special fibres of integral canonical models of Shimura varieties of Hodge type and the Shimura-ordinary loci will be presented.
Ngô Việt
Trung
VAST, Vietnam
Bhattacharya function of complete monomial ideals in two variables
Abstract.
Let R be a polynomial ring in d variables over a field k. Let I,J be two M-primary homogeneous ideals, where M is the maximal homogeneous ideal. Then one can consider the length function dim_k(R/I^mJ^n) for nonnegative integers m, n. Bhattacharya proved that this function is a polynomial P(m,n) for m, n large enough. In general, it is very difficult to compute the Bhattacharya polynomial. If I, J are monomials, the computation of the Bhattacharya function can be reduced to a problem of counting integral points of Minkowski sum of polytopes. In this talk I will show how to compute the Bhattacharya function in the case I,J are complete monomial ideals in two variables. Our results show that the coefficients of the Bhattacharya polynomial can be explicitly expressed in terms of the vertices of the Newton polyhedra of I and J. This is a report on a joint work with Hong Ngoc Binh (Hanoi, Vietnam).
Giovanni
Conforti
HU Berlin
What is a self-avoiding random walk?
Abstract.
In this talk we will give an informal introduction to random walks on graphs and see how typical quantities of interest can be computed with the help of the Markov property. We will also see how, in the case of the self avoiding random walk, very basic questions remain unanswered and the Markov property, in the usual sense, is lost. Motivating examples from polymer science and particular graphs and lattices where the self avoiding random walks have been intensively studied will be presented.
Mireille
Bousquet-Melou
LaBRI
Self-avoiding Walks
Stefan
Felsner
Generalized Schnyder woods and box contact graphs in higher dimensions
Barbara
Jung
HU Berlin
The star product of Green currents on the Siegel modular variety of degree two
Abstract.
To obtain the arithmetic degree of the Hodge bundle on (a compactification of) the Siegel modular variety A_2 of degree two, the fourfold intersection product of the bundle of modular forms equipped with the Petersson metric has to be computed. This leads to an integral over the regularized star product of corresponding Green currents on A_2. Choosing appropriate currents, we obtain a decomposition in computable integrals over cycles on A_2. We will carry out this decomposition, evaluate the integrals and compare with the expected value.
June 2013
Franz
Király
TU Berlin
What is matrix completion?
Abstract.
Matrix completion is the task of reconstructing missing entries in a matrix of known rank which has gained wide attention through the \$1,000,000 NetFlix prize and the subsequent class action lawsuit. In the talk I will briefly explain the problem and its many interrelated connections to different fields of mathematics and computer science such as statistics, machine learning, convex optimization, functional analysis, algebraic geometry, commutative algebra, graph theory and combinatorics — highlighting some interesting results and viewpoints which can serve as different but related starting points for approaching the problem of matrix completion.
Linda
Kleist
On the queue and track number of planar graphs
Debajyoti
Mondal
Acyclic Coloring with Few Division Vertices
David
Ouwehand
HU Berlin
Rigid cohomology at singular points and the computation of zeta functions
Abstract.
The topic of this talk is the computation of the action of Frobenius on the rigid cohomology of the complement of certain singular projective hypersurfaces over a finite field. Abbott, Kedlaya and Roe have developed a method that solves this problem for smooth hypersurfaces, but for singular hypersurfaces there are still many open questions left. We start by introducing a notion of equivalence of singularities for varieties over finite fields. This notion has the advantage that two equivalent singularities have isomorphic local rigid cohomology. Then we discuss a method for dealing with varieties having weighted homogeneous singularities by using an idea of Dimca. This is where the understanding of the local cohomology at singular points plays a key role.
Christian
Kreusler
TU Berlin
What is abstract functions and weak convergence?
Abstract.
Most of the modern theory of Partial Differential Equations is formulated via the help of abstract functions, i.e., functions taking values in Banach spaces of infinite dimension. We will discuss the spaces of those functions and problems that arise when working in infinite dimensions such as lack of compactness. The concept of weak convergence is one of the major tools of interest.
Thibault
Manneville
Counting edges of linear extension graphs
Etienne
Emmrich
TU Berlin
On monotone operators, evolution equations, and existence via discretization
Julia
Böttcher
London School of Economics
A Blow-up Lemma for arrangeable graphs
Abstract.
The Blow-up Lemma is an important tool for embedding spanning graphs H into dense graphs G. One of its limitations is, however, that it can only be used if the maximum degree of H is bounded by a constant. We recently established a generalisation of this lemma which can handle even H which are only c-arrangeable (and observe a very weak maximum degree bound). Here a graph is called c-arrangeable if its vertices can be ordered in such a way that the neighbours to the right of any vertex v have at most c neighbours to the left of v in total. In the talk I will explain this lemma, discuss several applications, and outline parts of the proof. Joint work with Yoshiharu Kohayakawa, Anusch Taraz and Andreas Würfl.
Lejla
Smajlovic
U Sarajevo
On Maass forms and holomorphic modular forms on certain moonshine groups
Abstract.
We present results on analytical and numerical study of Maass forms and holomorphic modular forms on moonshine groups of level N, where N is a squarefree positive integer. We derive "average" Weyl's law for the distribution of discrete eigenvalues of Maass forms from which we deduce the "classical" Weyl's law. The groups corresponding to levels N=5 and N=6 have the same signature; however, our analysis shows that there are infinitely more cusp forms for N=5. Furthermore, we deduce a Kronecker limit formula for parabolic Eisenstein series and express the "Kronecker limit function" as a geometric mean of product of classical eta functions. We also study holomorphic forms non-vanishing at the cusp and discuss the construction of the j-function(s).
Irina
Mustaţă
On visibility and k-visibility graphs
Louis H.Y.
Chen
National University of Singapore
From functional equations to probability approximations: The amazing Stein's Method
Dmitry
Shabanov
Moscow State University
Equitable two-colorings for simple hypergraphs
Abstract.
Equitable two-coloring for a hypergraph is a proper vertex coloring such that the cardinalities of color classes differ by at most one. The famous Hajnal-Szemerédi theorem states that for any graph G with maximum vertex degree d there is an equitable coloring with d + 1 colors. In our talk we shall discuss similar question for uniform hypergraphs and present a new bound in the Hajnal-Szemerédi-type theorem for the class of simple hypergraphs. The proof is based on the random recoloring method and the results of Lu and Székely concerning negative correlations in the space of random bijections.
Andrey
Kupavskii
Moscow State University
Two notions of unit distance graphs
Abstract.
A complete (unit) distance graph in Rd is a graph whose set of vertices is a finite subset of the d-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A (unit) distance graph in Rd is any subgraph of such a graph. We show that for any fixed d the number of complete distance graphs in Rd on n labelled vertices is 2(1 + o(1))dn log2n while the number of distance graphs in Rd on n labelled vertices is 2(1 - 1/⌊d/2⌋ + o(1))n2/2.
Elias
Pipping
FU Berlin
What is a subgradient?
Abstract.
The subgradient of a convex function can be seen as a generalisation of the gradient in that it does not require differentiability. After a brief introduction, I will say a word or two about its connection to nonsmooth constitutive laws and the Legendre transform.
Udo
Hoffmann
Interval Dimension and Bipartite Graphs
May 2013
Stefan
Felsner
GIGs and Dimension
David
Eisenbud
MSRI
Syzygies from Cayley to Kontsevich and beyond
Shaoul
Zemel
TU Darmstadt
A Gross-Kohnen-Zagier Type Theorem for Higher-Codimensional Heegner Cycles
Abstract.
The multiplicative Borcherds singular theta lift is a well-known tool for obtaining automorphic forms with known zeros and poles on quotients of orthogonal symmetric spaces. This has been used by Borcherds in order to prove a generalization of the Gross-Kohnen-Zagier Theorem, stating that certain combinations of Heegner points behave, in an appropriate quotient of the Jacobian variety of the modular curve, like the coefficients of a modular form of weight 3/2. The same holds for certain CM (or Heegner) divisors on Shimura curves. The moduli interpretation of Shimura and modular curves yields universal families (Kuga-Sato varieties) over them, as well as variations of Hodge structures coming from these universal families. In these universal families one defines the CM cycles, which are vertical cycles of codimension larger than 1 in the Kuga-Sato variety. We will show how a variant of the additive lift, which was used by Borcherds in order to extend the Shimura correspondence, can be used in order to prove that the (fundamental cohomology classes of) higher codimensional Heegner cycles become, in certain quotient groups, coefficients of modular forms as well. Explicitly, by taking the $m$th symmetric power of the universal family, we obtain a modular form of the desired weight 3/2+m.
Lydia
Scheel
Die Max-Algebra und ihre Anwendung in der Graphentheorie und Optimierung
Banafsheh
Farang-Hariri
HU Berlin
Local and global theta correspondence
Abstract.
I will start by describing the local theta correspondence also known as Howe correspondence over a local non-archimedean field. This correspondence relates different class of representations of two reductive groups (G,H) called a dual pair by means of the Weil representation. Then I will talk about the global theta correspondence and its relation with the local one. I will also explain the link between global theta correspondence and theta series in the theory of automorphic forms.
Nieke
Aerts
Not all (2,2)-tight graphs have an L-Contact representation, nor an L-Contact representation allowing degenerate L-shapes.
Henrik
Schrezenmaier
Ein Luftdruckparadigma zur Optimierung von Zerlegungen
Stefan
Keil
HU Berlin
Non-square order Tate-Shafarevich groups of non-simple abelian surfaces over the rationals
Abstract.
For an elliptic curve (over a number field) it is known that the order of its Tate-Shafarevich group is a square, provided it is finite. In higher dimensions this no longer holds true. We will present work in progress on the classification of all occurring non-square parts of orders of Tate-Shafarevich groups of non-simple abelian surfaces over the rationals. We will prove that only finitely many cases can occur. To be precise only the cardinalities k=1,2,3,5,6,7,10,13,14,26 are possible. So far, for all but the last three cases we are able to show that these cases actually do occur by constructing explicit examples.
Philipp
Bannasch
HU Berlin
Thetadivisoren auf elliptischen Modulflächen und ihr Schnittverhalten
Thomas
Hixon
Hook Graphs and More
Ariane
Beier
University of Potsdam
What is the Nash embedding theorem?
Abstract.
The notion of a Riemannian manifold evolved from more concrete objects like surfaces in three-dimensional Euclidean space. But how much more general is this concept? Nash's embedding theorem gives one answer to the question whether or not a Riemannian manifold can be isometrically embedded into Euclidean space. It provides surprising and, at first glance, inconsistent results which we want to illustrate by considering the example of a flat torus.
László
Székelyhidi
U Leipzig
Bending Surfaces and Turbulent Energy Cascades
Linda
Kleist
On a Geometric Partitioning Problem.
April 2013
Giovanni
De Gaetano
HU Berlin
Remarks on Determinants of Laplacians on Riemann Surfaces
Miguel
Grados
HU Berlin
What is an $L$-function?
Abstract.
$L$-functions are analytical objects containing meaningful information for the underlying context in which they are defined. For example, take the Riemann zeta function; since it can be written in terms of primes, it will encode arithmetical information of $\mathbb Z$. Also, the proof of the prime number theorem was possible thanks to the nice properties of $L$-functions. In this talk we will take a quick tour through different $L$-functions appearing in mathematics. We will point out some features they have in common, and finally we will arrive to an approach of what an $L$-function should be (in a broad context).
Irina
Mustaţă
On the Recognition of Orthogonal Ray Graphs
Niels
Lindner
HU Berlin
Bertini's theorem for weighted projective space over a finite field
Abstract.
The classical Bertini theorem states that a general hypersurface in complex projective space is smooth. Given a smooth subvariety X of projective space over a finite field, one can actually calculate the fraction of hypersurfaces whose intersection with X is again smooth. This number can be expressed in terms of the Zeta function of X. The question makes also sense when replacing 'projective space' with 'weighted projective space' and 'smooth' with 'quasismooth'. However, the nature of weighted projective space raises some new difficulties.
Max
Pfeffer
TU Berlin
What is a higher order tensor decomposition?
Abstract.
Low-rank matrix structures can be exploited in many ways. We seek to generalize this concept to higher order tensors by generalizing the Singular Value Decomposition (HOSVD). Several ideas have been put forward, each proving to have certain advantages and disadvantages. Different tensor decompositions will be briefly discussed and the more recent approach of TT tensors will be introduced. The alternating least squares (ALS) algorithm will be presented as one of the most basic yet reliable tools in tensor optimization.
Reinhold
Schneider
TU Berlin
Hierarchical Tensor Product Representations
Maurice
Liebner
Von perfekten Matchings zu stabilen Hochzeiten
Isabel
Beckenbach
Hall Bedingung für Hypergraphen
Veit
Wiechert
The Balanced Pair Conjecture for Antimatroids
Jan
Bruinier
TU Darmstadt
Heights of Kudla-Rapoport divisors and derivatives of L-functions
Udo
Hoffmann
Clique Minimal Separator Decomposition and Intersection Graphs of Trees in a Cactus.
Andrew
McDowell
Royal Holloway, University of London
Non-vertex-balanced factors in random graphs
Abstract.
For a fixed graph H, and a (larger) graph G, an H-factor of G is a vertex disjoint collection of copies of H, that cover all the vertices of G. The simplest example is when H is a single edge, where an H-factor is just a perfect matching. We are interested in the threshold functions for the existence of an H-factor in the Erdős-Rényi random graph The case where H is a single edge has been known since 1964 but the case where H is a triangle is far more difficult and remained unsolved until the 2008 paper by Johansson, Kahn, and Vu, which found thresholds for all strictly balanced graphs. In this talk we outline difficulties of the problem and the various known cases, before showing how a generalisation of the strictly balanced case allows for proving the threshold for non-balanced graphs.
March 2013
Ariadne
Bolz
Algebraische obere Schranken für Codes
Daniel
Heldt
The face flip markov chain is rapidly mixing on 2-orientations of plane quadrangulations of maximum degree 4
Nieke
Aerts
Straight Line Triangle Representations
Irina
Mustaţă
Orthogonal Ray Graphs and Trees
Aprameyo
Pal
Univ. Heidelberg
Functional equation of characteristic elements of Abelian varieties over function fields
Abstract.
In this talk we apply methods from the Number field case of Perrin-Riou & Zabradi in the Function field set up. In Zl- and GL2-case (l≠ p), we prove algebraic functional equations of the Pontryagin dual of Selmer group which give further evidence of the Main conjectures of Iwasawa Theory. We also prove some parity conjectures in commutative and non-commutative cases. As consequence, we also get results on the growth behaviour of Selmer groups in extension of Function fields.
Tobias
Buchwald
Domino Tilings auf dem Torus.
February 2013
Nieke
Aerts
Extremal Graphs Having No Independent Cutsets
Anita
Liebenau
FU Berlin
Orientation Games
Abstract.
In an orientation game, two players called OMaker and OBreaker take turns in directing previously undirected edges of the complete graph on n vertices. The final graph is a tournament. OMaker wins if this tournament has some predefined property P, otherwise OBreaker wins. We analyse the Oriented-cycle game, in which OMaker's goal is to close a directed cycle. Since this game is drastically in favour of OMaker, we allow OBreaker to direct (up to) b > 1 edges in each of his moves and ask for the largest b* such that OMaker has a winning strategy (the so called threshold bias). It is known that n/2 - 3 < b* < n - 2, where the upper bound was conjectured to be tight. In this talk I will discuss recent developments in the field of orientation games, including a sketch of an OBreaker strategy which improves the upper bound on b* to roughly 5n/6. Joint work with Dennis Clemens.
Dennis
Clemens
FU Berlin
What is a positional game?
Abstract.
Using some specific examples, we will introduce positional games, in particular, the class of maker-breaker games. Moreover, we will take a deeper look at the Erdős-Selfridge theorem from 1973 which often is seen as the starting point in the history of maker-breaker games.
Christian
Schröder
TU Berlin
What is a palindromic eigenvalue problem?
Abstract.
"Mom", "Dad", "I prefer pi", and " A man, a plan, a canal — Panama" are palindromes — they can be read form left to right and vice versa. Analogously, a polynomial is palindromic if its sequence of coefficients is the same in both directions (i.e., $2x^4 + 7x^3 + 5x^2 + 7x + 2$). In just a few more small steps, we get to palindromic matrix polynomials and their corresponding eigenvalue problems. The talk will be on palindromic eigenvalue problems: their introduction, where they arise, their properties and suitable algorithms.
Stefan
Felsner
Table Cartograms
Tibor
Szabó
FU Berlin
and
Tuan
Tran
FU Berlin
Ramanujan Graphs: The Construction
Vladimir
Dokchitser
Cambridge
Parity of ranks of elliptic curves
Abstract.
It is in general very difficult to compute ranks of elliptic curves over number fields, even if equipped with any conjectures that are available. On the other hand, the parity of the rank is (conjecturally) very easy to determine -- it is given as a sum of purely local terms, which have a reasonably simple classification. Since 'odd rank' implies 'non-zero rank' implies 'the curve has infinitely many points', this leads to a number of (conjectural!) arithmetic phenomena. The second part of the talk will concern the 'parity conjecture' - that the parity of the rank that is predicted by the Birch-Swinnerton-Dyer conjecture agrees with the prediction of the Shafarevich-Tate conjecture.
Silvia
de Toffoli
HU Berlin
What is a rational tangle?
Abstract.
"During the period from the end of the '60s through the beginning of the '70s, Conway pursued the objective of forming a complete table of knots. […] Therefore, he pulled another jewel from his bag of cornucopia and introduced the concept of tangle." - Murasugi A $n$-tangle is an embedding of a collection of $n$ arcs in a $3$-ball such that the endpoints are on specific 2$n$ points on the boundary sphere. The focus will be on $2$-tangles. A rational tangle is a $2$-tangle that is homeomorphic to the trivial tangle, which is formed by two unlinked arcs, vertical or horizontal. By taking the numerator closure of a tangle we obtain a knot and, in particular, the numerator of a rational tangle is a rational knot. Rational tangles are associated, as the name suggests, to rational numbers (union infinity) and there is a deep connection between them and the theory of continued fractions. Apart from their mathematical importance in knot theory, rational tangles are crucial to the study of DNA topology and, in general, to biological applications of knot theory.
Dorothy
Buck
Imperial College London
DNA knots: Why they're (biologically) important and (mathematically) interesting
Anil
Aryasomayajula
HU Berlin
Towards an extension of a key identity to Hilbert modular surfaces
Abstract.
In 2006, J. Jorgenson and J. Kramer came up with a beautiful identity relating the canonical and hyperbolic volume forms on a Riemann surface. In this talk, we report the progress of the on-going work in collaboration with Nahid Walji, towards an extension of this formula to Hilbert modular surfaces.
Ahmad
Afuni
FU Berlin
What is a monotonicity formula?
Abstract.
Monotonicity formulae are an indispensable tool for extracting information about solutions of ODEs and PDEs directly from the structure of the equations, especially when explicit solutions are not readily available. In this talk, we introduce the notion of a monotonicity formula and outline how such formulae may be used to draw certain conclusions about solutions of certain differential equations.
January 2013
Codruţ
Grosu
FU Berlin
and
Anita
Liebenau
FU Berlin
Ramanujan Graphs: PSL 2 ( q )
Özlem
Imamoglu
ETH Zurich
On Klein's j-invariant
Maryna
Viazovska
Univ. Köln
CM Values of Higher Green's Functions
Abstract.
Higher Green functions are real-valued functions of two variables on the upper half plane which are bi-invariant under the action of a congruence subgroup, have logarithmic singularity along the diagonal, and satisfy the equation Δ f = k(1 - k)f, where Δ is the hyperbolic Laplace operator and k is a positive integer. The significant arithmetic properties of these functions were disclosed in the paper of B. Gross and D. Zagier "Heegner points and derivatives of L-series" (1986). In particular, it was conjectured that higher Green's functions have "algebraic" values at CM points. In this talk we will present a proof of the conjecture for any pair of CM points lying in the same quadratic imaginary field. Moreover, we give an explicit factorization formula for the algebraic number obtained (up to powers of ramified primes).
Isabella
Thiesen
TU Berlin
What is a quasiconformal mapping?
Abstract.
Have you ever tried to find a conformal mapping between a square and a non-square rectangle that maps corners to corners? This won't work, so the German mathematician Grötzsch asked for the most conformal way instead. That was in 1928, and he laid the foundation for what became later known as quasiconformal mappings, a natural generalization of conformal mappings. Some people love to use quasiconformal mappings as a tool for proving theorems, and (more often) other people love to compute them for applications in computer graphics. Both groups use it for the same reason; in some sense they are almost as good as conformal maps but much more flexible. I will talk about the different approaches to quasiconformal mappings and try to get you interested in this topic.
Emre
Sertöz
HU Berlin
What is a $j$-invariant?
Abstract.
The elliptic curves (or complex tori) can be parametrized in 2 different ways. The first method parametrizes lattices in the complex plane in a rather obvious way. The second parametrization gives to each elliptic curve a more geometric value in the sense that this value corresponds more closely to how the curve is embedded in the plane. Then there is a function mapping the first parametrization to the other. This function is called the $j$-invariant and it is a modular form of weight zero, where number theory comes to join geometry and algebra. We will discuss briefly what modular forms are and what a fundamental domain is--all the absolute basics you need to know.
Udo
Hoffmann
Coloring Triangle-Free Rectangular Frame Intersection Graphs with O(log log n) colors.
Yoshiharu
Kohayakawa
University of São Paulo
On the Density of Universal Random Graphs
Abstract.
We shall discuss a polynomial time randomized algorithm that, on receiving as input a pair (H, G) of n-vertex graphs, searches for an embedding of H into G. If H has bounded maximum degree and G is suitably dense and pseudorandom, then the algorithm succeeds with high probability. Our algorithm proves that, for every integer d ≥ 3 and suitable constant C = Cd, as n → ∞, asymptotically almost all graphs with n vertices and ⌊Cn2 - 1/d log1/dn⌋ edges contain as subgraphs all graphs with n vertices and maximum degree at most d. This is joint work with Domingos Dellamonica Jr., Vojtěch Rödl and Andrzej Ruciński.
Jürg
Kramer
HU Berlin
Towards arithmetic intersections on mixed Shimura varieties
Daniel
Heldt
The Shannon capacity of C_5
Dennis
Clemens
FU Berlin
and
Lothar
Narins
FU Berlin
Ramanujan Graphs: Number Theory
Damian
Rössler
Toulouse
A direct proof of the equivariant Gauss-Bonnet formula on abelian schemes
Abstract.
We shall present a proof of the relative equivariant Gauss-Bonnet formula for abelian schemes, which does not rely on the Grothendieck-Riemann-Roch theorem. This proof can be carried through in the arithmetic setting (i.e. in Arakelov theory) and leads to interesting analytic questions.
December 2012
Frederik
Garbe
FU Berlin
and
Tuan
Tran
FU Berlin
Ramanujan Graphs: Algebraic Graph Theory
Özlem
Imamoglu
ETH Zürich, z.Zt. HU Berlin
On weakly harmonic Maass forms and their Fourier coefficients
Abstract.
Harmonic Maass forms have been studied extensively in last several years due to their connection to several arithmetic problems. Most notably in the half integral weight case, it can be shown that the Ramanujan's mock theta functions and generating functions of traces of singular moduli can be understood in terms of harmonic Maass forms. A simpler example is provided by the non-holomorphic Eisenstein series of weight 2. In this talk, after giving a short overview of the subject, I will show how to construct a distinguished basis of such forms in the case of weight 2 and study their relation of the regularized inner products of modular functions.
Florian
Frick
TU Berlin
What is $SU(2)$ and why does it double-cover $SO(3)$?
Abstract.
We will investigate the relationship between rotations in vector spaces of dimension at most four and multiplication of complex numbers or quaternions. We will see that complex unitary 2-by-2 matrices with determinant one, which is a subgroup of isometries of a complex two-dimensional vector space, and rotations of three-dimensional Euclidean space are closely related.
Jason
Cantarella
U Georgia
The geometry and topology of random polygons
Nieke
Aerts
A characterisation of Generic Circuits.
Tuan
Tran
FU Berlin
Subspace evasive sets
Abstract.
Let F be a finite field. A subset S ⊆ Fn is called (k, c)-evasive if it has intersection of size at most c with every k-dimensional affine subspace of Fn. A simple probabilistic argument shows that a random set S ⊂ Fn of size |F|(1 - ε)n will have intersection of size at most O(k/ε) with any k-dimensional affine subspace. Recently, Zeev Dvir and Schachar Lovett gave an explicit construction of a (k, (k/ε)k)-evasive set S ⊂ Fn of size |F|(1 - ε)n. In this talk we will present the construction and its application in extremal combinatorics.
Tony
Varilly-Alvarado
Rice University, currently at EPFL Lausanne
Failure of the Hasse principle on general K3 surfaces
Abstract.
Transcendental elements of the Brauer group of an algebraic variety, i.e., Brauer classes that remain nontrivial after extending the ground field to an algebraic closure, are quite mysterious from an arithmetic point of view. These classes do not arise for curves or surfaces of negative Kodaira dimension. In 1996, Harari constructed the a 3-fold with a transcendental Brauer-Manin obstruction to the Hasse principle. Until recently, his example was the only one of its kind. We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general K3 surface X of degree 2 over Q, together with a two-torsion Brauer class α that is unramified at every finite prime, but ramifies at real points of X. Motivated by Hodge theory, the pair (X,α) is constructed from a double cover of P2 × P2 ramified over a hypersurface of bi-degree (2, 2). This is joint work with Brendan Hassett.
Irina
Mustata
Efficient representations of unit grid intersection graphs
Lothar
Narins
FU Berlin
Cooperative Colorings and Independent Systems of Representatives
Abstract.
Introduced by Aharoni, Holzman, Howard, and Sprüssel, a cooperative coloring of a system of graphs G1, ..., Gn on the same vertex set is a choice of one independent set from each graph such that they together cover the vertex set. This is closely related to the notion of independent systems of representatives (also called independent transversals), and in fact these two concepts can be translated into one another. Aharoni, Holzman, Howard, and Sprüssel investigate conditions under which a system has a cooperative coloring, in part utilizing a topological result of Meshulam.
Miguel
Grados
Univ. Peruana de Ciencias Aplicadas
L-series of Elliptic Curves with Complex Multiplication
Abstract.
L-series of elliptic curves are complex functions that carry arithmetical information of the curve. From the analytic point of view, it is desirable that those functions possess additional properties such as Euler product expression, analytic continuation to the entire complex plane and a functional equation. Remarkably, such properties hold for the L-series associated to elliptic curves with complex multiplication. In this talk, we will elaborate on the role played by complex multiplication in the context of L-series. Additionally, since the above properties remind us of the Prime Number Theorem in arithmetic progressions, we will discuss how the analytic methods used in Dirichlet's proof can be adjusted to yield a similar theorem for the L-series of elliptic curves.
November 2012
Philippo
Lappicy
FU Berlin
What is a spacetime
singularity
Abstract.
Trying to answer questions about space, time and gravity leads us to the notion of singularities — where the theory doesn't hold. In this talk, we'll introduce the basics of General Relativity and how it is related to dynamics.
Alan
Rendall
MPI
Spacetime singularities and heteroclinic chains
Viola
Mészáros
Geometric graphs with few disjoint edges
Roman
Glebov
FU Berlin
Biased Hamiltonicity game on random boards
Abstract.
One of the first Maker-Breaker games that was analyzed when the concept was introduced is the Hamiltonicity game. Already Erdős and Chvátal found out that Maker can build a Hamilton cycle playing on the edge set of Kn for sufficiently large n. It took decades and several partial results of different authors before Krivelevich proved their conjecture and showed that the game in fact obeys the random graph intuition, showing that the critical bias is asymptotically log(n)/n. In this talk, we consider the Hamiltonicity game played on the edge set of the random graph G(n, p) for some appropriate p. Confirming (and even strengthening) a conjecture of Stojaković and Szabó, we show that this game also follows the random graph intuition, that is, the critical bias is asymptotically np/log(n) a.a.s.. As far time permits, I present the main steps of the proof. Joint work with A. Ferber, A. Naor, and M. Krivelevich.
Jan Steffen
Müller
Univ. Hamburg
A p-adic BSD conjecture for modular abelian varieties
Abstract.
In 1986 Mazur, Tate and Teitelbaum came up with a p-adic analogue of the conjecture of Birch and Swinnerton-Dyer for elliptic curves over the rationals. In this talk I will report on joint work with Jennifer Balakrishnan and William Stein on a generalization of this conjecture to the case of modular abelian varieties and primes p of good ordinary reduction. I will discuss the theoretical background that led us to the formulation of the conjecture, as well as numerical evidence supporting it in the case of modular abelian surfaces and the algorithms that we used to gather this evidence.
Faniry
Razafindrazaka
FU Berlin
What is a regular map?
Abstract.
The notion of a regular map is everywhere. It sits at the intersection of arithmetic geometry, hyperbolic geometry, topology, and computational group theory. Recently, it is also being used in mathematical visualization. Even people in string theory use the notion in some applications. In this talk, we use the geometrical definition related to its symmetry group. We will investigate problems that have been solved and those that are still open in the process of visualizing them.
Stefan
Felsner
Exploiting Air-Pressure to Map Floorplans on Point Sets
Codruţ
Grosu
FU Berlin
Fp is locally like C
Abstract.
Vu, Wood and Wood showed that any finite set S in a characteristic zero integral domain can be mapped to Fp, for infinitely many primes p, while preserving all algebraic incidences of S. In this talk we will show that the converse essentially holds, namely any small subset of Fp can be mapped to some finite algebraic extension of Q, while preserving bounded algebraic relations. This answers a question of Vu, Wood and Wood. Most of the talk will be devoted to the presentation of applications of the main result. For small subsets of Fp, we: show that the Szemerédi-Trotter theorem holds with optimal exponent 4/3, improve the previously best-known sum-product estimate, transfer results from C to Fp concerning sets with small doubling constant, and so on. We shall briefly give some insight into the proof of the main result, which is a simple application of elimination theory and is similar in spirit with the proof of the quantitative Hilbert Nullstellensatz.
Martin
Raum
ETH Zürich
Computational methods, Jacobi forms, and linear equivalence of special divisors
Abstract.
We will start by briefly discussing the influence of computations on mathematics. Some examples will make clear that computations have led to surprising insight into the area of modular forms, and continue doing so. After this general considerations, we will turn our attention to Jacobi forms. Their definition and their connection with ordinary modular forms will be explained in detail. A new algorithm allows us to compute Fourier expansions of Jacobi forms. We will translate this into information about linear equivalences of special divisors on modular varieties of orthogonal type.
Opening of the
Stochastic Analysis
Alina C.
Cojocaru
Univ. of Illinois at Chicago, z.Zt. Univ. Göttingen
Frobenius fields for elliptic curves
Abstract.
Let E be an elliptic curve defined over Q. For a prime p of good reduction for E, let πp be the p-Weil root of E and Q(πp) the associated imaginary quadratic field generated by πp. In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes p < x for which Q(πp) is isomorphic to a fixed imaginary quadratic field. I will discuss progress on this conjecture, in particular an average result confirming the predicted asymptotic formula. The latter is joint work with Henryk Iwaniec and Nathan Jones.
Ananyo
Dan
HU Berlin
What is a Hodge locus?
Abstract.
Given a family of smooth projective varieties $B$, the Hodge locus corresponding to a Hodge class $\gamma$ parametrizes all $b \in B$ where $\gamma$ remains a Hodge class. In this talk we discuss the definition of a Hodge class, variation of Hodge structures and the statement of the Hodge conjecture. We also look at the Lefschetz (1,1)-theorem which states that the Hodge conjecture is true in the case of divisors.
Inder
Kaur
HU Berlin
What is a Bruhat-Tits tree?
Abstract.
In this talk I will define a Bruhat-Tits tree and describe the construction of the tree associated to $PGL(2,F)$ for $F$ a local field. If time permits, I will also discuss some applications of the tree in representation theory.
Matjaz
Kovse
Betweenness
Christian
Wald
HU Berlin
What is a quaternion algebra?
Abstract.
Quaternion algebras over a field $K$ are special noncommutative $K$-algebras of dimension four. We will consider the classical example of the Hamilton quaternions followed by quaternion algebras over arbitrary fields and different characterizations. We will discuss basic properties and, if time permits, explain the notion of automorphic forms on quaternion algebras.
Joachim
Schwermer
U Wien
On arithmetically defined hyperbolic 3-manifolds
Kolja
Knauer
Partial Cubes: Lattices and Topology
Dennis
Clemens
FU Berlin
On balanced coloring games in random graphs
Abstract.
Let F be some graph. Consider first the following one player game, known as balanced Ramsey game: The Player has r colors and in each round r random edges (that were never seen before) on n vertices are presented to her. Immediately, the Player has to color all these edges differently. The game ends as soon as a monochromatic copy of F appears. Secondly consider the Achlioptas game: Here the Player only loses when she creates a copy of F in one distinguished color. The main question is how long these games last typically. In the talk, we will get an overview of the recent results proven by Luca Gugelmann and Reto Spöhel: They compare the typical time (threshold) for both games, settling an open problem by M. Krivelevich, R. Spöhel and A. Steger. Furthermore, they consider vertex analogues of both games and show that here the thresholds coincide for all graphs F.
October 2012
Niels
Lindner
HU Berlin
Cuspidal plane curves and their Alexander polynomials
Abstract.
The Alexander polynomial is a useful invariant of the fundamental group of a curve in the complex projective plane. It is strongly connected with the singularities of the curve. If one restricts to curves whose singular points are either ordinary double points or ordinary cusps, one can use calculations on the Mordell-Weil rank of certain associated elliptic three-folds to obtain a nice description of the Alexander polynomial in terms of syzygies of the ideal of cusps.
Mijail
Guillemard
TU Berlin
What is support vector machines?
Abstract.
Persistent homology is an important modern subject in computational topology, and it offers useful tools for signal and data analysis. This short talk is intended for a general audience, and we cover elementary aspects of persistent homology, including some basics on simplicial homology and the concept of persistent diagrams.
Udo
Hoffmann
Recognition of Simple-Triangle Graphs and Linear-Interval Orders is Easy
Giovanni
De Gaetano
HU Berlin
A regularized determinant for the hyperbolic Laplacian on modular forms
Abstract.
With the goal to generalize the work of T. Hahn, who was able to prove an arithmetic Riemann-Roch type formula for the Hodge bundle on a modular curve with log-singular metric, we define and compute a suitable metric for the determinant bundle of a power of the Hodge bundle. It will be a non-smooth analogue of the Quillen metric used by Gillet and Soulé to prove their arithmetic Riemann-Roch theorem.
Felix
Knöppel
TU Berlin
What is a smoke ring?
Abstract.
This talk will give a short introduction to smoke rings and the rich theory to which they are connected. Smoke rings are closed curves that evolve by the vortex filament equation. This equation is connected to the non-linear cubic Schrödinger equation, which is well-known in the theory of solitons.
Ulrich
Pinkall
TU Berlin
Decomposing Smoke into Smoke Rings
Andrei
Asinowski
Orders induced by segments in floorplans, and (2-14-3, 3-41-2)-avoiding permutations
Bill
Duke
UCLA
Mock-modular forms of weight one and Galois representations
Katharina
Jochemko
Arithmetic of marked order polytopes and monotone triangle reciprocity
September 2012
Tillmann
Miltzow
Unique Bichromatic Matchings
Stefan
Felsner
Coloring rectangle intersection graphs
August 2012
Joachim
Schröder
Eigensequences; Kirchhoff and Graphs
George B.
Mertzios
An Intersection Model for Multitolerance Graphs: Efficient Algorithms and Hierarchy
George
Mertzios
The Intersection of Tolerance and Cocomparability Graphs
Viola
Mészáros
Compatible matchings and planar straight line graphs
Irina
Mustata
The jump number of 2-directional orthogonal ray graphs
July 2012
Hannah
Enders
HU Berlin
What is
Ext and its applications?
Abstract.
From the module of homomorphisms $Hom(M,N)$ between two modules $M$ and $N$, one can derive other modules $Ext^i (M,N)$, for all $i$. These modules contain certain information about $M$ and $N$. We will show how to construct $Ext$ explicitly and give some further results on how to use it.
Lars
Kühne
ETH Zürich
An effective result of André-Oort type
Abstract.
"The André-Oort Conjecture (AOC) states that the irreducible components of the Zariski closure of a set of special points in a Shimura variety are special subvarieties. Here, a special variety means an irreducible component of the image of a sub-Shimura variety by a Hecke correspondence. The AOC is an analogue of the classical Manin-Mumford conjecture on the distribution of torsion points in abelian varieties. I will present a rarely known approach to the AOC that goes back to Yves André himself: Before the model-theoretic proofs of the AOC in certain cases by the Pila-Wilkie-Zannier approach, André presented the first non-trivial proof of the AOC in case of a product of two modular curves. In my talk, I discuss results in the style of André's method, allowing to actually compute all special points in a non-special curve of a product of two modular curves and more..."
Adrián
Gonzalez Casanova
TU Berlin
What is Brownian motion?
Abstract.
Rather than giving a precise mathematical statement to describe Brownian motion, here is a great quote from "Stochastic Calculus," by Richard Durrett. "If you run Brownian motion in two dimensions for a positive amount of time, it will write your name. Of course, on top of your name it will write everybody else's name, as well as all the works of Shakespeare, several pornographic novels, and a lot of nonsense." In this seminar, we will better understand one of the most fascinating and complex mathematical objects: the Brownian motion.
Peter
Mörters
U Bath
Tossing coins, matching points, and shifting Brownian motion
Roman
Glebov
FU Berlin
On a conjecture of Erdős and Sós
Abstract.
Erdős and Sós asked the following question: given a hypergraph on sufficiently many vertices with positive edge density on every linearly large vertex set, is it true that it contains a given subgraph? In particular, they raised this question for the case of 3-uniform hypergraphs with the subgraph of interest being the hypergraph K on 4 vertices and 3 edges. Rödl showed that, somewhat surprisingly, this lower density condition is not sufficient, giving an increasing sequence of hypergraphs not containing K and having density almost ¼ on every linearly large vertex set. In this talk, we show that the constant ¼ is optimal. The proof uses the computational flag algebra method. Joint work in progress with Daniel Král’ and Jan Volec.
Barbara
Jung
HU Berlin
Computations towards the arithmetic self intersection number of the bundle of modular forms on A2
Abstract.
To compute the self intersection number of the bundle of modular forms on A2, one has to choose sections tactically, as one has to integrate over forms and subvarieties defined by them. I will present a computation of one of the arising terms via induction A1 → A2.
June 2012
Giovanni
de Gaetano
HU Berlin
What is a group representation?
Abstract.
Representation theory is the main bridge between abstract and linear algebra. Moreover, representations can be found in virtually any field in mathematics and can generalize many commonly used objects. In this talk, we will approach the fundamentals of the theory, with specific attention to the representations of groups. If time permits, we will clarify some notions useful for the following Friday talk by Prof. Bridson, such as finitely presented groups and ${\rm SL}(n,\mathbb{Z})$ representations.
Martin
Bridson
Oxford U
Discrete groups: A story of geometry, complexity, and imposters
Codruţ
Grosu
FU Berlin
Following a question of Boris Bukh
Abstract.
Let f(x) be some polynomial with T non-zero real coefficients. Suppose we square this polynomial. What is a lower bound q(T) for the number of non-zero coefficients of f(x)2? It was a question of Erdős from 1949 that q(T) goes to infinity when T goes to infinity. This was solved by Schinzel in 1987, who proved q(T) > log log T. The result was recently (2009) improved by Schinzel and Zannier to log T. This is the proof I am going to present. Bukh asked whether one can show q(T) > T1 - ε, for some ε > 0.
Jürg
Kramer
HU Berlin
Remark on a question of Parshin
Felix
Günther
TU Berlin
What is the Fabricius-Bjerre theorem?
Abstract.
The Fabricius-Bjerre theorem states that for a generic curve in the plane, the number of crossings plus half the number of inflections plus the number of opposite-side double tangencies is equal to the number of same-side double tangencies. We will define each of the quantities referred to in the theorem and look at some examples before we give the original (and very beautiful) proof of Fabricius-Bjerre himself. In the case of polygons with vertices in general position, similar definitions are given for crossings, opposite-side and same-side double tagencies, and inflection edges. We will sketch the proof of Tom Banchoff for the polygonal version of the Fabricius-Bjerre theorem, which is based on a deformation argument. In the end, we will look at generic curves in the sphere, and give the Spherical Fabricius-Bjerre formula.
Tom
Banchoff
Brown U
Folds, Intersections and Inflections for Smooth and Polyhedral Surfaces: Distinguishing Cylinders from Möbius Bands
Nieke
Aerts
Straight Line Triangle Representation of a Graph
Alexandru
Tomescu
TU Berlin
Extensional acyclic digraphs and set graphs
Abstract.
A digraph is called 'extensional' if its vertices have pairwise distinct (open) out-neighborhoods. Extensional acyclic digraphs originate from set theory where they represent transitive closures of hereditarily finite sets. We will present some results concerning the underlying graphs of extensional acyclic digraphs, which we call 'set graphs'. Even though we argue that recognizing set graphs is NP-complete, we show that set graphs contain all connected claw-free graphs and all graphs with a Hamiltonian path. In the case of claw-free graphs, we provide a polynomial-time algorithm for finding an extensional acyclic orientation. Extensional digraphs can also be characterized in terms of a slight variation of the notion of separating code, which we call open-out-separating code. Concerning this, deciding if a digraph admits an open-out-separating code of given size is NP-complete. Joint work with Martin Milanic and Romeo Rizzi
Hannah
Enders
HU Berlin
On perfect sheaves of modules
Abstract.
In 1957, Rees defined perfect modules as modules where the homological invariants grade and projective dimension coincide. Starting from this point, we will investigate under which conditions a definition of perfect sheaves of modules is possible. Furthermore, we will show that perfect sheaves of modules inherit some interesting properties of perfect modules.
Juan
Orduz
HU Berlin
What is
a Dirac operator?
Abstract.
The main idea of this talk is to introduce the notion of a Dirac Operator from the physical problem of finding the quantum equation for a relativistic electron. After that we will explore how these ideas can be generalized in order to find a general geometric setting for these operators and how they can be used to construct a bridge between analysis and topology via index theorems.
Peter
Allen
London School of Economics
and
Julia
Böttcher
London School of Economics
Tight cycles in dense uniform hypergraphs
Wouter
Castryck
Univ. Leuven
Curve gonalities and Newton polygons
Abstract.
Let Delta be a lattice polygon, i.e. the convex hull in R2 of a finite number of points of Z2 (called "lattice points"). Assume that it is not contained in a line. Then it is well-known that a generic Laurent polynomial f(x,y) having Delta as its Newton polygon defines a curve whose genus equals the number of lattice points in the interior of Delta. In this talk we will search for combinatorial interpretations for other discrete invariants, such as the Clifford index, Clifford dimension, and the gonality. The latter is by definition the minimal degree of a morphism to P1.
Mikola
Lysenko
UW Madison
What is a group action?
Abstract.
This short expository talk is an attempt to answer the "What?" and "Why?" of group theory. Because no familiarity with abstract algebra is assumed, we will start from first principles in order to keep this presentation self contained. The main goal here is to impart some conceptual understanding, not to study any single problem in much depth. We will focus on motivation, and survey a few interesting applications.
Karen
Vogtmann
Cornell U
Outer Spaces
Luca
Gugelmann
ETH Zürich
Random Hyperbolic Graphs
Abstract.
Many large real-world networks have degree sequences which follow a power-law (i.e. the number of vertices of degree k is of order k-a for some fixed a) and have large clustering coefficients (the average probability that two neighbors of a vertex are themselves adjacent). Unfortunately these properties seem difficult to reproduce in mathematically tractable random graph models. There are models for graphs with power-law degree sequences (e.g. preferential attachment) or models for graphs with large clustering coefficients (e.g. Watts-Strogatz) but to our knowledge none which reproduce both properties and yet still remain mathematically tractable. Recently Papadopoulos et al. introduced a random geometric graph model which is based on hyperbolic geometry and for which they claimed empirically and with some preliminary mathematical analysis that it satisfies both properties above. This model consists (in its simplest version) of n points sampled uniformly at random from a hyperbolic disc of radius R = R(n) which are connected if their hyperbolic distance is at most R. We (Konstantinos Panagiotou, Ueli Peter and the speaker) analyze this model rigorously and compute exact asymptotic expressions for the expected number of vertices of degree k (up to the maximum degree). We also prove concentration around these values. Further we compute asymptotic expressions for the average and maximum degree and a constant lower bound for the clustering coefficient.
Stefan
Felsner
Covering partial cubes with cuts
Henning
Thomas
ETH Zürich
Mastermind
Abstract.
Most of you probably know the game of Mastermind: one player (the "codemaker") makes up a secret code word z = (z1, z2, ..., zn) over an alphabet of size k, where in the classical board game (n = 4, k = 6) the symbols are represented by pegs of different colors. The other player (the "codebreaker") has to identify the code word by asking questions of the form x = (x1, x2, ..., xn) over the same alphabet. The codemaker answers a question with a distance measure to the secret code by revealing (i) the number a(m, x) of positions i for which zi = xi, in the original board game depicted by black pegs, and (ii) the maximum a(m, q) over all permutations of q, originally depicted by white pegs. In this talk we present some known upper and lower bounds as well as a new strategy which improves the currently best-known upper bounds for the case k = Θ(n) from O(n log n) questions to O(n log log n). This is joint work with Benjamin Doerr, Reto Spöhel and Carola Winzen from MPI Saarbrücken.
Andrea
Cattaneo
Univ. Mailand
Elliptic CY threefolds over surfaces
Abstract.
In this talk I want to give a classification of the elliptic CY threefolds we can find in a projective bundle over a base surface B. More in detail, if L is an ample line bundle on B, then I want to give explicit bounds on the pairs (a,b) such that in the bundle P(O + La +Lb) the generic anticanonical variety is smooth. I will also give a detailed description in the case B = P2. As an application of this classification, I will switch to physics and show a nice result concerning string theory, which generalizes a result by Aluffi-Esole.
Julio
Backhoff
HU Berlin
What is
a problem in mathematical finance?
Abstract.
In this talk, we introduce stochastic financial models for both discrete time and continuous time and discuss some of their defining properties. Then we present some of the most relevant problems encountered in financial mathematics.
Torsten
Ueckerdt
L-Contact and Segment-Contact is the same
May 2012
Lothar
Narins
FU Berlin
Home Base Hypergraphs and Ryser's Conjecture
Abstract.
Ryser's Conjecture states that any r-partite r-uniform hypergraph has a vertex cover of size at most r - 1 times the size of the largest matching. For r = 2, the conjecture is simply König's Theorem. It has also been proven for r = 3 by Aharoni using topological methods. Our ambitious goal is to try to extend Aharoni's proof to r = 4. We are currently still far from this goal, but we start by characterizing those hypergraphs which are tight for the conjecture for r = 3. These we call home base hypergraphs. Our proof of this characterization is also based on topological machinery, particularly utilizing results on the (topological) connectedness of the independence complex of the line graph of a graph. Joint work with Penny Haxell and Tibor Szabó.
Christian
Liedtke
Univ. Bonn
On the Birational Nature of Lifting
Abstract.
Whenever a variety X lifts from characteristic p to characteristic zero, say over the Witt ring, then many classical results over the complex numbers hold for X, and certain "characteristic p pathologies" cannot occur, simply because one can reduce modulo p (I will discuss this in examples). But then, lifting results are difficult, and generally, varieties do not lift. However, in many situations, it is possible or easier to lift a birational model of X, maybe even one that has "mild" singularities (again, I will give examples). So, a natural question is whether the liftability of such a birational model implies that of our original X. We will show that this completely fails in dimension at least 3, that this question is surprisingly subtle in dimension 2, and that it is trivial in dimension 1.
Dror
Atariah
FU Berlin
What is
a configuration space?
Abstract.
In a nutshell, a configuration space is one where each point represents a unique pose/configuration of some physical system. In this talk we will introduce a special instance of a configuration space. For the physical system, we will consider the case of a planar polygonal robot which is free to rotate and translate amid planar polygonal obstacles. By means of explicit parameterization of the configuration space, we will obtain a clear picture of the geometrical properties of the space and, in particular, of the configuration space obstacles. The talk will be accompanied with a short video visualizing the discussed case.
Kolja
Knauer
Drawing outerplanar graphs with few slopes
Boris
Bukh
University of Cambridge, CMU
Generalized Erdős-Szekeres theorems
Abstract.
We introduce a generalization of the Erdős-Szekeres theorem, and show how it can be applied to construct a strengthening of weak epsilon-nets. We also discuss the decision problem for Erdős-Szekeres-type for arbitrary semialgebraic predicates.
Shahrad
Jamshidi
FU Berlin
What is
a differential inclusion?
Abstract.
Differential equations with discontinuous right hand sides crop up when modelling switches or relays in control systems. In order to deal with these discontinuities we need to generalise the concept of solution so that the differential equations are satisfied almost everywhere. Differential inclusions incorporate the discontinuities of the differential equation in order to obtain a well defined solution. In this talk we focus on the mathematical complications that arise with the discontinuous right hand side and how they are overcome using differential inclusions.
Tuan
Tran
TU Berlin
On the edge polytopes of finite graphs
Abstract.
Given a finite graph G. The edge polytope P(G) of G is defined as the convex hull of the column vectors of the incidence matrix of G. In this talk we will discuss the following topics: A description of the low-dimensional faces of the polytope P(G) Non-linear relations between the components of the f-vector of P(G) Asymptotic growth rate of the maximum number of facets of a d-dimensional edge polytope. This is the author's thesis, written under the supervision of Günter Ziegler
Anna
von Pippich
HU Berlin
On the spectral zeta function of a hyperbolic cusp or cone
Abstract.
In a joint project with G. Freixas we aim at establishing an arithmetic Riemann-Roch isometry for singular metrics. In this talk we report on an analytic ingredient, namely the computation of the regularized determinant of the hyperbolic Laplacian on a cusp or cone with Dirichlet boundary conditions.
Daniel
Heldt
Conductance, congestion and canonical paths of Markov chains
Simon
Brendle
Stanford U
Der Satz von Alexandrov in gekrümmten Räumen
Anita
Liebenau
FU Berlin
A new bound on the largest tournament Maker can build
Abstract.
Two players, usually called Maker and Breaker, play the following positional game. Given a tournament T (a complete directed graph) on k vertices, they claim edges alternately from the complete graph Kn on n vertices. Maker also has to choose a direction for every edge she picks. Maker wins this game as soon as her graph contains a copy of T. In 2008, Beck showed that the largest clique Maker can build (that is, disregarding directions) is of order kc = (2 - o(1)) log n. Given n large enough, we show that Maker is able to build a tournament of order k = (2 - o(1)) log n = kc - 10. That is, building a tournament is almost as easy for Maker as building a clique of the same size. This improves the lower bound of 0.5 log n (Beck 2008) and log n (Gebauer 2010). In particular, this result shows that the so called "random graph intuition" fails for the tournament game. Joint work with Dennis Clemens and Heidi Gebauer.
Nuno
Freitas
Univ. Barcelona
The generalized Fermat-type equations x^5+y^5 = 2^z^p or 3^z^p via Q-curves
Abstract.
In order to attack the generalized Fermat equation Ax^p+By^q = Cz^r the modular approach to Diophantine equations that initially led to the proof of Fermat's Last Theorem has been progressively refined. In this talk we will explain how several pieces of the strategy need to be generalized in order to solve equations of the form x^5+y^5 = Cz^p. In particular, we will show how we can use two simultaneous Frey-curves defined over Q(√5) to solve the previous equations for a set o primes with density 3/4.
Kristian
Rother
What is
software development?
Abstract.
Scientists who depend on programming for their work face uncertainty; they need to adapt their programs as their research projects evolve. This typically makes it impossible to design a complete software up front. How, then, can you write good software under this condition of uncertainty? Software development is the engineering solution to that question. This talk will present three general methodologies to writing software: Waterfall (old), Agile (new), and Lean Development (very new). In addition, best practices for developing software faster, making it more reliable, and communicating with peers are briefly presented. The goal of this talk is to enable you to apply development techniques in your programming practice and evaluate their usefulness.
Stefan
Felsner
A note on proportional contact representations
Yury
Person
FU Berlin
Twins in Words
Abstract.
Suppose we are given a 0-1-sequence S of length n and we want to find two long identical non-overlapping (disjoint) sequences (twins) in S. Which length can one always guarantee? I will answer this question (asymptotically) and discuss some generalizations of it. Joint work with Maria Axenovich and Svetlana Puzynina.
Udo
Hoffmann
Recognition of Simple Triangle Graphs
April 2012
Jennifer
Rasch
HU Berlin
What is
a path-following method?
Abstract.
Mathematical modeling of real-world phenomena often leads to formulations of problems in infinite dimensions containing variational inequalities. In this talk, we will focus on so-called elasto-plastic problems which require the pointwise bounding of the gradient of the displacement, i.e., the stress on a body at each point in the presence of a given force. We will then derive the first order optimality conditions. To use the semismooth Newton method to solve the problem numerically, we require a regularized penalization of our problem. Numerical path-following strategies will be developed and analytical results can be numerically verified.
Irina
Mustata
Unit grid intersection graphs: Properties and relationships to other graph classes
Christian
Reiher
Universität Rostock
The clique density problem
Abstract.
It is well known that by a theorem of Turán every graph on n vertices possessing more than (r - 2)n2/(2r - 2) edges has to contain a clique of size r, where r > 3 denotes some integer. Given that result, it is natural to wonder what one can say about the number of r–cliques that have to be present in a graph on n vertices with at least γ·n2 edges, where γ > (r - 2)/(2r - 2) denotes some real parameter. In awareness of the structure of the graph that is extremal for Turán’s original problem, it is quite tempting to suggest that the answer is provided by first choosing some appropriate integer s > r - 1 depending on γ alone and then constructing a complete (s + 1)–partite graph all of whose vertex classes except for one possibly smaller one are of equal size. This gives a number of r–cliques that is proportional to nr and it is an amazingly difficult problem to decide whether the factor of proportionality thus obtained is indeed (asymptotically) best possible. This question has first been asked by Lovász and Simonovits in the 1970s, and has remained widely open until very recently when Razborov introduced new analytical ideas and solved the case r = 3. Shortly afterwards, Nikiforov solved the case r = 4. In the talk, we will present the recent solution to this clique density problem, and discuss some related ideas of Lovász and Razborov concerning graph limits.
Emmanuel
Ullmo
Université Paris-Sud, Orsay
The hyperbolic Ax-Lindemann conjecture for projective Shimura varieties and some applications to the André-Oort conjecture
Romain
Nguyen
FU Berlin
What is a dissipative weak solution?
Abstract.
Hydrodynamical models are descriptions of fluids based on transport equations for macroscopic quantities such as flow field, temperature, density, etc. Two distinct terms are found in these equations: advection terms, which describe transport by the fluid flow itself and are reversible and independent on the nature of the fluid, and diffusion terms, which describe the irreversible transport due to disordered molecular motion. In many interesting cases, after non-dimensionalizing the equations, one finds that diffusion coefficients are extremely small. For example, the momentum diffusion coefficient which applies when pouring a glass of water is about 10^{-5}! When dealing with such cases, although it is very tempting to omit diffusion terms, we know that the water quickly comes to rest, its momentum being irreversibly dissipated. How could this process be described without dissipative terms in the equations?? We will show that asking this simple question leads to the mathematical concept of a dissipative weak solution, and to some of the most difficult open problems in mathematical fluid dynamics.
Edriss
Titi
UC Irvine
The Navier-Stokes, Euler and Other Related Equations
Daniel
Reichman
The Weizmann Institute
Random permutations and random subgraphs
Abstract.
We describe how random permutations give rise to random subgraphs of undirected graphs with interesting properties. We present applications to bootstrap percolation, proving existence of large k-degenerate graphs in bounded degree graphs as well as a new framework for designing algorithms for the independent set problem. Joint work with Uri Feige
Jakob
Scholbach
Univ. Münster
Motives and Arakelov theory
Abstract.
Beginnend mit einer kurzen Einführung in die stabile Homotopiekategorie von Schemata werden wir eine neue Kohomologietheorie namens Arakelov-motivischer Kohomologie vorstellen. Diese kann als Variante (und Verallgemeinerung) von arithmetischen K- und Chow-Gruppen angesehen werden. Wir diskutieren einige Eigenschaften wie den arithmetischen Satz von Riemann-Roch sowie, falls Zeit bleibt, die Beziehung zu speziellen L-Werten.
Maciek
Korzec
TU Berlin
What is
a spectral method?
Abstract.
Besides the usual techniques, such as finite difference methods and finite element methods, spectral methods are another important tool for numerically solving partial differential equations. If the solutions are sufficiently smooth these methods yield spectral accuracy and hence one can achieve a faster convergence rate than with any local polynomial based interpolation method. In this talk the general framework of spectral methods, i.e., collocation methods based on trigonometric interpolation, is introduced. Example nonlinear partial differential equations are solved numerically.
Viola
Mészáros
Graph Sharing Games
March 2012
Thomas
Hixon
Cyclic segment graphs and diagonal hook graphs
Kristian
Dannowski
Chromatic Invariants of Shift Graphs and Kneser Graphs
Bartosz
Walczak
The chromatic number of geometric intersection graphs
February 2012
Fabian
Lenhardt
FU Berlin
What is the fundamental group and covering space?
Abstract.
We define the fundamental group of a topological space, which is one of the most fundamental topological invariants. One of the main tools for computing fundamental groups is the theory of covering spaces; we give a fast introduction and try to compute some fundamental groups in the end.
Dennis
Clemens
FU Berlin
Fast strategies in Maker-Breaker games played on random boards
Abstract.
A Maker-Breaker game is defined as follows: Given a board X and a set of winning sets F ⊂ 2X, two players, called Maker and Breaker, alternately take elements of X. If Maker occupies an element of F completely until the end of the game, he wins. Otherwise Breaker is the winner. In the seminar I will present results about Maker-Breaker games played on the edge set of a sparse random graph G ~ Gn,p. We will consider the the perfect matching game, the Hamiltonicity game and the k-connectivity game. For p = logd(n)/n (with d large enough) we will see that Maker a.a.s. can win these games as fast as possible, i.e. in n/2 + o(n), n + o(n) and kn/2 + o(n) moves, respectively. Joint work with Asaf Ferber, Anita Liebenau and Michael Krivelevich.
Torsten
Ueckerdt
How many bends for one additional direction?
Asaf
Ferber
Tel Aviv University
The weak and strong k-connectivity games
Abstract.
In this talk we consider the weak and strong k-connectivity game played on the edge set of the complete graph. In the weak k-connectivity game, two players, Maker and Breaker, alternately claim free edges of the complete graph. Maker wins as soon as the graph consists of his edges is a (spanning) k-connected graph. If Maker doesn't win by the time all the edges of Kn are claimed then Breaker wins the game. In the strong version of this game, two players Red and Blue, alternately claim free edges of Kn (Red is the first player to move). The winner is the first player to build a spanning k-connected graph. We prove that in the weak k-connectivity game Maker has a winning strategy within nk/2 + Θ(k2) moves and that Red has a winning strategy in the analogues strong game. Joint work with Dan Hefetz.
Alice
Garbagnati
Mailand
Symplectic and non symplectic automorphisms of K3 surfaces
Abstract.
Fixed a particular family of K3 surfaces, the automorphisms group of a general member is very often unknown. In order to understand properties of the automorphisms groups of K3 surfaces it seems better to fix a particular group and to find the family of K3 surfaces admitting that group as subgroup of the full automorphisms group. The aim of this talk is to present some results on moduli spaces of K3 surfaces admitting a certain finite group $G$ as subgroup of the group of the automorphisms. We consider both groups acting symplectically on the surface (i.e. preserving the nowhere vanishing holomorphic two form) and group acting purely non symplectically (i.e. there is no elements in the group which preserve the nowhere vanishing holomorphic two form). One of the main results we present is that there exist some pairs of groups $(G,H)$ such that $G$ is a subgroup of $H$ and, under some hypothesis, a K3 surface $X$ admits $G$ as subgroup of the automorphisms group if and only if it admits $H$ as subgroup of the automorphisms group. This phenomenon happens in three distinct situations: both $G$ and $H$ act symplectically on the K3, both $G$ and $H$ act purely non symplectically on the K3, $G$ acts purely non symplectically and $H$ contains elements which are symplectic and elements which are non symplectic. Some of the results presented are obtained in collaboration with Alessandra Sarti.
Christian
Bogner
HU Berlin
What is a Feynman integral?
Abstract.
Computations in perturbative Quantum Field Theory typically involve a certain kind of integral, the so-called Feynman integrals. Without assuming prior knowledge of particle physics, I will point out some of the main ideas on why we need these integrals and how they arise from certain graphs. In the end I will briefly point out some recently developed interrelations with modern mathematics.
Hao
Chen
The distance geometry for the kissing balls
Heidi
Gebauer
ETH Zürich
Constructing a Non-Two-Colorable Hypergraph with Few Edges
Abstract.
We say that a given k-uniform hypergraph is non-two-colorable if every red/blue-coloring of its vertices yields an edge where all vertices have the same color. It is a long-standing open problem to determine the minimum number m(k) such that there exists a non-2-colorable k-uniform hypergraph with m(k) hyperedges. Erdős showed, using the probabilistic method, that m(k) ≤ O(k2·2k), and by a result of Radhakrishnan and Srinivasan, m(k) ≥ Ω(√(k/ln k) 2k). These are the best known bounds. This talk will be about an explicit construction of a non-two-colorable hypergraph with 2k(1 + o(1)) edges.
Özlem
Imamoglu
ETH Zürich
Periods of modular forms
Samuel
Drapeau
HU Berlin
What is risk?
Abstract.
Risk as a notion, even if very intuitive, is still ambiguous. In this seminar, we will briefly discuss the characteristics of risk and how it can then be modeled and studied mathematically. Finally, we discuss the insights we gain from various interpretations of this approach.
Michael
Kupper
HU Berlin
Risk, model uncertainty and nonlinear expectations
January 2012
Peter
Scholze
Univ. Bonn
Die Hasse-Weil-Zeta-Funktion von Modulkurven
Abstract.
Ein alter Satz besagt, dass die Zeta-Funktion von Modulkurven geschrieben werden kann als Produkt von L-Funktionen von Modulformen und Hecke-Charakteren. Insbesondere folgt, dass sie eine meromorphe Fortsetzung hat und die erwartete Funktionalgleichung erfüllt. Der ursprüngliche Beweis beruht auf Arbeiten von Eichler, Shimura, Langlands, Deligne, und Carayol. Wir erklären die Methode von Langlands, und zeigen, wie diese erweitert werden kann, um an Primstellen schlechter Reduktion einen neuen, vereinfachten Beweis dieses Satzes zu liefern.
Özge
Ekin
FU Berlin
What is the connection between Kantian intuition of mathematical
objects and diagrams?
Abstract.
In this talk, I will reveal the connection between particular representations (symbols, diagrams) and the Kantian intuition of mathematical objects. I will construct an interpretation of Kant's philosophy of mathematics that explains the requirement of pure intuitions, space and time and reveals the a priori nature of mathematics in Kant's doctrine. Kantian characterization of mathematics exposes a different reasoning model from the current method of mathematics, namely the formal sentential reasoning. I argue that by understanding this approach and by recognizing the roles of diagrams in mathematics, it is possible to realize the advantages of heterogeneous reasoning in mathematics.
Veit
Wiechert
The dimension of posets with outerplanar cover or planar comparability graphs
Penny
Haxell
University of Waterloo
On Ryser’s Conjecture
Abstract.
We discuss some old and new results and ideas on the following long-standing conjecture. Let H be an r-partite r-uniform hypergraph. A matching in H is a set of disjoint edges, and we denote by ν the maximum size of a matching. A cover of H is a set of vertices that intersects every edge. It is clear that there exists a cover of H of size rν, but it is conjectured that there is always a cover of size at most (r - 1)ν.
Evelina
Viada
Basel
Anomalous Varieties and the effective Mordell-Lang Conjecture
Abstract.
We consider an algebraic variety $V$ embedded in a product of elliptic curves $E^N$. It may happen that components of the intersection of $V$ with a proper algebraic subgroup of $E^N$ have dimension larger than expected. Such components are called $V$ anomalous varieties. The non density of all $V$ anomalous varieties in $V$, for all $V$ not contained in any algebraic subgroup, implies the Mordell-Lang Conjecture. Effective bounds for the height and degree of the maximal $V$ anomalous varieties gives the effective Mordell-Lang Conjecture. We will discuss these implications. We will give some new results for varieties of codimension 2 and some cases of the effective Mordell-Lang Conjecture for curves.
Lothar
Narins
HU Berlin
What is an independent
transversal?
Abstract.
In this talk I introduce independent transversals and give a combinatorial proof of their existence for certain classes of graphs.
Penny
Haxell
U Waterloo
Independent transversals, or, choosing faculty committees
Marie
Albenque
Constellations and multicontinued fractions: application to Eulerian triangulations
Dmitry
Shabanov
Moscow State University
Colorings of uniform hypergraphs with large girth and their applications in Ramsey theory
Abstract.
Extremal problems concerning colorings of uniform hypergraphs arose in connection with classical theorems of Ramsey Theory. Our talk is devoted to the problem of estimating the maximum vertex degree of a uniform hypergraph with large girth and large chromatic number. This problem is not completely solved even in the case of graphs. We shall present some new results and show their applications for finding quantitative bounds in Ramsey's theorem and Van der Waerden's theorem. The proofs use a continuous-time random recoloring process based on Poisson stochastic processes.
Peter
Toth
HU Berlin
A Geometric Proof of the Tamely Ramified Geometric Abelian Class Field Theory
Abstract.
Unramified geometric abelian class field theory establishes a connection between the Picard group and the abelianized etale fundamental group of a smooth, projective, geometrically irreducible curve over a finite field. We begin with a fairly detailed discussion of the unramified theory concentrating on Deligne's geometric proof. Then we turn to the tamely ramified theory, which transforms the classical situation to the open complement of a finite set of closed points of the curve, establishing a connection between a generalized Picard group and the tame fundamental group of the curve with respect to this finite set of closed points and present a geometric proof for the tamely ramified theory.
Stefan
Felsner
The graphs that can be drawn with one bend per edge
Jacob
Fox
MIT
Chromatic number, clique subdivisions, and the conjectures of Hajós and Erdős-Fajtlowicz
Abstract.
A famous conjecture of Hajos from 1961 states that every graph with chromatic number k contains a subdivision of the complete graph on k vertices. This conjecture was disproved by Catlin in 1979. Erdős and Fajtlowicz further showed in 1981 that a random graph on n vertices almost surely is a strong counterexample to the Hajós conjecture. They further conjectured that in a certain sense a random graph is essentially the strongest possible counterexample among graphs on n vertices. We prove the Erdős-Fajtlowicz conjecture. Joint work with Choongbum Lee and Benny Sudakov.
Amador
Martin-Pizarro
Lyon, z.Z. HU Berlin
On Schanuel's conjecture and CIT
Abstract.
Schanuel's conjecture states that given n complex elements (x_1,...,x_n) linearly independent over \\(mathbb{Q}\\), then \\mathrm{tr.deg} (x_1,...,x_n, \exp(x_1),...,\exp(x_n) )\geq n. In recent research, Zilber has shown that there is a unique \emph{universal} field (up to isomorphism) of cardinality continuum equipped with a group homomorphism \exp : \\(mathbb{G}_a\\) to \\(mathbb{G}_m\\) where Schanuel's conjecture holds and such that the exponential-closure of a finite set is countable. The question remains whether (\\(mathbb{C}\\),\exp) is that field and in particular whether the key obstacle is Schanuel's conjecture itself. In order to prove some of the results, Zilber introduced a weak version of the \emph{Conjecture of Intersection with Tori}, which states that there is only finitely many cosets of tori (uniformely) for a given closed subvariety V of \\mathbb{G}_m\\ describing all possible atypical intersections of V with any proper torus. In this talk, we will present some of the ideas in Zilber's work as well as an approach to weak CIT in positive characteristic by introducing Hasse-Schmidt iterative derivations in a separably closed field and relate the above to the existence of infinite Mersenne primes.
Wang-Q
Lim
HU Berlin
What is a shearlet?
Abstract.
Shearlets were introduced as means to sparsely encode anisotropic singularities of multivariate data while providing a unified treatment of the continuous and digital realm. In this talk, recent results on the construction of compactly supported shearlet systems will be presented, in particular, showing that these shearlet frames provide optimally sparse approximations of piecewise smooth 2D as well as 3D functions. Finally, we will discuss various applications of shearlets such as image restoration, data separation and numerical PDEs.
Gitta
Kutyniok
TU Berlin
Imaging Science meets Compressed Sensing
Nieke
Aerts
Matchings that yield 'good' triangle representations
Gérard
Freixas i Montplet
Univ. Paris 6
On Faltings theorem for abelian schemes over arithmetic surfaces
Abstract.
In this talk I will review the statement of the Tate conjecture for abelian varieties. In the number field case this is the celebrated theorem of Faltings. Faltings himself showed how to reduce the case of abelian schemes over higher dimensional bases to abelian schemes over a number field. His proof uses Hodge theory. We will give another approach that avoids Hodge theory but relies on higher dimensional Arakelov geometry. This will be the occasion to state some generalizations of diophantine statements in this theory. The contents of the talk will be based on joint work with Jean-Benoit Bost.
December 2011
Yury
Person
FU Berlin
A randomized Version of Ramsey's theorem
Abstract.
The classical theorem of Ramsey states that for all integers k ≥ 2 there exists an integer R(k) such that no matter how one colors the edges of the complete graph KR(k) with two colors, there will always be a monochromatic copy of Kk. Here we consider the following problem recently suggested by Allen, Böttcher, Hladky and Piguet. Let R(n, q) be a random subset of all copies of F on a vertex set Vn of size n, in which every copy is present independently with probability q. For which functions q = q(n) can we color the edges of the complete graph on Vn with r colors such that no monochromatic copy of F is contained in R(n, q)? We also discuss the usual randomization of Ramsey's theorem for random (hyper-)graphs and its connection to the problem above. This is joint work with Luca Gugelmann, Angelika Steger and Henning Thomas.
Irina
Mustata
2-Directional Orthogonal Ray Graphs
Jeng-Daw
Yu
Taiwan National Univ.
The irregular Hodge filtration
Abstract.
This is a report of work in progress with Esnault. Motivated by various cohomology theories of exponential sums over finite fields, we propose a Hodge-type filtration on the cohomology attached to an exponentially twisted de Rham complex over a complex smooth (quasi-projective) variety. I will indicate some good properties of this filtration and list some natural questions.
Antoine
Laurain
HU Berlin
What is propagating interfaces and level set methods?
Abstract.
Level Set Methods are numerical techniques which allow to model the evolution of interfaces between geometrical domains. Unlike parametrization methods, complex geometrical changes such as sharp corners, appearance of holes, and merging together may be modeled using level set methods. The techniques have a wide range of applications, including problems in fluid mechanics, combustion, structural optimization, computer animation, image processing, and the shape of soap bubbles.
James
Sethian
UC Berkeley
Advances in Advancing Interfaces: Efficient Algorithms for Inkjet Plotters, Coating Rollers, Semiconductors, Retinopathy Diagnosis, and Chemical Pathway Analysis
Mihyun
Kang
FU Berlin
The Bohman-Frieze process near criticality
Abstract.
The Erdős-Rényi random graph process begins with an empty graph on n vertices and edges are added randomly one at a time to a graph. A classical result of Erdős and Rényi states that the Erdős-Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 and a giant component emerges. In this talk we discuss the so-called Bohman-Frieze process, a simple modification of the Erdős-Rényi process. The Bohman-Frieze process begins with an empty graph on n vertices. At each step two random edges are present and if the first edge would join two isolated vertices, it is added to a graph; otherwise the second edge is added. We show that the Bohman-Frieze process has a qualitatively similar phase transition to the Erdős-Rényi process in terms of the size and structure of the components near the critical point. (Joint work with Perkins and Spencer)
Matthias
Schütt
Univ. Hannover
Modularity of Maschke's octic and Calabi-Yau threefold
Abstract.
Maschke's octic surface is the unique invariant of a particular group of size 11520 acting on projective fourspace. Recently Bini and van Geemen studied this surface and two Calabi-Yau threefolds derived from it as double octic and quotient thereof by the Heisenberg group. In particular they computed a decomposition of the cohomology in terms of the group and conjectured its modularity. We will sketch how to actually prove this for all three varieties. The proofs rely on automorphisms of the varieties and in one case on isogenies of K3 surfaces.
Barbara
Jung
HU Berlin
What is a lattice in a
Lie group good for?
Abstract.
The talk of Igor Rivin will have discrete subgroups of matrix groups as a topic. But why? In the seminar I will give two examples of such lattices. I will connect them with elliptic curves and with each other, so at the end we see how to understand elliptic curves without algebraic geometry.
Andrew
Treglown
Charles University in Prague
Perfect packings in dense graphs
Abstract.
We say that a graph G contains a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. For all 'non-trivial' graphs H, the decision problem whether a graph G has a perfect H-packing is NP-complete. Thus, there has been significant attention on obtaining sufficient conditions which ensure a graph G contains a perfect H-packing. The Hajnal-Szemerédi theorem states that a graph G whose order n is divisible by r has a perfect Kr-packing provided that the minimum degree of G is at least (1 - 1/r)n. In this talk our focus is on strengthening the Hajnal-Szemerédi theorem: Given integers n, r and d, we haracterise the edge density threshold that ensures a perfect Kr-packing in a graph G on n vertices and of minimum degree at least d. We also give two conjectures concerning degree sequence conditions which force a graph to contain a perfect H-packing. This is joint work with Jozsef Balogh and Alexandr Kostochka.
Timo
Strunk
Labellings and decompositions of planar and toroidal quadrangulations
Igor
Rivin
Temple U
Probability, geometry and algorithms of matrix groups
November 2011
Stefan
Keil
HU Berlin
Shafarevich-Tate groups of non-square order
Abstract.
The order of the Shafarevich-Tate group (=sha) of an elliptic curve over a number field is, if it is finite, a square number. For abelian varieties in higher dimensions this is no longer the case. However, for principally polarized abelian varieties over a number field this is almost true, since then the order of sha is a square or twice a square. The only known example of an abelian surface over QQ having order of sha not equal to a square or twice a square has order of sha equal to three times a square. We will explain how to construct abelian surfaces over QQ having order of sha equal to five or seven times a square.
Jonad
Pulaj
ZIB
What is the Lagrangian relaxation of an integer program?
Abstract.
The Lagrangian relaxation is a well known technique which provides bounds for integer programs through penalty adjustments. We give an overview of the method and illustrate its application to multi-period network design.
Viola
Mészáros
The upper bound for separated matchings
David
Conlon
University of Oxford
On two strengthenings of Ramsey's theorem
Abstract.
Ramsey's theorem states that every 2-colouring of the edges of the complete graph on {1, 2, ..., n} contains a monochromatic clique of order roughly log n. We prove new bounds for two extensions of Ramsey's theorem, each demanding extra structure within the monochromatic clique. In so doing, we answer a question of Erdős and improve upon results of Rödl and Shelah. This is joint work with Jacob Fox and Benny Sudakov.
Jürg
Kramer
HU Berlin
A note on scattering matrices
Andre
Mialebama
FU Berlin
What is a Gröbner basis in a polynomial ring?
Abstract.
Our main goal in this talk is to define a Gröbner basis in a polynomial ring over a field and show how it is important to solve the ideal membership problem and other problems.
Daniel
Heldt
Three families of graphs with bad conductance for the face-flip Markov chain on α-orientations
Julia
Wolf
École Polytechnique, Paris
Large sets with little structure
Abstract.
There has been much activity in the past twelve months regarding upper bounds on the density of sets containing no 3-term arithmetic progressions. We shall give an overview of some of these developments and subsequently examine the corresponding lower bounds for this problem. This includes joint work with Ben Green and Yuncheng Lin.
Philipp
Habegger
Univ. Frankfurt am Main
Beyond the André-Oort Conjecture (joint with Jonathan Pila)
Abstract.
An isomorphism class of elliptic curves defined over C can be identified with a complex number by virtue of Klein's j-function. The so-called singular j-invariants are particularly interesting from an arithmetic point of view. These come from elliptic curves with complex multiplication. A particular case of the André-Oort Conjecture describes the distribution of points on subvarieties of affine n-space whose coordinates are singular j-invariants. Here the conjecture is known due to work of André, Edixhoven, and Pila. Pink's more general conjecture describes points on subvarieties that satisfy moduli theoretic properties which are generally weaker than asking for complex multiplication. This includes examples such as points in affine n-space whose coordinates are pairwise isogenous elliptic curves. I will present progress into the direction of Pink's Conjecture. Our method of proof relies on the theory of o-minimal structures which has its origins in model theory; the general strategy was developed originally by Zannier. In the talk I will explain what an o-minimal structure is and how it interacts with arithmetic components of our proof.
Guillem
Perarnau
UPC Barcelona
Identifying codes for regular graphs
Abstract.
Given a graph G, an identifying code can be defined as a dominating set that identifies each vertex of G with a unique code. They have direct applications on detecting and locating a "failure" of a vertex in networks. We are mainly interested in bounding the size of a minimal code in terms of the maximum or minimum degree of G. The first part of the talk is devoted to give a new result that provides an asymptotically tight approximation to a conjecture stated by Foucaud, Klasing, Kosowski and Raspaud (2009) for a large class of graphs. In the second part we will talk about graphs with girth five, where better upper bounds can be given. Moreover, we use them to compute the minimal size of a code for random regular graphs with high probability. This is a joint work with Florent Foucaud.
Ingrid
Daubechies
Duke University
Distinguishing the "hand" of the master?
Ulrich
Terstiege
Univ. Duisburg-Essen
Intersections of special cycles on unitary Rapoport-Zink spaces of signature (1,n-1)
Abstract.
We discuss results on intersections of special cycles on unitary Rapoport-Zink spaces that can be applied to the conjectures of Kudla and Rapoport on intersections of special cycles on unitary Shimura varieties and to the arithmetic fundamental lemma conjecture of W. Zhang.
Andreas
Hochenegger
FU Berlin
What is a derived category?
Abstract.
The derived category is an important object in algebraic geometry. The derived category associated to a variety is an important invariant that is even stronger than cohomology. I will try to give a gentle introduction by showing how the definition of this category evolved.
David
Ouwehand
HU Berlin
What is a zeta function?
Abstract.
Zeta functions are objects that arise in many areas of mathematics. This talk will be about the Dedekind zeta function of a number field and the Hasse-Weil zeta function of a smooth curve over a finite field; the goal is to explain how these zeta functions contain (respectively) arithmetic and geometric information. If time permits, I will also talk about the zeta functions of schemes that are of finite type over the integers. These generalize the previous types of zeta functions.
Dieter
Jungnickel
Universität Augsburg
Klassische Designs und ihre Codes
Abstract.
Wir betrachten die endliche projektive bzw. affine Geometrie der Dimension n über dem endlichen Körper GF(q) mit q Elementen, also PG(n, q) bzw. AG(n, q). Die klassischen Designs zu einer derartigen Geometrie sind die Inzidenzstrukturen, die aus allen Punkten sowie allen d-dimensionalen Unterräumen der Geometrie gebildet werden (für ein d mit 1 ≤ d ≤ n - 1); sie werden üblicherweise als PGd(n, q) bzw. AGd(n, q) bezeichnet. Da es (exponentiell) viele Designs mit denselben Parametern wie die klassischen Designs gibt, möchte man diese schönen Beispiele unter allen Designs mit diesen Parametern charakterisieren. Vor nahezu 50 Jahren hat Hamada eine codierungstheoretische Charakterisierung der klassischen Designs vorgeschlagen; jedoch ist die nach ihm benannte Vermutung trotz einiger Fortschritte in den letzten Jahren weiterhin weitgehend offen. Der Vortrag beschäftigt sich mit einer alternativen (von Hamada inspirierten, aber deutlich komplexeren) Möglichkeit, die klassischen Designs codierungstheoretisch zu charakterisieren. In einer gerade zur Publikation angenommenen gemeinsamen Arbeit mit V.D.Tonchev ist dies für alle projektiven und für viele affine Fälle (nämlich für d = 1 sowie (n - 2)/2 < d ≤ n - 1) gelungen; dabei konnten die affinen Fälle auf eine geometrische Vermutung reduziert werden, die sicherlich für alle d gilt, aber bisher nur im genannten Bereich verifiziert werden konnte. Unsere Beweismethoden bestehen aus einem Zusammenspiel von kombinatorischen, geometrischen und codierungstheoretischen (also algebraischen) Argumenten. Dabei spielen einige sehr interessante (aber leider verhältnismäßig wenig bekannte) Ideen aus der Codierungstheorie (Spur-Codes, Galois-abgeschlossene Codes, Erweiterungscodes) eine entscheidende Rolle; diese Begriffe sind nicht vorausgesetzt, sondern werden im Vortrag erklärt.
Giovanni
De Gaetano
HU Berlin
On the fundamental group of the affine line in positive characteristic
Abstract.
In the first part of the talk we introduce the notion of fundamental group of a scheme by a pure categorical point of view. This construction is a generalization of the fundamental group of a topological space and of the Galois group of a field. Then, using a little étale topology, we move the notions of 'loop' and 'neighborhood' from the topological to the arithmetic context. In the last part we see what kind of coverings of the affine line arise when we allow a certain ramification index to be divisible by the characteristic of the field, and we outline a strategy for the solution of the problem. Eventually we will understand that the affine line, in positive characteristic, is very far from being simply connected.
October 2011
Cesar
Ceballos
FU Berlin
What is tropical
geometry?
Abstract.
Tropical geometry is a relatively recent area of mathematics with strong applications to algebraic geometry, mirror symmetry, combinatorics, mathematical biology and enumerative geometry among others. It manipulates, in a purely combinatorial way, geometric objects that take over the role of classical algebraic varieties. In this talk, we present an elementary introduction to the subject.
Hannah
Markwig
U Saarland
What corresponds to Broccoli in the real world?
Christoph
Koch
TU München
On Kuratowski's Theorem and the Kelmans-Seymour conjecture
Abstract.
Kuratowski's well-known theorem gives a precise characterization of planar graphs - there are exactly two types of forbidden subgraphs in planar graphs: subdivisions of K3,3 and subdivisions of K5. Thus, a natural question to arise is the following: Are there classes of graphs such that we need only check for one type of these subdivisions in order to determine planarity? In the past decades there have been many results of that kind, but still some problems remain unsolved. One of them is the Kelmans-Seymour conjecture: Does every 5-connected non-planar graph contain a subdivided K5? This talk will illustrate two main approaches towards this conjecture. In particular, there have been some partial results by various people within the last year.
Marie
Albenque
Computing numbers with balls and urns
Remke N.
Kloosterman
HU Berlin
Zariski triples and equisingular deformations of cuspidal curves
Abstract.
In this talk we construct a Zariski triple, i.e., three plane curves $C_1,C_2,C_3$ of the same degree, with the same number and type of singularities such that the fundamental groups of $\mathbb{P}^2\setminus C_i$ are pairwise non-isomorphic. This is done by calculating the Alexander polynomial of $C_i$. We use this example of a Zariski triple to construct families of singular plane curves such that their equisingular deformation space has larger dimension than expected. In the end we show that if $C$ is a curve of degree at least 13 with non-constant Alexander polynomial then the tangent space of the equisingular deformation space has larger dimension than expected.
Maciek
Korzek
TU Berlin
What is diffusion?
Abstract.
One of the most fundamental differential operators appearing in partial differential equations (PDEs) is the Laplace operator. Its understanding is essential to be able to treat models describing real-world problems. One of the most basic PDEs relates the rate of change of some quantity to the Laplace operator applied to the same function: the diffusion equation $u_t = k \nabla^2 u$, also known as heat equation. In this talk several examples for diffusion will be explained. A connection between random walks and continuous diffusion will be established, a Gaussian filter will be linked to diffusion in image processing and the anisotropic diffusion equation $u_t = \nabla \cdot k(x) \nabla u$ will be used to improve an image by advocating diffusion in small slope regions only. In this way the edges in an image remain intact while noise or similar image failures are diffused out. Finally the heat equation will be derived in a bulk material, and also on a regular surface, resulting in a surface diffusion equation, $u_t = \nabla_{\!s} \cdot k(x) \nabla_{\!s} u$.
Thomas
Hixon
Weak conflict free colourings
Charlie
Elliott
U Warwick
Evolving interfaces and surfaces
Tibor
Szabó
FU Berlin
Sharp threshold for bounded degree spanning trees with many leaves or a long bare path
Abstract.
We show that any given n-vertex tree with bounded maximum degree and linearly many leaves is contained in the binomial random graph G(n, p) asymptotically almost surely for some p = (1 + o(1))log n/n. Furthermore we also show that G(n, p) contains asymptotically almost surely every bounded degree spanning tree T that has a path of linear length whose vertices have degree two in T. This represents joint work with Dan Hefetz and Michael Krivelevich.
Kolja
Knauer
The hull number of partial cubes
Torsten
Ueckerdt
Crossing-free curves within pseudodiscs
September 2011
Jan
Hladky
Warwick Mathematics Institute
Loebl-Komlós-Sós Conjecture and embedding trees in sparse graphs
Abstract.
We prove an approximate version of the Loebl-Komlós-Sós Conjecture which asserts that if half of the vertices of a graph G have degrees at least k then G contains each tree T of order k + 1 as a subgraph. The proof of our result follows a strategy common to approaches which employ the Szemerédi Regularity Lemma: the graph G is decomposed, a suitable combinatorial structure inside the decomposition is found, and then the tree T is embedded into G using this structure. However the decomposition given by the Regularity Lemma is not of help when G sparse. To surmount this shortcoming we use a general decomposition technique (a variant of which was introduced by Ajtai, Komlós, Simonovits and Szemerédi to resolve to Erdős-Sós conjecture) which applies also to sparse graphs: each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the Regularity Lemma), and an expander-like part. Joint work with János Komlós, Diana Piguet, Miki Simonovits, and Endre Szemerédi.
Bartłomiej
Bosek
and
Grzegorz
Matecki
Generalization of Dilworth's Theorem and Semiantichain Conjecture
Stefan
Felsner
Squaring the Torus
Alex
Sheive
The Group Theory of Spinpossible
Irina
Mustata
Jump number(s) of partially ordered sets
Maximilian
Werk
Rhombische Pflasterungen von Dreiecken
Roman
Glebov
FU Berlin
Rainbow matchings in hypergraphs
Abstract.
We investigate the following problem: given a (large) integer r and a (sufficiently large) integer t, we are presented an r-uniform r-partite multihypergraph that consists of f matchings of size t. What is the extremal number f = f(t, r), such that we know that there exists a t-matching with every hyperedge coming from a different matching? We present an upper bound of order tr, improving a previous bound of Alon. We also present an observation leading to the intuition that the "extremal" examples live on a small number of vertices. The aim of the talk is to equate our knowledge in the problem and to motivate further joint research in this and closely related problems. Joint work in progres with Benny Sudakov, Tibor Szabó, Yury Person and Anita Liebenau.
August 2011
Daniel
Heldt
Sampling Schnyder-woods of planar triangulations with max degree ≤6
Benjamin
Matschke
FU Berlin
Equivariant topology methods and the colored Tverberg problem
Abstract.
The first part of this talk will be a quick survey on equivariant topology methods and how to use them in discrete geometry. The second part is about a particular application, the colored Tverberg problem, which is joint work with Pavle Blagojević and Günter Ziegler.
Nieke
Aerts
Graphs that yield safe communication schemes
Thomas
Hixon
The Tron Problem
Marie
Albenque
Planar maps and continued fractions
July 2011
Kolja
Knauer
Cayley graphs of semigroups
Markus
Hihn
HU Berlin
What is a matroid? And how is it related to quantum field theory?
Abstract.
In quantum field theory, the study of periods associated to Feynman graphs is of growing interest. However, some graphs define the same period (if it exists). The question which graph defines the same period is answered by their corresponding cycle matroid. This is an easy construction, but they are also appearing in other topics in quantum field theory: the falsification of Kontsevich conjecture and Feynman integrals with tensor structure.
Thomas
Picchetti
How to compute a squaring?
Kevin
Milans
University of South Carolina
Doug
Ulmer
Georgia Inst. of Technology
Arithmetic of the Legendre curve in a Kummer tower
Abstract.
Let k be a finite field of odd characteristic, K=k(t), and K_d=k(t^{1/d}). We consider the arithmetic of the Legendre elliptic curve E: y^2=x(x-1)(x-t) over the fields K_d. A remarkable elementary construction gives many points on E over K_d for suitable values of d. Less elementary considerations lead to interesting problems and results on the full Mordell-Weil group E(K_d), on heights, and on the Tate-Shafarevich group of E over K_d.
Marcel
Ortgiese
TU Berlin
What is a Green's function?
Abstract.
The concept of a Green's function arises when trying to solve certain partial differential equations. We will highlight the interplay between analysis and probability theory and see how to solve these equations in a probabilistic way and in particular we will give an interpretation of the corresponding Green's functions.
Bjorn
Poonen
MIT
Random maximal isotropic subspaces and Selmer groups
Abstract.
We show that the p-Selmer group of an elliptic curve is naturally the intersection of two maximal isotropic subspaces in an infinite-dimensional locally compact quadratic space over F_p. By modeling this intersection as the intersection of a random maximal isotropic subspace with a fixed compact open maximal isotropic subspace, we can explain the known phenomena regarding distribution of Selmer ranks, such as the theorems of Heath-Brown, Swinnerton-Dyer, and Kane for 2-Selmer groups in certain families of quadratic twists, and the average size of 2- and 3-Selmer groups as computed by Bhargava and Shankar. The random model is consistent with Delaunay's heuristics for Sha[p], and predicts that the average rank of elliptic curves is at most 1/2. This is joint work with Eric Rains.
Wendelin
Werner
U Paris Sud - Orsay
Random surfaces and their geometry
Uri
Zwick
Tel Aviv University
All-Pairs Shortest Paths in O(n2) Expected Time
Abstract.
We present an All-Pairs Shortest Paths (APSP) algorithm whose expected running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from [0, 1] is O(n2). This resolves a long standing open problem. The algorithm is a variant of the dynamic all-pairs shortest paths algorithm of Demetrescu and Italiano. The analysis relies on a proof that the expected number of locally shortest paths in such randomly weighted graphs is O(n2). We also present a dynamic version of the algorithm that recomputes all shortest paths after a random edge update in O(log2 n) expected time. Joint work with Yuval Peres, Dmitry Sotnikov and Benny Sudakov.
Stefan
Felsner
Trapezoidal dissections and Markov chains
Christian P.
Robert
On Approximate Bayesian Computation (ABC) methods
June 2011
Benny
Sudakov
UCLA
Nonnegative k-sums, fractional covers, and probability of small deviations
Abstract.
More than twenty years ago, Manickam, Miklos, and Singhi conjectured that for any integers n ≥ 4k, every set of n real numbers with nonnegative sum has at least (n - 1 choose k - 1) k-element subsets whose sum is also nonnegative. In this talk we discuss the connection of this problem with matchings and fractional covers of hypergraphs, and with the question of estimating the probability that the sum of nonnegative independent random variables exceeds its expectation by a given amount. Using these connections together with some probabilistic techniques, we verify the conjecture for n ≥ 33k2. This substantially improves the best previously known exponential lower bound on n. If time permits we will also discuss applications to finding the optimal data allocation for a distributed storage system and how our approach can be used to prove some theorems on minimum degree ensuring the existence of perfect matchings in hypergraphs. Joint work with N. Alon and H. Huang, last part is also joint with P. Frankl, V. Rodl and A. Rucinski.
Jawaherul
Alam
Proportional Contact Representations with Rectilinear Polygons
Christine
Robinson
Univ. of Illinois at Chicago
On a converse theorem and a Saito-Kurokawa lift for Siegel wave forms
Ambros
Gleixner
ZIB
What is linear programming?
Abstract.
Linear programming is arguably the single most essential technique in theory and practice of mathematical optimisation. We will introduce the underlying duality theory including complementary slackness and the central path. As an example for the powerful algorithms available to solve linear programs, we will sketch a primal-dual interior point method. This algorithm solves a linear program by following the central path to an optimal solution.
Bernd
Sturmfels
UC Berkeley
The Central Curve in Linear Programming
Raphael
Traut
Zeichnen von Ordnungen; Eine Experimentalstudie
Hanno
Lefmann
TU Chemnitz
Edge Colorings of Graphs and Fixed Forbidden Monochromatic Subgraphs
Abstract.
Given a fixed positive integer r and a fixed graph F, we consider for host-graphs H on n vertices, n large, the number cr,F(H) of r-colorings of the set of edges of H such that no monochromatic copy of F arises. In particular, we are looking at the maximum cr,F(n) of cr,F(H) over all host-graphs H on n vertices. For forbidden fixed complete graphs F = Kℓ the quantity cr,F(n) has been investigated by Yuster, and Alon et al. and they proved for r = 2 or r = 3 colors that the maximum number of r-colorings is achieved by the corresponding Turán graph for F, while for r ≥ 4 colors this does not hold anymore. Based on similar results for some special forbidden hypergraphs, one might suspect that such a phaenomenon holds in general. However, it seems this is not valid for (at least some) classes of forbidden bipartite graphs.
Nicolai
Beck
FU Berlin
What is a sheaf?
Abstract.
A basic object in mathematics is the set of certain functions on a space, for example continuous functions on a topological space or differentiable functions on a manifold. A presheaf is a first generalization of this notion, which is easily defined but can have unexpected properties. Sheaves are presheaves which behave nicely. This definition leads to some technical difficulties. However, it is just this difficulty which allows us to define sheaf cohomology. Finally one can show that for many spaces sheaf cohomology agrees with singular cohomology.
Anita
Liebenau
FU Berlin
A survey on the 'Cops-and-robbers problem'
Abstract.
Cops and robbers - in theory almost as thrilling as in an action film. The game was introduced by Nowakowski and Winkler, and by Quilliot more than thirty years ago: A number of cops moves along the edges of a graph and try to catch the robber. First, the cops move (at most) one step, then the robber. The cop number c(G), introduced by Aigner and Fromme in 1982, is the minimal number of cops needed in G in order to catch the robber. Numerous questions connected with this game have been posed and partly answered: What kind of graphs can be covered by one cop alone (the so-called cop-win graphs)? How many cops are enough to catch the robber on a planar graph? What can we say about graphs embeddable in orientable surfaces of genus g? How many cops do you need/are enough on general graphs? Variations include a drunken robber (who moves randomly, and not in an intelligent way), a very fast robber (who may move along longer paths in one step), and non-perfect information. In the talk, I will define all necessary concepts, and give an overview of what has been done in these last three decades. Further, I will sketch a proof by Scott and Sudakov (2011) of one of the most recent upper bounds on c(G).
George
Walker
Bristol
Zeta functions of plane curves
Abstract.
The problem of computing zeta functions of varieties over finite fields has received considerable interest in recent years, particularly for the case of curves. I shall outline a new algorithm, based on $p$-adic cohomology and the work of Lauder, which may be applied to a wide range of classes of smooth curves. In particular, it matches in complexity Kedlaya's algorithm for hyperelliptic curves and its generalizations to superelliptic curves. I shall focus on its application to smooth plane curves, where the algorithm improves on the complexity of the previously best known algorithm almost by a factor of $g$, the genus.
Fabian
Müller
HU Berlin
What is a moduli space?
Abstract.
In modern algebraic geometry, moduli spaces provide a way of describing the set of isomorphism classes of various kinds of objects, such as curves, maps or vector bundles. While the points of the moduli space correspond just to these isomorphism classes, these spaces can be endowed with a much richer algebraic structure reflecting the way in which the objects under consideration behave in families. The talk will give a low-level introduction to the concepts of fine and coarse moduli spaces, classifying maps and universal families. With an eye towards the subsequent talk by Valery Alexeev, we will also take a short look at the issues one encounters when one tries to compactify such spaces.
Valery
Alexeev
U Georgia
Polytopes, tilings, and compact moduli of algebraic varieties
Endre
Szemerédi
Rényi Institute of Mathematics
Long Arithmetic Progression in Sumsets
Abstract.
We are going to give exact bound for the size of longest arithmetic progression in sumset sums. In addition we describe the structure of the subset sums and give applications in number theory and probability theory. (this part is partially joint work with Van Vu)
Torsten
Ueckerdt
Introducing the bar visibility number of a graph
Ananyo
Dan
HU Berlin
Noether-Lefschetz locus on projective hypersurfaces
Abstract.
Noether's theorem states that a generic degree $d$ smooth hypersurface in $ ackslashegin{P}^3ackslashegin{end}$ has picard number 1. We define the Noether-Lefschetz to be the smooth hypersurfaces in $ackslashegin{P}^3ackslashegin{end}$ of degree $d$. In my talk we study the geometry of this locus including when is an irreducible component of this locus non-reduced.
May 2011
Gérard
Freixas i Montplet
Univ. Paris 6
Arithmetic Riemann-Roch theorem and Jacquet-Langlands correspondence
Abstract.
In this talk we will review the arithmetic Riemann-Roch theorem for pointed curves and we will show how to combine it with the Jacquet-Langlands correspondence, in order to get equalities between certain arithmetic self-intersection numbers on modular and Shimura curves.
Beatrice
Bugert
WIAS
What is the direct method in the calculus of variations?
Abstract.
One of the fundamental problems in the calculus of variations consists of finding a function $u$ minimizing the integral functional \[ I(u) = \int_\Omega f(x, u(x), Du(x)) \ dx \] over all the functions $u$ satisfying $u = u_0$ on the boundary $\partial \Omega$ of $\Omega$, where $u_0$ is a given function. Euler--often referred to as the founder of the calculus of variations--treated this problem by deducing the so-called Euler-Lagrange equation from the integral functional. He proved that in the case of convex functionals solutions of this equation are already minimizers of $I(u)$. As this method is hard to implement for higher dimensional integrals (i.e., not one-dimensional ones), there was a great need to find an alternative method avoiding the Euler-Lagrange equations. It was Riemann who finally succeeded at this task and introduced the so-called direct method in the calculus of variations, which provides the existence of minimizing functions $u$ directly from the properties of the functional $I$. This talk will give an overview of Riemann's method for convex functionals and show how it has further developed over almost two centuries under the influence of the Italian mathematicians Tonelli and De Giorgi.
Ulisse
Stefanelli
IMATI
Evolution = Minimization ?
Aaron
Greicius
HU Berlin
Images of Galois representations attached to l-Tate modules
Abstract.
Let A/K be a principally polarized abelian variety with trivial endomorphism ring. In many cases (for example, when dim(A) is 2,6 or odd), the various l-adic Galois representations to GSp associated to the l-Tate modules of A/K are surjective for all but finitely many primes l. In such situations one hopes to find an algorithm for finding, or at least bounding, the finite set of exceptional primes where the Galois representation fails to be surjective. We will consider existing algorithms for elliptic curves and abelian surfaces over *Q* and endeavor to extend these to more general number fields. Time permitting, we will also examine the situation when the endomorphism ring of A is larger than *Z*.
Timothy
Gowers
Cambridge
The internet and new ways of doing mathematics
Timothy
Gowers
University of Cambridge
A combinatorial proof of the density Hales-Jewett theorem
Irina
Mustata
Tolerance graphs as trapezoid graphs and NP-completeness
Vincenz
Busch
Univ. Hamburg
Beilinson's conjecture for K_2 of a superelliptic curve
Abstract.
The Beilinson conjecture generalize and unify multiple theorems and conjectures in arithmetic geometry, e.g. the class number formula and the BSD conjecture. In the case of K_2 of algebraic curves the effects of the Beilinson conjectures can actually be observed in concrete calculations. In this talk, we will cover an example of such calculations in the case of a superelliptic curve.
Daniel
Heldt
TU Berlin
What is a graph?
Abstract.
A short introduction to graphs, which contains some pictures (of graphs) as well as (some of) the definitions in graph theory, like cliques, bipartite graphs, spanning trees, colorings, adjacency matrices, and so on.
Dan
Spielman
Yale U
Approximating Graphs and Solving Systems of Linear Equations
Dan
Spielman
Yale University
On some problems that have been bugging me
Dennis
Clemens
FU Berlin
The uniqueness conjecture of Markoff numbers and equivalent problems: Part II
Marie
Albenque
On the enumeration of simple symmetric quadrangulations
Lenny
Taelman
Univ. Leiden
Special values in characteristic p
Abstract.
I will present a theorem which is a characteristic-p-valued function field analogue of the class number formula and the Birch and Swinnerton-Dyer conjecture. In these special value formulas the multiplicative group (for the CNF) and elliptic curves (for BSD) are replaced by Drinfeld modules. Reference: [http://arxiv.org/abs/1004.4304].
Thomas
El Khatib
TU Berlin
What is
Who?
Thomas El Khatib (TU Berlin)
When?
2011/05/06, 16:00
Where?
TU Berlin, at the BMS Lounge, MA 212
About what?
In the 17th century, Leibniz and Newton invented calculus using infinitesimally small quantities. 200 years later, Bolzano, Cauchy and Weierstraß made calculus rigorous by introducing the modern epsilon-delta-formulation of limits, bereaving mathematics of intuitively appealing objects. Still, another 150 years later, students of physics and engineering are still taught to think in terms of infinitesimals, with or without the warning never to mention them in the presence of a mathematician.
In this talk, I will discuss some ways of rigorously introducing infinitesimals in modern day mathematics, and — if time admits — I will talk about a model theoretic approach by Robinson/Zakon in more detail.
Abstract.
In the 17th century, Leibniz and Newton invented calculus using infinitesimally small quantities. 200 years later, Bolzano, Cauchy and Weierstraß made calculus rigorous by introducing the modern epsilon-delta-formulation of limits, bereaving mathematics of intuitively appealing objects. Still, another 150 years later, students of physics and engineering are still taught to think in terms of infinitesimals, with or without the warning never to mention them in the presence of a mathematician. In this talk, I will discuss some ways of rigorously introducing infinitesimals in modern day mathematics, and — if time admits — I will talk about a model theoretic approach by Robinson/Zakon in more detail.
Daniel
Heldt
Groebner bases for order theorists
Dennis
Clemens
FU Berlin
The uniqueness conjecture of Markoff numbers and equivalent problems: Part I
Abstract.
In 1879/80 Andrei A. Markoff studied the minima of indefinite quadratic forms and found an amazing relationship to the Diophantine equation x2 + y2 + z2 = 3xyz, which today is named the Markoff equation. Later Georg Frobenius conjectured that each solution of this equation is determined uniquely by its maximal component. Defining these components as the so-called Markoff numbers this conjecture means that for each Markoff number m there exists up to permutation exactly one triple (m, p, q) with maximum m solving the Markoff equation. Till this day it is not known whether the conjecture is true, or not. However, we are able to prove uniqueness for certain Markoff numbers and we can find different statements, such as a statement in Markoff's theory on the minima of quaratic forms, being equivalent to the uniqueness conjecture of the Markoff numbers. At first I will give a short overview on the properties of Markoff numbers as well as some easier results concerning the uniqueness conjecture. We will see how the solutions of Markoff's equation, called Markoff triples, can be computed recursively such that we will get a tree of infinitely many Markoff triples. Then, by using correspondences to other trees we will get different statements from Number Theory, Graph Theory and Linear Algebra being equivalent to the uniqueness conjecture. The second part of my presentation will be concerned with the best known result on this conjecture proven by J. O. Button in 2001. With a bijection between Markoff triples and elements in certain number fields we will see how the uniqueness problem can be reformulated as a question in Ideal Theory. Taking results on ideals and continued fractions we finally can conclude a criterion proving uniqueness for a big subset of Markoff numbers.
Dennis
Eriksson
Univ. Göteborg
Multiplicities of discriminants
Abstract.
I will discuss some recent formulas for multiplicities of discriminants of polynomials, generalising in one direction those of Ogg's formula for the discriminant of elliptic curves. In the particular case of discriminants of planar curves we obtain more precise information, and we can relate it to finite contributions of Arakelov intersection numbers.
April 2011
Thomas
Lehmann
ZIB
What is the Kuhn-Tucker theorem?
Abstract.
The Kuhn-Tucker theorem is the foundation of nonlinear programming. Requiring a constraint qualification, it states necessary conditions for local solutions of nonlinear programs. Even more, these (Karush-)Kuhn-Tucker conditions are the basis of many efficient nonlinear programming algorithms.
John M.
Sullivan
TU Berlin
Ropelength criticality for knots and links
Veit
Wiechert
Planar Posets and Dimension 2
Hartwig
Mayer
HU Berlin
On Zhang's admissible intersection theory
Abstract.
In 1993, S.-W. Zhang introduced (based on previous work of T. Chinburg and R. Rumely) an intersection theory for smooth, irreducible curves $X/K$ over a local field $K$ by defining a potential theory on the dual of the reduction graph $R(X)$ of $X$. This theory was used to give first answers to the Bogomolov conjecture. In the first part of this talk, we present Zhang's intersection theory and its arithmetic implication in case of a modular curve. In the second part, we present recent developments in extending Zhang's potential theory to Berkovich curves, and end with some open questions concerning a higher-dimensional analogue, which would be of great interest in the context of Arakelov theory.
Thomas
Hixon
Crossing a bridge at night
Anil
Aryasomayajula
HU Berlin
Estimates of the Canonical Green's Function
Abstract.
In this talk, I will try to present the estimates obtained for the canonical Green's function associated to the canonical metric, on non-compact finite volume hyperbolic Riemann surfaces of genus g > 1, in terms of invariants from hyperbolic geometry.
Anilatmaja
Aryasomayajula
HU Berlin
What is a modular form?
Abstract.
Modular forms are complex analytic functions on the upper half plane satisfying a certain kind of functional equation and growth condition. In this talk we will see how the theory of modular forms answers a classical problem in number theory, namely: "Which natural numbers can be represented as the sum of four squares, and in how many ways can that be done?".
Katrin
Bringmann
U Köln
Mock theta functions, probability and cellular automata models
Kolja
Knauer
A graph-theoretical axiomatization of oriented matroids
March 2011
Alex
Bartel
POSTECH, Pohang Univ., Korea
Some applications of integral group representations in number theory
Abstract.
Certain quotients of regulators of number fields or of abelian varieties can be interpreted as purely representation theoretic invariants. I will introduce a technique for analysing such invariants and will apply it to some questions on the arithmetic of number fields and of elliptic curves. One of the main results will be a curious identity linking the fine integral Galois module structure of certain units of number fields and of Mordell-Weil groups to sizes of class groups and of Tate-Shafarevich groups, respectively. There will be plenty of examples along the way, and also some mystery and food for future thought. The talk will be accessible to graduate students with a very modest background in number theory and ordinary representation theory.
Julia
Rucker
On Rectangle Contact Representations
Kai
Lawonn
The Colin de Verdiere Graph Parameter
Verena
Richter
Pflasterung orthogonaler Polygone mit Rechtecken
February 2011
Fiona
Skerman
Australian National University
Row and column sums of random 0-1 matrices
Abstract.
Construct a random m × n matrix by independently setting each entry to 1 with probability p and to 0 otherwise. We study the joint distribution of the row sums s = (s1, ..., sm) and column sums t = (t1, ..., tn). Clearly s and t have the same sum, but otherwise their dependencies are complicated. We prove that under certain conditions the distribution of (s, t) is accurately modelled by (S1, ..., Sm, T1, ..., Tn), where each Sj has the binomial distribution Binom(n, p'), each Tk has the binomial distribution Binom(m, p'), p' is drawn from a truncated normal distribution, and S1, ..., Sm, T1, ..., Tn are independent apart from satisfying Σj=1,...,m Sj = Σk=1,...,n Tk. We also consider the case of random 0-1 matrices where only the number of 1s is specified, and also the distribution of s when t is specified. In the seminar I will include details of one of the bounding arguments used in this last case. This bounding argument is an application of the generalised Doob's martingale process. These results can also be expressed in the language of random bipartite graphs. Joint work with Brendan McKay.
Stefan
Felsner
Contact Representations of Planar Graphs with Weights
Katherine
Edwards
McGill University
Packing T-joins in Planar Graphs
Abstract.
Let G be a graph and T an even sized subset of its vertices. A T-join is a subgraph of G whose odd-degree vertices are precisely those in T, and a T-cut is a cut δ(S) where S contains an odd number of vertices of T. It has been conjectured by Guenin that if all T-cuts of G have the same parity and the size of every T-cut is at least k, then G contains k edge-disjoint T-joins. We discuss some recent progress on this conjecture and related results.
Matthias
Lenz
TU Berlin
What is a matroid?
Stefan W.
von Deylen
FU Berlin
What is a weak derivative?
Abstract.
...or: Plugging objects into an equation that it was absolutely not thought for. Of course, all of you know the partial integration rule: $\int f' g = -\int f g'$ plus some boundary term we will impudently ignore. In your Analysis I course, both functions needed to be differentiable. But what if $f$ were piecewise differentiable? Perhaps even with jumps in the function values? Or unbounded, yet still integrable? In this talk, we will think about other ways to define $f'$ while keeping this equation valid. More precisely, we require nothing from real calculus, only this single equation. Surprisingly, this requirement, all alone in the world, is not as lonely, lost and feeble as it seems, but already leads to a huge and beautiful theory (which could answer many questions from partial differential equations or calculus of variations, but that would be going too far). I will first introduce the most general notion for weak derivatives, the language of distributions. But you may recall your analysis courses on continuity or connectedness: What seems the most natural general definition is, in the domain of analysis, often a very nasty-behaving object. We will take a short look at the pitfalls we foolishly did not exclude in that first attempt and quickly move on to Sobolev spaces. As much as we can fit in half an hour, this will be the ultimate answer to life, universe and the Dirichlet problem.
Torsten
Ueckerdt
The Isometric Dimension of Median Graphs via a Generalization of Birkhoff's Representation Theorem
Oliver
Friedman
LMU München
On Exponential Lower Bounds for Solving Infinitary Payoff Games and Linear Programs: Part II
Abstract.
Policy iteration is one of the most important algorithmic schemes for solving problems in the domain of determined game theory such as parity games, stochastic games and Markov decision processes, and many more. It is parameterized by an improvement rule that determines how to proceed in the iteration from one policy to the next. It is a major open problem whether there is an improvement rule that results in a polynomial time algorithm for solving one of the considered game classes. Simplex algorithms for solving linear programs are closely related to policy iteration algorithms. Like policy iteration, the simplex algorithm is parameterized by a so-called pivoting rule that describes how to proceed from one basic feasible solution in the linear program to the next. Also, it is a major open problem whether there is a pivoting rule that results in a (strongly) polynomial time algorithm for solving linear programs. We describe our recent constructions for parity games that give rise to superpolynomial and exponential lower bounds for all major improvement rules, and how to extend these lower bounds to more expressive game classes like stochastic games. We show that our constructions for parity games can be translated to Markov decision processes, transferring our lower bounds to their domain, and finally show how the lower bounds for the MDPs can be transferred to the linear programming domain, solving problems that have been open for several decades.
Anna-Maria
von Pippich
HU Berlin
The Weierstrass $\wp$-function and transcendence questions
Jannik
Matuschke
TU Berlin
What is P vs. NP?
Abstract.
Complexity theory is a branch of theoretical computer science that deals with the limits of efficient computability. Complexity classes allow us to compare the computational hardness of problems from a theoretical point of view. We will give formal definitions of the classes P (“problems that can be solved efficiently by a deterministic computer as we all know it”) and NP (“problems that can be solved efficiently by a non-deterministic computer that can guess the answer and only needs to verify its correctness”) and motivate the importance of the great open question whether or not P = NP.
David P.
Williamson
Cornell U
What Computers Can Compute
Marie
Albenque
A bijection between fractional trees and pentagulations of girth 5
Lejla
Smajlović
Univ. of Sarajevo
On the distribution of the zeros of the derivative of the Selberg zeta function
Han
Lie
HU Berlin
What is the exit problem?
Abstract.
Consider the behaviour of a deterministic dynamical system in a domain containing a stable attracting equilibrium — for example, a particle trapped in a potential well, or a marble dropped into a large bowl. Given suitable initial conditions, the system will move towards the equilibrium and stay there once it has arrived at the equilibrium. However, if the dynamical system is perturbed by white noise, is it possible that the system might exit the domain of attraction? If it is possible, what conditions do we need? Where and when is the system most likely to exit the domain? These are the questions involved in the “exit problem”, which is often encountered in large deviations theory. In this talk we will take a different perspective from large deviations theory, and instead apply ideas from control theory to answer these questions in the context of linear systems.
Jürg
Kramer
HU Berlin
On a result of Hoffstein-Lockhart
Abstract.
Information not provided.
January 2011
Felix
Breuer
FU Berlin
What is math blogging?
Abstract.
Social media are omnipresent today! But what is the role of social media in mathematics? This question is still wide open and in this talk I want to give a glimpse of the developments in this area. The focus will lie on mathematical blogs, but mathoverflow, the polymath project and the use of blogs in other sciences will also be addressed. Moreover, I will use the opportunity to present Mathblogging.org, an aggregator for mathematical blogs that can serve both as an index as well as a starting point for exploring the mathematical blogosphere.
Bernd
Wegner
Zentralblatt für Mathematik
Zentralblatt MATH - the gateway to mathematics literature
Dmitry
Shabanov
Moscow State University
The Problem of Erdős and Hajnal Concerning Colorings of Hypergraphs and its Generalizations
Abstract.
The talk is devoted to the classical exremal combinatorial problem that was stated by P.Erdős and A.Hajnal in the 60-s. The task is to find the value m(n, r) equal to the minimum number of edges in an n-uniform non-r-colorable hypergraph. This problem has a lot of generalizations and is closely related to the classical problems of Ramsey theory. We shall discuss the known bounds in the Erdős-Hajnal problem, present some new results and pay special attention to the probabilistic methods, by which these results have been obtained.
Richard
Sieg
Dissections of polygons and the cylinder into triangles of equal areas
Barbara
Jung
HU Berlin
On the volume formula of the fundamental domain of the Siegel modular group
Natasa
Djurdjevac
FU Berlin
What is a random walk on a network?
Abstract.
Networks are widely used to characterize and model a broad range of complex systems in various fields from biology to the social sciences. Important information about the topology and dynamics of the network can be obtained by analyzing random walks on the network. This talk will give the theoretical background of the random-walker-based approach for complex networks and address some of the current challenges, such as partitioning of networks and identifying important nodes.
Tibor
Szabó
FU Berlin
The Local Lemma is tight for SAT II
Abstract.
I will give a construction of unsatisfiable k-SAT formulas, where each variable is contained in only a few clauses. The numerical value of "few" is asymptotically best possible. Joint work with Heidi Gebauer and Gabór Tardos.
Irina
Mustata
Grid Intersection Graphs
Jay
Jorgenson
City College New York/BMS
Progress in bounds for Fourier coefficients of modular forms
Stefan
Keil
HU Berlin
What is a Diophantine equation?
Abstract.
As a warmup to Professor Funke's lecture, we introduce Diophantine equations and present a famous example of how elliptic curves can be employed in their solution: namely, Fermat's Last Theorem.
Tibor
Szabó
FU Berlin
and
Gabór
Tardos
Rényi Institute of Mathematics
The Oberwolfach Problems
Jens
Funke
U Durham
Hecke and Langlands - A little bit of number theory
Stefan
Felsner
Contact representations of planar graphs with cubes
Jens
Funke
Univ. of Durham/BMS
On a theorem of Hirzebruch and Zagier
Torsten
Ueckerdt
Online Coloring of Bounded-Tolerance Graphs
December 2010
Tibor
Szabó
FU Berlin
The Local Lemma is tight for SAT
Abstract.
We construct unsatisfiable k-CNF formulas where every clause has k distinct literals and every variable appears in at most (2/e + o(1))2k/k clauses. The lopsided Loca Lemma shows that our result is asymptotically best possible. The determination of this extremal function is particularly important as it represents the value where the k-SAT problem exhibits its complexity hardness jump: from having every instance being a YES-instance it becomes NP-hard just by allowing each variable to occur in one more clause. We also consider the related extremal function l(k) which denotes the maximum number, such that every k-CNF formula with each clause containing k distinct literals and each clause having a common variable with at most l(k) other clauses, is satisfiable. We establish that l(k) = (1/e + o(1))2k The SAT-formulas are constructed via special binary trees. In order to construct the trees a continuous setting of the problem is defined, giving rise to a differential equation. The solution at 0 diverges, which in turn implies that the binary tree obtained from the discretization of this solution has the required properties. Joint work with Heidi Gebauer and Gabór Tardos.
Thomas
Hixon
On the crossing number of complete and complete bipartite graphs
Brendan
Creutz
Univ. Bayreuth
Large Tate-Shafarevich Groups
Abstract.
For an abelian variety A over a number field k, the Tate-Shafarevich group of A/k parameterizes principal homogeneous spaces for A/k which have points over every completion of k. It is conjectured that this group is finite. However, there are several results in the literature which show that this group can be arbitrarily large. We will discuss some of these and show, in particular, that the p-torsion in the Tate-Shafarevich group of any principally polarized abelian variety over k is unbounded as one ranges over extensions of k of degree O(p).
Roland
Friedrich
HU Berlin
What is cognitive neuroscience?
Abstract.
In this short talk we shall focus on probably the single most important tool in doing research in cognitive neuroscience, namely, functional magnetic resonance imaging, (fMRI). We shall explain how with this method questions concerning higher cognitive processes in the brain are investigated and what kind of inferences can be made.
Yury
Person
FU Berlin
Extremal Hypergraphs for Hamilton Cycles
Abstract.
We study sufficient conditions for various Hamilton cycles in k-uniform hypergraphs and obtain both Turán- and Dirac-type results. In particular, we show that the only extremal 3-uniform hypergraph (for n moderately large) not containing a loose Hamilton cycle on n vertices consists of the complete hypergraph on n - 1 vertices and an isolated vertex (thus answering a question of Woitas). More generally, we determine extremal hypergraphs for so-called l-tight Hamilton cycles and we give first sufficient conditions on the minimum degree δ of type c(n choose k - 1), with fixed c < 1 and n sufficiently large, that ensure the existence of Hamilton cycles. Joint work with Roman Glebov and Wilma Weps.
Rafael
Nunez
UCSD
What then is Mathematics? A view from Cognitive Science
Daniel
Heldt
Some remarks on the behavior of a local operating Markov chain on the set of k-heights
Jose Ignacio
Cogolludo
Zaragoza
On the connection between the topology of plane curves, the position of their singular points, and elliptic threefolds
Anita
Liebenau
FU Berlin
What is the probabilistic method?
Abstract.
To give an idea of the power of probabilistic arguments in graph theory, I will show two results of different flavour. The first comprises the statement that for an arbitrary graph G there a is partition of the vertices such that the resulting bipartite graph has at least half the number of edges of G. For the second, I will introduce the random graph model G(n,p) and the notion of a threshold function. We will then take a short look at the property of having isolated vertices. Hopefully, this will pave the way to a better understanding of the talk by Joel Spencer for people from outside the area.
Joel
Spencer
NYU
Der Zauberer von Budapest - Paul Erdös and the Rise of Discrete Mathematics
Michael
Krivelevich
Tel Aviv University
The Size Ramsey Number of a Directed Path
Abstract.
Given a (di)graph H and an integer q ≥ 2, the size Ramsey number re(H, q) is the minimal number m for which there is a (di)graph G with m edges such that every q-coloring of G contains a monochromatic copy of H. We study the size Ramsey number of the directed path of length n in oriented graphs, where no antiparallel edges are allowed. We give nearly tight bounds for every fixed number of colors, showing that for every q ≥ 2, the corresponding number re(H, q) has asymptotic order of magnitude n2q - 2 + o(1). A joint work with Ido Ben-Eliezer (Tel Aviv U.) and Benny Sudakov (UCLA).
November 2010
Luis
Dieulefait
Universitat de Barcelona
Modularity by propagation: Serre's conjecture and non-solvable base change for GL(2)
Barbara
Jung
HU Berlin
What is the origin of elliptic functions?
Abstract.
Elliptic functions are double-periodic meromorphic functions on C, that means basically $$f(a+ib)=f((a+n)+i(b+m)) \qquad \forall a, b \in \mathbb{R},\; n, m \in \mathbb{Z}.$$ So they define a function on a torus. But what has this to do with an ellipse? To find the answer, we go on a journey to the origins of Riemann surfaces in the times of Euler and Lagrange and see the theory beautifully arising from the observation of integrals over some simple curves we already know from school.
Jean-Benoît
Bost
U Paris-Sud Orsay
The work of Jacobi on elliptic functions: an introduction for today's mathematician
Veit
Wiechert
Planar posets and dimension
Klebert
Kentia
HU Berlin
What is a multifunction?
Abstract.
Multifunctions, or multivalued mappings or correspondences, are “functions” that assign to a fixed point one or several values. They can be viewed as set-valued mappings and turn out to be very interesting objects in many areas of mathematics (e.g. optimization, probability, functional analysis). This motivates the need for a nice and thorough analysis of these objects. During this analysis, some questions arise when one indeed considers correspondences as mappings taking values in the power set of a given set. Of particular importance is how one defines a useful (mainly in application) concept of measurability (i.e. conservation of information by inverse image) for such mappings. If this is at all possible, then can one associate to a measurable correspondence a suitable notion of integral? An even more interesting question is that of the existence of a measurable selection of a correspondence (i.e. a measurable function that takes values in the values of the correspondence). The purpose of this talk will be to attempt to address some of these questions.
Remke
Kloosterman
HU Berlin
Mordell-Weil groups, Syzygies and Brill-Noether theory
Abstract.
For a particular class of elliptic threefolds with base P^2 we discuss the relation between the rank of the Mordell-Weil group and the syzygies of the singular locus of the discriminant curve of the elliptic fibration. From this we deduce that for each high rank elliptic threefold we find examples of triples (g,k,n) such that the locus {[C] M_g | C admits a g^2_{6k} and the image of C has at least n cusps} has much bigger dimension than expected.
Heinrich
Mellmann
HU Berlin
What is a selflocator?
Abstract.
Whatever we do, wherever we go, most of the time we know “where” we are—in Berlin, at home, in front of the fridge, etc. For a successful interaction with the environment (e.g., switching on the TV set) the knowledge of one's own position relative to the object is essential. Thus, an artificial being, like a robot, which should perform similar to a human also has to know its environment and especially its position in it. In this talk we introduce some mathematical groundings for Bayesian modeling and discuss, in particular, the Monte-Carlo particle filter which can be used to model the position of a robot in a dynamic world. We will also illustrate the gray theory with interesting videos and examples from robot soccer.
Tobias
Mueller
Line arrangements and geometric graph classes
Stefan
Keil
HU Berlin
The Sato-Tate Conjecture and the L-Function Method
Abstract.
In the last years Tayler et al. showed that the m-th symmetric power of the L-functions of l-adic Galois-representations of an elliptic curve over QQ are potential automorphic. Therefore they fulfill certain meromorphic properties which are sufficient to use the so called L-function method to prove the Sato-Tate Conjecture (over QQ). We will talk in general about equidistribution and the L-function method and its application (e.g. the prove of Chebotarev's density theorem). We will explain why one can apply the L-function method for the Sato-Tate Conjecture. If there is time we will discuss the exceptional case of CM-curves.
Ágnes
Cseh
TU Berlin
What is a stable marriage?
Abstract.
This definitely important question can be answered with the help of graph theory. The stable marriage theorem of Gale and Shapley states that for some men and women there always exists a stable marriage scheme, that is, a set of pairs such that no man and woman mutually prefer each other to their partners in the matching. The stable marriage problem can be extended in several directions, one of the most recent topics deals with network flows. Besides sketching some theorems and unanswered questions we will give some useful hints to find a stable partner in real life.
Marie-Françoise
Roy
U Rennes
Certificates of positivity
Mathew
Francis
The class of segment graphs
Anilatmaja
Aryasomayajula
HU Berlin
Bounds on canonical Green's function
October 2010
Nahid
Walji
HU Berlin
The Lang Trotter conjecture on average and congruence class bias
Jean-Philippe
Labbé
Technische University Berlin
What is a Catalan number?
Abstract.
Catalan numbers form a sequence of natural numbers that counts a plethora of objects in mathematics. Thus, they appear constantly in enumerative combinatorics and many related fields like algebra, discrete geometry, representation theory and more. I will present their origin, discuss about some recursive and closed formulas and finally show some objects that are counted by Catalan numbers.
Don
Zagier
MPI Bonn
Mathematics in Another World - A report on Japanese mathematics in the Edo period
September 2010
Stefan
Felsner
d-Schnyder structures
August 2010
Torsten
Ueckerdt
CAT-Enumeration of the Ideals of a Planar Poset
July 2010
Stefan
Felsner
Square Tilings and Extremal Length
Aner
Shalev
Hebrew U Jerusalem
Words: From Number Theory to Finite Simple Groups
Daniel
Heldt
A glimpse at the kernel method and Knuth's approach to the ballot problem
Kaie
Kubjas
FU Berlin
What is phylogenetic algebraic geometry?
Abstract.
Phylogenetic algebraic geometry studies algebraic varieties arising from evolutionary trees. In this talk I will explain based on examples how to construct these algebraic varieties.
Gábor
Mészáros
HU Berlin
Road Coloring Problem
Abstract.
The road coloring problem deals with the question whether there exists an edge-coloring of a network such that by using special instructions, one can reach or locate an object or destination from any other point within the network. Precisely, if you have a directed graph G with regular out-degree k, the main question is, whether you can color the edges such a way, that: all k edges leaving any vertex have distinct colors and there is a suitable sequence w of colors, such that no matter which vertex we choose as start-point we get the same end-point, reading the word w label to label and choosing the suitable edge (according to w), on which we can travel to the next vertex. It was first conjectured in 1970 by Weiss and Adler, that one can always find an appropriate coloring with a suitable word, if the digraph is strongly connected and aperiodic, that is, there is no integer k > 1 that divides the length of every cycle of the graph. In 2007 Trahtman proved the conjecture (now it is called Road coloring theorem). In my talk I will present a partial result proved by Kari (2002), which says that the theorem is true if the graph is eulerian and regular, that is, all vertices have in and outdegree k.
Morten
Risager
Univ. Kopenhagen
Non-vanishing of Fourier coefficients, Poincaré series, and central values of L-functions.
Abstract.
We discuss Fourier coefficients of modular forms at cusps and non-cuspidal values. We show, without to much effort, that 'generically' these coefficients are all non-vanishing. Yet it is highly non-trivial to prove that for a specific point z_0 the coefficients are non-vanishing. In the simplest case of the discriminant function the non-vanishing of Fourier coefficients at infinity is an old conjecture of Lehmer's. We show how the non-cuspidal analogue of this conjecture is true for certain CM-points. We then discuss how this has applications to non-vanishing of certain Poincaré series and to non-vanishing of certain central values of L-functions. This is joint work with Cormac O'Sullivan.
Hanna
Döring
TU Berlin
What is large deviations?
Abstract.
A classical result in probability theory is the law of large numbers. For a long sequence of coin tosses we expect half the coins to show head. Large deviations describe the probability of so-called rare events, that differ from its expected behaviour. Starting with the example of coin tossing we learn about Cramér's theorem and point out some applications.
Alison
Etheridge
U Oxford
The pain in the torus: modelling populations in a spatial continuum
June 2010
Gérard
Freixas
Univ. Paris 6
Generalizing analytic torsion
Shahrad
Jamshidi
FU Berlin
What is a hybrid automaton?
Abstract.
A thermostat maintains a constant temperature via a switching mechanism. This discrete switch and the evolution of the temperature can be modelled using a hybrid automaton. In this talk we will discuss the advantages and disadvantages of this modelling method.
Anusch
Taraz
TU Munich
On co-prime labellings of trees
Abstract.
A conjecture by Entringer from around 1980 states that the vertices of every n-vertex tree can be labelled with numbers 1 up to n in such a way that adjacent vertices get co-prime labels. In joint work with P.E. Haxell and O. Pikhurko we recently managed to prove this conjecture for sufficiently large n. I will discuss the proof in this talk.
Petr
Gregor
Linear extension diameter of subposets of Boolean lattice induced by two levels
Emerson
Leon
TU Berlin
What is Pólya theory?
Abstract.
Pólya enumeration methods help to count objects up to some symmetry. I will briefly explain how it works and then show couple of applications, were other combinatorial objects will be involved.
Doron
Zeilberger
Rutgers U
The Number of Isomers of Nicotine
Wilma
Weps
FU Berlin
On Size Ramsey Number of Graphs with Bounded Maximum Degree
Abstract.
The size Ramsey number r̂(G) of a graph G is the smallest number of integers r̂ such that there exists a graph F with r̂ edges for which every red-blue-edge-coloring contains a monochromatic copy of G. Josef Beck raised the question whether for a graph G with bounded maximum degree d there exists a constant c(d) depending only on the maximum degree such that r̂(G) < c(d)·n, with n the number of vertices of G. It has shown to be true that such a constant exists if G is a cycle or a tree. In my talk I will present a counterexample by Rödl and Szemerédi which shows that the statement above already fails if d = 3.
Frank
Monheim
Univ Tübingen
Iterierte Integrale automorpher Formen
Stephan
Müller
HU Berlin
What is the classifying space of a category?
Abstract.
I will describe an easy way to associate a (nice) topological space to a (small) category — its classifying space. Using homotopy theory we can discover Quillen's higher $K$-theories.
Balázs
Szegedy
University of Toronto
Limits of discrete structures (recent directions)
Abstract.
This talk is a status report on a recently developing subject in the frame of which big finite structures are viewed as approximations of infinite analytic structures. This framework enables one to use differential calculus, measure theory, topology and other infinite tools to analyze finite structures.
Osmanbey
Uzunkol
Konstruktion elliptischer Kurven mit vorgegebener Ordnung
Andreas
Stein
Univ. Oldenburg
Elliptic Curves and Cryptography - Some (new) attacks to the elliptic curve discrete logarithm problem
Abstract.
In recent years, elliptic curves have become objects of intense investigation because of their significance to public-key cryptography. The major advantage of ECC is that the cryptographic security is believed to grow exponentially with the length of the input parameters. This implies short parameters, short digital signatures, and fast computations. We provide a survey of elliptic curves over finite fields and their interactions with algorithmic number theory. Our main focus will be the discussion of various interesting attacks to the so-called elliptic curve discrete logarithm problem (ECDLP) and their mathematical background as well as their important impact on public-key cryptography. For several attacks, results on algebraic curves, especially hyperelliptic curves, are needed.
Ciro
Ciliberto
U Roma 2 - Tor Vergata
Geometric aspects of polynomial interpolation in more variables
Gábor
Mészáros
HU Berlin
Path Size Ramsey Numbers
Abstract.
The size Ramsey number r̂(G) of a graph G is the smallest integer r̂ such that there is a graph F of r̂ edges with the property that any two-coloring of the edges of G yields a monochromatic copy of G. An obvious upper bound is known for the size Ramsey number of an arbitrary graph G, namely r̂(G) ≤ (r(G) choose 2) where r(G) denotes the Ramsey number of G. For paths Erdős offered 100$ for a proof or disproof of the following conjectures: r̂(Pn)/n → ∞ r̂(Pn)/n2 → 0 In 1983 Beck proved that for every sufficiently large value of n r̂(Pn) < 900·n, which result answered Erdős's question. In my talk I will present Beck's proof and discuss some related results in the area, including some interesting counterexamples which tell us that Beck's result is not necessarily true for general trees.
Stefan
Müller-Stach
Univ. Mainz
Numerical characterizations of Shimura subvarieties
May 2010
Wendelin
Werner
U Paris-Sud
Zufall und Stabilität
Hartwig
Mayer
HU Berlin
Arakelov theory on the modular curve X_1(N) (Part II)
Marcus
du Sautoy
U Oxford
Through the looking glass: groups from a number theoretic perspective
Roman
Glebov
FU Berlin
On-line Ramsey Numbers II
Abstract.
In Ramsey theory, we mostly look at the number r(t) denoting the minimum number of vertices sich that a two-coloring of the edges of the clique on r(t) many vertices necessarily produces a monochromatic t-subclique (we also say an r(t)-clique "arrows" a t-clique). The size Ramsey number r'(t) is the smallest number of edges such that there exists a graph with r'(t) edges arrowing an r(t)-clique. In my two talks I present the on-line version of this problem according to a resent paper by David Conlon. In a two-players-game, Builder draws edges one-by-one, and Painter colors them as each appears. Builder's aim is to force Painter to draw a monochromatic t-clique. In my second talk on the on-line Ramsey numbers, I present a specific upper bound for r''(t) of order 4t - c(log2(t))/(loglog(t)).
Kolja
Knauer
Lattices and Set Systems
Hartwig
Mayer
HU Berlin
Arakelov Theory on X_1(N)
Aaron
Greicius
HU Berlin
What is the big deal with math and music?
Abstract.
The connection between math and music has fascinated scientists for millenia. The ancient Greek investigated relations between numerical ratios and musical scales, nowadays new composition techniques have led to applications of set theory, abstract algebra and number theory in musical theory. This talk will shed some light on the implicit and explicit math in music and vice versa.
Roman
Glebov
FU Berlin
On-line Ramsey Numbers I
Abstract.
In Ramsey theory, we mostly look at the number r(t) denoting the minimum number of vertices sich that a two-coloring of the edges of the clique on r(t) many vertices necessarily produces a monochromatic t-subclique (we also say an r(t)-clique "arrows" a t-clique). The size Ramsey number r'(t) is the smallest number of edges such that there exists a graph with r'(t) edges arrowing an r(t)-clique. In my two talks I present the on-line version of this problem according to a resent paper by David Conlon. In a two-players-game, Builder draws edges one-by-one, and Painter colors them as each appears. Builder's aim is to force Painter to draw a monochromatic t-clique. The minimum number of edges which Builder must draw is the on-line Ramsey number r''(t). The main result presented in the first talk is the fact that for in finitely many values of t, r''(t) is exponentially smaller than r'(t).
Stefan
Felsner
Transitive orientation and vertex partitioning
Aaron
Greicius
HU Berlin
Abelian varieties with large adelic image of Galois
Jerry
Gagelman
TU Berlin
What is a Fourier Series?
Abstract.
Fourier's method of analyzing differential equations has inspired so many ideas in other areas of mathematics that the name has become seemingly ubiquitous — in harmonic analysis, representation theory, number theory, and so on. In this talk I'll motivate one perspective on Fourier series through basic linear algebra. From only the most minimal theory (e.g., basic calculus), I'll explain a couple things that Fourier series reveal.
Andrew
Stuart
U Warwick
Inverse Problems
Yury
Person
HU Berlin
A conjecture of Erdős on graph Ramsey numbers
Abstract.
For a given graph G, let r(G) be the minimum number n such that any two-coloring of the edges of the complete graph Kn on n vertices yields a monochromatic copy of G. For example it is known that r(Kk) is at least 2k/2 and at most 22k and despite efforts of many researchers the constant factors in the exponents remain the same for more than 60 years! For a graph G with m edges and no isolated vertices Erdős conjectured that r(G) < 2C√(m) (this would be then best possible up to a constant factor in view of the above mentioned bounds for r(Kk)). Very recently, Benny Sudakov proved this conjecture. In my talk I will present Sudakov's proof and discuss some related results and propblems in the area.
Daniel
Heldt
Walking on a path (with loops) II
Moritz
Minzlaff
TU Berlin
p-adic Cohomology of Curves and the Calculation of Zeta Functions
April 2010
Mimi
Tsuruga
FU Berlin
What is a Killing vector field?
Abstract.
A Killing vector field is a vector field on a Riemannian manifold that preserves the metric. We will go through some basic concepts in differential geometry including Riemannian metric, isometries, and Lie derivatives, and note some nice properties of Killing fields.
Oleg
Pikhurko
Carnegie Mellon University
An analytic approach to stability
Abstract.
This is an attempt to understand how the recently developed theory of graph limits may apply to finite problem of extremal graph theory. We formulate the notion of a stable problem (meaning, roughly speaking, that almost extremal graphs have structure close to that of an extremal graph) and give an equivalent characterization in terms of graph limits. As an application, we present a new proof of the Erdős-Simonovits stability theorem.
Till
Miltzow
Punkte mit grosser Quadrantentiefe
Alessandro
Verra
Univ. Roma 3
K3 surfaces and some moduli spaces related to curves
Abstract.
Information for the abstract is not provided in the input.
Annie
Raymond
TU Berlin
What is a Theta Body?
Abstract.
The cold war is raging outside and our young hero, László Lovász, stumbles upon a body that has been stabbed! Or, in math terms: in 1979, Lovász found an upper bound for the Shannon capacity of a graph and introduced the theta body, a powerful relaxation of the stable set polytope $\text{STAB}(G)$.
Lex
Schrijver
CWI and U Amsterdam
Bounding Stable Sets in Graphs and Codes
Tibor
Szabó
FU Berlin
A combinatorial model for the diameter of a polyhedra
Abstract.
The Hirsch Conjecture states that the diameter of a d-dimensional polytope with n facets is at most n - d. The best general upper bound due to Kalai and Kleitman is n1 + log d. For constant dimension Larman showed that the diameter is linear in n. Recently, Eisenbrand, Hähnle, Razborov, and Rothvoß introduced a combinatorial model for the problem which admits both of the upper bound proofs, gives rise to a superlinear lower bound. The lower bound is proved using the Lovász Local Lemma. In the talk we present this paper.
Torsten
Ueckerdt
Coroutines and Hamiltonian Paths in Lattices
Anna
von Pippich
HU Berlin
Kronecker Limit Formula
Eyal
Ron
FU Berlin
What is Hilbert's 16th problem?
Abstract.
Ever think that the Millennium problems were the first of their kind to be posed? Not at all! In 1900, David Hilbert, the godfather of math at that time, posed a list of no less than 23 unsolved problems in various math disciplines. These problems received remarkably large attention from the math community and a solution of one bestowed the solver with huge appreciation — and, much more importantly — a modest field medal. As of today, ten of the problems have been completely solved. Another seven were "solved", where the quotation marks denote that the solution is either not fully accepted, or, worse, nobody is really sure what Hilbert meant when posing the problem. The remaining six problems still lay in the dark, waiting for a brave mathematician to one day come and save (=solve) them. An example? the 16th problem: the problem of the topology of algebraic curves and surfaces. There are two equivalent phrasings of the 16th problem. The incomprehensible one — at least for me — and the one that concerns phase plane analysis and dynamical systems tools. What are these? what's the problem? and how can you solve it and earn an easy ticket to a good post-doc position and a comfortable tenure? All of these, and a myriad of other questions, will be answered during this seminar talk! Bonus: It seems that bounty-problems generate much more media interest than regular ones. Therefore, I will personally offer a 100-Euro prize to anybody that solves this problem during the seminar talk!* *In return the solver would be obliged to include my name as a co-author in the resulting paper. Fair is fair, right?
Hendrik
Lüthen
Abzählprobleme für Pfade im Gitterstreifen
March 2010
Torsten
Ueckerdt
Enumerating all Ideals of a Poset - Some Special Cases
Torsten
Ueckerdt
How far is a graph from being an interval graph?
Anita
Liebenau
Cambridge University
Balog-Szemerédi-Gowers Theorem
Abstract.
The Balog-Szemerédi-Gowers theorem is a result in the field of additive combinatorics and gives a relationship between the sizes of sum sets and partial sum sets. Surprisingly, the proof is purely graph theoretical and relies on a statement about paths of length 3 in a bipartite graph. It is the object of this talk to develop this relationship as well as giving sketches of the proofs of the corresponding statements in graph theory. I will assume only basic definitions of graph theory and no previous knowledge about additive combinatorics.
Kolja
Knauer
Edge-intersection graphs of grid paths
Wilma
Weps
FU Berlin
Nonrepetitive colorings of graphs
Abstract.
Nonrepetitive coloring of graphs is a graph theoretic variant of the nonrepetitive sequences of Thue. A (in)finite sequence is called k-nonrepetitive (k ≥ 2) if it contains no k-repetition, i.e. a block B of the form B = CC...C (k-times) with C being a nonempty block. A vertex coloring f of a graph G is called nonrepetitive if each path P in G has a 2-nonrepetitive sequence f(P) of colors. Defining the Thue number of a graph to be the smallest number of colors needed for such a coloring, one is interested in estimating this graph invariant. I will present a theorem by Alon et al. proving the Thue number is bounded by the maximum degree of a graph; furthermore, a theorem by Kündgen and Pelsmajer showing the Thue number is bounded by the treewidth of a graph from above. However, for most classes of graphs the exact value of the Thue number remains unknown.
Julia
Rucker
Triangle contact representations of plane graphs
Gábor
Tardos
Simon Fraser University, Rényi Institute
Conflict free coloring of neighborhoods in graphs
Abstract.
A (not necessarily proper) vertex coloring of a graph is called conflict-free (with respect to the neighborhoods) if the neighborhood N(x) of any vertex x contains a vertex whose color is not repeated in N(x). We consider how many colors are needed in the worst case for conflict-free coloring of an n vertex graph. Surprisingly the answer depends very strongly on whether one considers the neighborhood N(x) to contain or exclude x itself. The results are joint with János Pach.
Agelos
Georgakopoulos
Hyperbolic graphs, fractal boundaries, and graph limits
February 2010
Stefan
Felsner
Antichains of (k+k)-free posets
Heidi
Gebauer
ETH Zürich
Game Theoretic Ramsey Numbers
Abstract.
The Ramsey Number, R(k), is defined as the minimum N such that every 2-coloring of the edges of KN (the complete graph on N vertices) yields a monochromatic k-clique. For 60 years it is known that 2(k/2) < R(k) < 4k, and it is a widely studied open problem to find significantly better bounds. In this talk we consider a game theoretic variant of the Ramsey number: Two players, called Maker and Breaker, alternately claim an edge of KN. Maker's goal is to completely occupy a Kk and Breaker's goal is to prevent this. The game theoretic Ramsey Number R'(k) is defined as the minimum N such that Maker has a strategy to build a Kk in the game on KN. In contrast to ordinary Ramsey numbers, R'(k) has been determined precisely -- a result of Beck. We will sketch a new, weaker result about R'(k) and use it to solve some related open problems.
Tomasz
Krawczyk
On-line dimension of orders
Bartosz
Walczak
Parity in graph sharing games
Chad
Schoen
Duke University, z.Zt. MPIM Bonn
A numerical test of the generalized Birch and Swinnerton-Dyer Conjecture
Abstract.
The generalized Birch and Swinnerton-Dyer Conjecture in its crudest form asserts that the rank of a Chow group is equal to the order of vanishing of an L-function at the center of the critical strip. We discuss joint work with Jaap Top and Joe Buhler in which the conjecture was put to a modest test.
Carsten
Lange
FU Berlin
What is a graph
associaedron?
Abstract.
Graph associahedra are a special breed of polytopes but appear in many different branches of mathematics. We start with basic definitions of a convex polytope, see many examples and explore some fundamental properties.
Victor
Buchstaber
Manchester/Moscow
Combinatorics of simple polytopes and differential equations
Tom
Trotter
On the Size of Maximal Antichains and the Number of Pairwise Disjoint Maximal Chains
Barbara
Jung
Univ. Mainz
A version of Luna's theorem for symplectic varieties
Abstract.
In 1973, Domingo Luna proved the existence of an etale slice for certain group actions on varieties: He showed that the algebraic quotient X//G of a variety X by a reductive group G can etale locally be described by the action of the stabilizer G_x of a point x on a subvariety S of X. A symplectic variety carries the additional structure of a non degenerate 2-form. To obtain a suitable quotient in the category of symplectic varieties, one has to pass to a certain subvariety Z_X first before taking the algebraic quotient. For this symplectic quotient, Luna's theorem had to be modified in such a way that we found a symplectic subvariety S of X such that there is an etale morphism from the symplectic quotient Z_S//G_x of S by G_x to the symplectic quotient ZX//G of X by G.
January 2010
Oliver
Sander
FU Berlin
What is a multigrid?
Abstract.
All known solvers for elliptic finite element problems get slower with increasing problem size. All solvers? No! Linear multigrid can solve even large problems in optimal time. This makes it the method of choice for many challenging application problems.
Torsten
Ueckerdt
One way to generalize interval graphs: edge-intersection graphs of elbows in the plane grid
Stephen
Meagher
Univ. Freiburg
Equations defining isogeny classes of ordinary Abelian varieties
Abstract.
The moduli space of Abelian varieties can be described in terms of equations which derive from relations holding between the theta null values of Abelian varieties. In characteristic p one can look at the action of the Frobenius and Vershiebung morphisms on these theta nulls when the variety is ordinary. From the work of Tate it is known that the characteristic polynomial of an iterate of the Frobenius morphism describes the isogeny type of an Abelian variety. Using this we describe relations between the level p theta nulls in an isogeny class of ordinary Abelian varieties. This is joint work with Robert Carls.
Hartwig
Mayer
HU Berlin
What is a language game?
Abstract.
Around the end of 19th century a new paradigm arose in philosophy known as the "linguistic turn". Philosophers, like Gottlob Frege, were focused on analyzing language and formalized its logical structure in order to distinguish between absurd, meaningless, and meaningful sentences hoping to make misunderstandings disappear. Wittgenstein's work was strongly influencing this kind of philosophy. In his early stage he was a supporter of an "ideal language". After finishing his "Tractatus Logico-Philosophicus" and keeping silent for some while (he believed he had solved all essential questions) Wittgenstein again became interested in philosophy. But now his interest was devoted to "normal language". Instead of searching for some ideal structure behind language he took it then as it is in the first place, namely as a performative act - a language game. I will talk about some basic ideas of Wittgenstein's later philosophy which will give some background for his understanding of mathematics.
Andrew
Treglown
University of Birmingham
Hamilton cycles in directed graphs
Abstract.
Since it is unlikely that there is a characterization of all those graphs which contain a Hamilton cycle it is natural to ask for sufficient conditions which ensure Hamiltonicity. One of the most general of these is Chvátal's theorem that characterizes all those degree sequences which ensure the existence of a Hamilton cycle in a graph: Suppose that the degrees of a graph G are d1 ≤ ... ≤ dn. If n ≥ 3 and di ≥ i + 1 or dn - i ≥ n - i for all i < n/2 then G is Hamiltonian. This condition on the degree sequence is best possible in the sense that for any degree sequence violating this condition there is a corresponding graph with no Hamilton cycle. Nash-Williams conjectured a digraph analogue of Chvátal's theorem quite soon after the latter was proved. In the first part of the talk I will discuss an approximate version of this conjecture. A Hamilton decomposition of a graph or digraph G is a set of edge-disjoint Hamilton cycles which together cover all the edges of G. A conjecture of Kelly from 1968 states that every regular tournament has a Hamilton decomposition. We recently proved the following approximate version of Kelly's conjecture: Every regular tournament on n vertices contains (1/2 - o(1))n edge-disjoint Hamilton cycles. I will discuss some of our techniques as well as some related open problems in the area. This is joint work with Daniela Kühn and Deryk Osthus.
Felix
Mühlhölzer
Göttingen
Surprises in Mathematics
Kolja
Knauer
Generalizing the order polytope
Fredrik
Strömberg
TU Darmstadt
On newforms and multiplicity of the spectrum for Gamma_0(9)
Abstract.
Through numerical investigations it was discovered (by two independent groups of researchers) some years ago that the spectrum of the Laplace-Beltrami operator on $\Gamma_0(9)$ possessed a peculiar feature. Namely, that there did not seem to be any eigenvalues with multiplicity one, that is, even the newforms (in the sense of Atkin-Lehner extended to non-holomorphic forms) appeared to be degenerate. The underlying phenomenon providing this multiplicity became only recently clear. I will give an overview of the proof of a precise version of this observation. The proof involves a mix of spectral theory, in terms of the Selberg trace formula, together with Hecke operators and symmetries of a (slightly overlooked) congruence group of level 3, $\Gamma^3$. I will also discuss how this relates to some of the multiplicity one results for automorphic representations.
Plamen
Turkedjiev
HU Berlin
What is a
semimartingale?
Abstract.
The theory of integration is well known to most mathematics students. We are interested here in the theory of 'good integrators'. The Riemann-Stieltjes construction of the integral allows only integrators that have finite variation to be integrators of continuous integrands. This is a problem for even the most fundamental stochastic process-the Brownian Motion-and it seems impossible to build a theory of stochastic integration with this approach. In this talk, I first show how the Riemann-Stieltjes approach fails, and then introduce the semimartingale as the general 'good integrator' for stochastic integration, and finally recover the integral for the Brownian Motion. For this talk, it will be useful for the audience to know basic measure theory and probability theory (in particular, the meaning of the term 'convergence in probability'). Due to the nature of the topic, some ideas are very technical. I will sketch these ideas very lightly. The intention of the talk is to give audience a grasp of the particularities of the theory of stochastic integrals.
Reto
Spöhel
ETH Zürich
Upper bounds for asymmetric Ramsey properties of random graphs
Abstract.
Consider the following problem: Is there a coloring of the edges of the random graph Gn,p with two colors such that there is no monochromatic copy of some fixed graph F? A celebrated result by Rödl and Rucinski (1995) states a general threshold function p0(F, n) for the existence of such a coloring. Kohayakawa and Kreuter (1997) conjectured a general threshold function for the asymmetric case (where different graphs F1 and F2 are forbidden in the two colors), and verified this conjecture for the case where both graphs are cycles. Implicit in their work is the following more general statement: The conjectured threshold function is an upper bound on the actual threshold provided that i) the two graphs satisfy some balancedness condition, and ii) the so-called KŁR-Conjecture is true for the sparser of the two graphs. We present a new upper bound proof that does not depend on the KŁR-Conjecture. Together with earlier lower bound results [Marciniszyn, Skokan, S., Steger (2006)], this yields in particular a full proof of the Kohayakawa-Kreuter conjecture for the case where both graphs are cliques.
Daniel
Heldt
Walking on a path (with loops) I
Ricardo
Menares
EPFL Lausanne
Functoriality in the Arakelov geometry of arithmetic surfaces. Applications to Hecke operators on modular curves
Abstract.
In the context of arithmetic surfaces, J.-B. Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L^2_1. We study the behavior of these groups under pullback and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are selfadjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. Using the Gross-Zagier formula and a calculation due to U. Kuehn we compute these invariants in terms of special values of L series.
Tobias
Pfeiffer
FU Berlin
and
Dror
Atariah
FU Berlin
What is a geodesic on a Riemannian manifold?
Abstract.
What does it mean to "go straight" on a sphere? What is the shortest distance between two points in a space other than the ordinary Euclidean $\mathbb{R}^n$? These two questions, and many more, are of geometrical nature, and are treated within the framework of differential geometry. The key object that is used is the manifold, which we will define in this talk. We will start from the broadest definition of a topological manifold, and end at the Riemannian one. Then we will give a basic idea and definitions of what a geodesic on a Riemannian manifold is, together with some examples.
Yury
Person
HU Berlin
Minimum H-decompositions of graphs
Abstract.
Let φH(G) be the minimum number of graphs needed to partition the edge set of G into edges (K2) and edge-disjoint copies of H. The problem is what graph G on n vertices maximizes φH(G)? Bollobás showed that for H = Kr, r ≥ 3 the only maximizer is the Turán graph. Pikhurko and Sousa extended his result for general H with χ(H) = r ≥ 3 and proved an upper bound being tr - 1(n) + o(n2), where tr - 1(n) is the number of edges in (r - 1)-partite Turán graph on n vertices. They also conjectured that φH(G) = ex(n, H). In the talk, we verify their conjecture for odd cycles, and show that the only graph maximizing φC2l + 1(G) is the Turán graph, for large n. Joint work with Lale Özkahya.
Martin
Rumpf
Bonn
Geodesics in the the space of shapes
Anna
Huber
MPII Saarbrücken
Performance and Robustness of Randomized Rumor Spreading Protocols
Abstract.
Randomized rumor spreading is a classical protocol to disseminate information in a network. Initially, one vertex of a finite, undirected and connected graph has some piece of information ("rumor"). In each round, every one of the informed vertices chooses one of its neighbors uniformly and independently at random and informs it. At SODA 2008, a quasirandom version of this protocol was proposed. There, each vertex has a cyclic list of its neighbors. Once a vertex has been informed, it chooses uniformly at random only one neighbor. In the following round, it informs this neighbor and in each subsequent round it informs the next neighbor from its list. I will talk about recent results on the performance and robustness of these two protocols, in particular about the runtime on random graphs and about the robustness on the complete graph.
Wojciech
Samotij
University of Illinois at Urbana-Champaign
Counting graphs without a fixed complete bipartite subgraph
Abstract.
A graph is called H-free if it contains no copy of H. Denote by fn(H) the number of (labeled) H-free graphs on n vertices. Since every subgraph of an H-free graph is also H-free, it immediately follows that fn(H) ≥ 2ex(n, H). Erdős conjectured that, provided H contains a cycle, this trivial lower bound is in fact tight, i.e. fn(H) = 2(1 + o(1))ex(n, H). The conjecture was resolved in the affirmative for graphs with chromatic number at least 3 by Erdős, Frankl and Rödl (1986), but the case when H is bipartite remains wide open. We will give an overview of the known results in case χ(H) = 2 and present our recent contributions to the study of H-free graphs. This is joint work with Jozsef Balogh (University of California, San Diego).
Cornelia
Dangelmayr
Neues zum Satz von Hanani und Tutte
Ananyo
Dan
HU Berlin
Classifying the biggest components of the Noether-Lefschetz locus
Abstract.
Information not provided.
December 2009
Joscha
Diehl
HU Berlin
What is a rough path?
Abstract.
Consider a $\mathbb{R}^d$-valued continuous function on $[0,1]$ that has finite length (i.e. finite variation). One can define integration with respect to such a function via the classical Riemann-Stieltjes integral. Rough path theory enables us to define integration with respect to functions of infinite length. It turns out that these paths must first be endowed (non-canonically) with more information, which leads to paths not taking values in $\mathbb{R}^d$, but some bigger space (a certain Lie group). One important area of application is stochastic analysis, where most processes are of infinite variation. Nonetheless this talk will focus on deterministic aspects and aims to be understandable with knowledge of only undergraduate mathematics.
Andrea
Hoffkamp
Funktionales Denken im propädeutischen Analysisunterricht - Ansätze und empirische Befunde zu einem qualitativen Zugang durch Computereinsatz II
Peter K.
Friz
TU Berlin
Rough paths and the Gap Between Deterministic and Stochastic Differential Equations
Andrea
Hoffkamp
Funktionales Denken im propädeutischen Analysisunterricht - Ansätze und empirische Befunde zu einem qualitativen Zugang durch Computereinsatz I
Michael
Krivelevich
Tel Aviv University
Intelligence vs Randomness: the power of choices
Abstract.
Consider the following very standard experiment: n balls are thrown independently at random each into n bins (if you are practically inclined, think about distributing n jobs at random between n machines). It is quite easy to see that the maximum load over all bins will be almost surely about ln n/lnln n. If however each ball is allowed to choose between two randomly drawn bins, the typical maximum bin load drops dramatically to about lnln n, as shown in a seminal paper of Azar, Broder, Karlin and Upfal from 1994 - an exponential improvement! The above described result is just one manifestation of a recently widely studied phenomenon, where a limited manipulation of the otherwise truly random input is capable to advance significantly various goals. In this talk I will describe results of this type, mainly focusing on the so called controlled random graph processes, where at each stage an algorithm is presented with a collection of randomly drawn edges and is allowed to manipulate this collection in a certain predefined way. Models to be defined and discussed include the Achlioptas process and Ramsey-type games on random graphs.
Stefan
Felsner
Lower bounds for on-line chain partitioning
Silvia
de Toffoli
TU Berlin
What is a Seifert
surface?
Abstract.
Giving the basic definitions and explaining the some important ideas, we will introduce the field of knot theory. We will explain the relation between a knot and its diagram and how to find invariants which allow us to distinguish different knots (or links). Thanks to an easy invariant we will prove the existence of non-trivial knots! We will introduce the concept of Seifert Surface of a knot: An orientable, compact connected surface whose boundary is the knot. We will show an algorithm to create a Seifert Surface starting form an arbitrary projection of a knot. We will see that this algorithm will be useful to calculate the genus, a knot invariant, of a certain class of knots. If time will permit we will talk of the signature of a knot, its relation with the unknotting number (Gordian distance), and the general context in which Seifert was working when he introduced his surface.
Jack
van Wijk
Eindhoven
Knots, Maps, and Tiles: Three Mathematical Visualization Puzzles
Bartłomiej
Bosek
and
Tomasz
Krawczyk
A subexponential upper bound for the on-line chain partitioning problem
Bisi
Agboola
UCSB z.Zt. HU Berlin
Restricted Selmer groups and special values of p-adic L-functions
Abstract.
I shall discuss conjectures of Birch and Swinnerton-Dyer type involving special values of the Katz 2-variable p-adic L-function that lie outsied the range of p-adic interpolation.
November 2009
Inna
Lukyanenko
TU Berlin
What is a quantum
group?
Abstract.
The term "Quantum Group" is due to V. Drinfeld and it refers to special Hopf algebras, which are the non-trivial deformations ("quantizations") of the enveloping Hopf algebras of semisimple Lie algebras or of the algebras of regular functions on the corresponding algebraic groups. These objects first appeared in physics, namely in the theory of quantum integrable systems, in the 1980's, and were later formalized independently by Vladimir Drinfeld and Michio Jimbo. In this talk I will explain the main idea of deformation and introduce the simplest and historically the first example of a Quantum Group: $U_q(\mathfrak{sl}(2))$.
Angelika
Steger
ETH Zürich
A deletion method for local subgraph counts
Abstract.
For a given graph H let X denote the random variable that counts the number of copies of H in a random graph. For subgraph counts one can use Janson’s inequality to obtain upper bounds on the probability that X is smaller than its expectation. For the corresponding upper tail, however, such bounds are not obtained easily and are known to not hold with similarly small probabilities. In this talk we thus consider the following variation of the problem: we want to find a subgraph that on the one hand still contains at roughly E[X] many H-subgraphs, and on the other hand has the property that every vertex (and more generally every small subset of vertices) is contained in ‘not too many‘ H-subgraphs. This is joint work with Reto Spöhel and Lutz Warnke.
Stefan
Felsner
Coding and Counting Arrangements of Pseudolines
Amaury
Thuillier
Lyon 1
Non-Archimedean analytic geometry and Arakelov theory
Abstract.
Arakelov geometry on an algebraic variety X over Q usually combines intersection theory on a model of X over the integers with analysis on the corresponding complex analytic variety. Relying on Berkovich's approach of p-adic analytic geometry, it is possible to replace integral models by (real) analysis at each non-Archimedean place of Q. This extension of Arakelov geometry is particulary suitable for the study of canonical heights, or to formulate and prove p-adic equidistribution theorems. I will explain this without assuming a specific knowledge of Berkovich theory.
Nicola
Tarasca
HU Berlin
What is a moduli
space?
Abstract.
Moduli Spaces are of vital importance in Algebraic Geometry. Given algebraic varieties with some fixed properties, one tries to understand the isomorphism classes and to put an algebraic structure on them. In this talk I will introduce Elliptic Curves, and I will show the classic example of Moduli Space of Elliptic Curves.
Roman
Glebov
FU Berlin
Extremal Graphs for Clique-Paths
Abstract.
In this talk, we deal with a Turán-type problem: given a positive integer n and a forbidden graph H, how many edges can there be in a graph on n vertices without a subgraph H? And how does a graph look like, if it has this extremal edge number? The forbidden graph in this talk is a clique-path, that is an k-path, where each edge is blown up to an r-clique, r ≥ 3. We determine both the extremal number and the extremal graph for sufficiently large n.
Klaus
Hulek
Hannover
Geometry of Moduli Spaces
Torsten
Ueckerdt
The Balloon Popping Problem
Peter
Bruin
Univ. Leiden
Arakelov theory and height bounds
Abstract.
In the work of Bas Edixhoven and others on computing two-dimensional Galois representations associated to modular forms over finite fields, part of the output of the algorithm is a certain polynomial with rational coefficients that is approximated numerically. Arakelov's intersection theory on arithmetic surfaces is applied to modular curves in order to bound the heights of the coefficients of this polynomial. I will explain the connection between Arakelov theory and heights, indicate what quantities need to be estimated, and give methods for doing this that lead to explicit height bounds.
Carsten
Schultz
What is Morse theory?
Abstract.
A Morse function on a differentiable manifold is a real valued function with only nice critical points. The idea behind Morse Theory is to use a function like this to obtain global information on the manifold. We will have a look at some instances of this approach.
Stefan
Felsner
Planar Bipartite Posets
Mark
Blume
HU Berlin
Losev-Manin moduli spaces and toric varieties associated with root systems
Philipp
Zumstein
ETH Zürich
Polychromatic Colorings of Plane Graphs
Abstract.
A vertex k-coloring of a plane graph G is called polychromatic if in every face of G all k colors appear. Let p(G) be the maximum number k for which there is a polychromatic k-coloring. For a plane graph G, let g(G) denote the length of the shortest face in G. We show p(G) ≥ (3g(G) - 5)/4 for every plane graph G and on the other hand for each g we construct a plane graph H with g(H) = g and p(H) ≤ (3g + 1)/4. Furthermore, we show that the problem of determining whether a plane graph admits a vertex coloring by 3 colors in which all colors appear in every face is NP-complete, even for graphs in which all faces are of length 3 or 4 only. The investigation of this problem is motivated by its connection to a variant of the art gallery problem in computational geometry. Joint work with Noga Alon, Robert Berke, Kevin Buchin, Maike Buchin, Peter Csorba, Saswata Shannigrahi, and Bettina Speckmann.
Noemi
Kurt
What is a self-avoiding random walk?
Abstract.
Imagine you are a tourist, visiting New York for the first time, going for a stroll in Manhattan. Since you don't know the city, at each crossroad you decide at random where to go next — left, right or straight — with one restriction: you never want to go back to a place you've visited before. After $n$ crossroads, how far will you be from your starting point? What kind of path will you have walked along? In this talk, we will present the probabilistic model for self-avoiding walk, and tell you what probabilists know about the answers to these questions.
Ragni
Piene
Oslo
Counting curves: the hunting of generating functions
Daniel
Heldt
Sensitivity of 3-heights on a path
Judith
Ludwig
Univ. Tübingen
Parabolic cohomology and rational periods
October 2009
Bruno
Benedetti
What is a facet
massacre?
Abstract.
We explain what it means to collapse away the faces of a given complex. This simple idea has applications in combinatorics, topology, commutative algebra and even physics. If time permits, we'll have pizza earlier than scheduled.
Christian
Schön
HU Berlin
Some \\Gamma_1(N) modular forms and their connection to the Weierstrass \\wp-function
Dominik
Scheder
ETH Zürich
Deterministic Local Search for 3-SAT
Abstract.
In an attempt to derandomize Schoening's famous algorithm for k-SAT, Dantsin et al. proposed a deterministic k-SAT algorithm based on covering codes and deterministic local search. The deterministic local search procedure was subsequently improved by several authors. I will present the main ideas behind the algorithm and the latest improvements, one of them by myself. At the heart of my improvement is the idea that recursive search-trees can sometimes be Copy-Pasted to save time.
Hernan
Leovey
HU Berlin
What is algorithmic
differentiation?
Abstract.
Practically all calculus based numerical methods for nonlinear computations are based on truncated Taylor expansions of problem-specific functions. The basic assumption of AD is that the function to be differentiated is at least conceptually evaluated by a sequence of "elemental" statements. We will see how the basic forward mode of Automatic Differentiation works and also some advanced applications to the field of high dimensional integration.
Tomislav
Doslic
University of Zagreb
Computing the bipartite edge frustration of some classes of graphs
Abstract.
The bipartite edge frustration of a graph is defined as the smallest number of edges that have to be deleted from the graph in order to obtain a bipartite spanning subgraph. The quantity is, in general, difficult to compute; however it can be efficiently computed for certain classes of graphs. I will speak about computing the bipartite edge frustration of some planar graphs, in particular fullerenes, and of some composite graphs.
Andreas
Griewank
HU Berlin
Derivative based Optimization
Till
Miltzow
Points with Some Quadrant Depth
Kira
Samol
Univ. Mainz
Dwork congruences and reflexive polytopes
René
Birkner
FU Berlin
What is a Gröbner basis?
Abstract.
For ideals $I$ of some algebra $k[x_1,...,x_n]$ a set of generators is in general not unique. When considering the technique of term orders and initial ideals one can compute a subset of the ideal, the so-called Gröbner Basis, with respect to this term order. A reduced form of this Gröbner Basis is in fact unique for a given term order and a generates the ideal. These Gröbner Bases have proven to be useful for many applications from solving polynomial equations to moduli spaces.
Tibor
Szabó
FU Berlin
Minimum degree of minimal ramsey graphs
Abstract.
A graph G is called H-Ramsey if any two-coloring of the edges of G contains a monochromatic copy of H. An H-Ramsey graph is called H-minimal if no proper subgraph of it is H-Ramsey. We investigate the minimum degree of H-minimal graphs, a problem initiated by Burr, Erdős, and Lovász. We determine the smallest possible minimum degree of H-minimal graphs for numerous bipartite graphs H, including bi-regular bipartite graphs and forests. We also make initial progress for graphs of larger chromatic number.
Kolja
Knauer
Polynomial Time Recognition of Cocircuit Graphs of Uniform Oriented Matroids
Torsten
Ueckerdt
Enumerating all Ideals of a Poset
September 2009
Stefan
Felsner
Squarings of Quadrangulations
Vincent
Pilaud
A failed construction of the multiassociahedron
Mareike
Massow
Swap Colors in Linear Extension Graphs
August 2009
Daniel
Heldt
Layer decomposition of planar graphs
July 2009
Balchandra
Thatte
Reconstruction of Pedigrees
Julia
Pap
Kernels and Sperner's Lemma
H.
Shiga
Chiba University
and
J.
Estrada Sarlabous
Havana
A Jacobi type formula in two variables with application to a new AGM; Non-hyperelliptic curves of genus 3 and the DLP
Abstract.
The index calculus algorithm of Gaudry, Thomé, Thériault, and Diem makes it possible to solve the Discrete Logarithm Problem (DLP) in the Jacobian varieties of hyperelliptic curves of genus 3 over F_q in O(q^{4/3}) group operations. On the other hand, applied to Jacobian varieties of non-hyperelliptic curves of genus 3 over F_q, the index calculus algorithm of Diem requires only O(q) group operations to solve the DLP. This vulnerability to faster index calculus attacks of the non-hyperelliptic curves of genus 3 has discouraged the use of Jacobian varieties of non-hyperelliptic curves of genus 3 as a basis of DLP-based cryptosystems. A recent work of B. Smith introduces the idea of exploiting this vulnerability to faster index calculus attacks of the non-hyperelliptic curves of genus 3 to discard a non-negligible subset of hyperelliptic curves of genus 3 over F_q. I would like to expose some highlights of this approach of B. Smith (the mathematical ingredients are nice: isogenies of Jacobian varieties, Recilla's trigonal construction, etc.) and to discuss the interest of studying non-hyperelliptic curves of genus 3 in the context of DLP-based cryptosystems.
Cedric
Villani
ENS Lyon/Princeton
Legacy of Boltzmann
Daniel
Heldt
Computational aspects of mixing heights
Anna
von Pippich
HU Berlin
The arithmetic of elliptic Eisenstein series
Peter
Krautzberger
FU Berlin
What is topological dynamics?
Abstract.
Topological dynamics is the abstract version of dynamical systems consisting of nothing else but a compact, Hausdorff space $X$ and a continuous function $f$ on $X$. Since this setting is very basic, I hope to offer an easy introduction to the basic phenomena, e.g. recurrence, proximality, maybe even chaos, while allowing for the discussion of some deeper results, e.g. the Ellis-Auslander Theorem.
Matjaz
Kovse
Topological representations of planar partial cubes
June 2009
Robin
Scholz
U Potsdam
What is the square root of a graph?
Abstract.
For a certain class of finite graphs we consider the concept of square root of a graph. This concept, as well as the examined class of graphs, arises from a special decision problem. We will step by step develop criteria which characterize the graphs that have a square root. This is essential for the solution of the original problem.
Kolja
Knauer
Antimatroids, Polytopes and ULDs
Remke Nanne
Kloosterman
HU Berlin
Average rank of elliptic n-folds
Abstract.
For elliptic curves over number fields it is conjectured that the half the curves have rank 1 and half the curves have rank 0. Similarly, if C/F_q is a curve then it is conjectured the half the elliptic curves over F_q(C) have rank 0 and half the curves have rank 1. In this talk we show that the situation is different if one considers elliptic curves over F_q(V), with dim(V)>1.
Laura
Hinsch
FU Berlin
What is the field with one element?
Abstract.
The field with one element is a recurring legend. In this talk I will try to explain where the idea came from, what it could be, what it might be and what it cannot be.
Margaret
Wright
New York University
Non-derivative optimization: Mathematics, engineering, or heuristics?
Torsten
Ueckerdt
(Linear) Induced Forests in Planar Graphs
Albert
Shiryaev
Steklov Institute of Mathematics
From von Mises' COLLECTIVE to Kolmogorov's COMPLEXITY
Stefan
Felsner
Condorcet Domains and Arrangements of Pseudolines
Axel
Werner
ZIB Berlin
What is a flag vector?
Abstract.
We'll define $f$- and flag vectors (mainly of polytopes), consider a few examples and see why flag vectors of polytopes span a space of affine dimension given by the Fibonacci numbers. If time permits, we'll also consider linear inequalities for the flag vectors which can be used to 'map' polytpes and see some maps of 3-, 4- and 5-dimensional polytopes.
Pavel
Valtr
On Triconnected and Cubic Plane Graphs on Given Point Sets
Ronald
van Luijk
Leiden
Explicit two-coverings of Jacobians of genus two
May 2009
Hendrik W.
Lenstra
Leiden
Modelling finite fields
Stefan
Felsner
Rectangular layouts: associated graphs and constructions.
Anil
Aryasomayajula
HU Berlin
A fundamental identity of metrics on hyperbolic Riemann surfaces of finite volume
Nathan
Ilten
FU Berlin
What is a Goppa code?
Abstract.
Reed-Solomon codes are widely used in coding theory due to their good parameters and ease of decoding. Goppa codes can in a way be viewed as a generalization of Reed-Solomon codes which, for the most part, maintains these nice properties. In this talk, I hope to briefly recall some basics of coding theory and the definition of Reed-Solomon codes. In what appears to be tagential, I will talk a little about divisors in algebraic curves. However, the definition of Goppa codes should bring everything back together. Time permitting, I will talk about what makes these codes so great.
Shabnam
Akhtari
MPI Bonn
Representation of Integers by Binary Forms
Abstract.
Suppose F(x,y) is an irreducible binary form with integral coefficients, degree n >= 3 and discriminant D_F \neq 0. Let h be an integer. The equation F(x,y)=h has finitely many solutions in integers x and y. I shall discuss some different approaches to the problem of counting the number of integral solutions to such equations. I will give upper bounds upon the number of solutions to the Thue equation F(x,y)= h. These upper bounds are derived by combining methods from classical analysis and geometry of numbers. The theory of linear forms in logarithms plays an essential rule in this study.
Yvon
Vignaud
TU Berlin
What is entropy?
Abstract.
Entropy is a state function measuring microscopic disorder of a system. Its importance is related to its nature of "arrow of time": considering an isolated system, its entropy is increasing over time, unlike its energy which is a conserved quantity. It is a rather subtle and yet fundamental notion, which is especially useful in information theory or statistical mechanics. We aim to follow an intuitive approach so as to help grasping the concept of entropy.
Keith
Ball
UC London
The second law of probability: entropy growth in the central limit theorem
Florent
Becker
Curry-Hähnchen zu Mittag (An Introduction to Curry-Howard and the Coq Proof-Assistant)
Anna
Posingies
HU Berlin
What is the Dedekind zeta function?
Abstract.
Euler's zeta function in which the sum is taken over all natural numbers is well-known. There is a corresponding function called Dedekind zeta function for number fields where the sum is taken over all ideals of the ring of integers. We will introduce the Dedekind zeta function with a detailed definition of all number theoretical objects that occur and illustrate these objects with examples.
Benedict
Gross
Harvard
Values of zeta functions at negative integers and the dimensions of spaces of modular forms
Tobias
Friedel
Root Systems and Generalized Associahedra
Anna
Posingies
HU Berlin
A Kronecker Limit Formula for Fermat Curves
April 2009
Sarah
Li
Adjacency Posets
Hartwig
Mayer
HU Berlin
Arithmetic self-intersection number of the dualizing sheaf on modular curves
Joscha
Gedicke
HU Berlin
What is the adaptive finite element method?
Abstract.
The adaptive finite element method (AFEM) is a powerful tool for numerical simulations. It is used to solve various problems of different kind numerically with high accuracy but less computational effort. This introductory talk explains key terms such as linear finite elements, a posteriori error estimators and adaptive mesh refinement. For a simple Poisson model problem the basic analytical setting such as the variational formulation is explained. The linear finite elements lead to a numerical discretization of the variational equation. In order to save computation time it is important to involve adaptivity into the algorithms. It will be explained what adaptivity means and how adaptive mesh refinement and a posteriori error estimates lead to the AFEM.
Michael
Hintermüller
HU Berlin
Use Level Sets and Relax!
Mareike
Massow
Convex Partitions of the Permutahedron
Jan
Foniok
Constraint Satisfaction and Category Theory: Constructing Tractable Templates
March 2009
Daniel
Heldt
A Short Introduction to Block Cipher Algorithms
Anton
Dochtermann
Graph Homomorphisms and Reflection Positivity
Torsten
Ueckerdt
On Quadrant-Depth - Closing the Gap
Melanie
Win Myint
Compactness proofs for infinite graphs
February 2009
Stefan
Felsner
On Quadrant-Depth
Kolja
Knauer
Generalized Chip Firing
Plamen
Turkedjiev
HU Berlin
What is Brownian motion?
Abstract.
Brownian Motion is a canonical process in stochastic analysis. Properties and existence will be discussed. Slides are available here.
Mareike
Massow
Linear Extension Diameter of Boolean Lattices
Aaron
Greicius
HU Berlin
Elliptic curves with surjective adelic Galois representation
Abstract.
Let E/K be an elliptic curve over a number field K. Let G_K be the absolute Galois group of K. The action of G_K on the torsion points of E gives rise to an \emph{adelic} Galois representation $$\rho:G_K\rightarrow Aut(E_{tor}(\overline{K}))\simeq \GL_2(\hat{\mathbb Z}).$$ In 1972 Serre proved that the image of \rho is open, hence of finite index, when E/K is non-CM. The question naturally arises then whether this index is ever equal to 1. In other words, is it possible for \rho to be surjective? By examining the maximal closed subgroups of $GL_2(\hat{\mathbbZ})$, we come up with simple necessary and sufficient conditions for this to be the case. This allows us to find examples of number fields $K\ne\mathbb{Q}$ and elliptic curves E/K for which \rho is indeed surjective.
Mareike
Massow
TU Berlin
What is Helly's theorem?
Abstract.
Helly's Theorem is one of the most famous results of a combinatorial nature about convex sets. It states that if we have $n$ convex sets in $\mathbb R^d$, where $n>d$, and the intersection of every $d+1$ of these sets is nonempty, then the intersection of all sets is nonempty. In preparation of Gil Kalai's BMS talk, we will see a basic proof of this theorem using (a basic proof of) Radon's Lemma. Hopefully we will also have a look at some application(s).
Gil
Kalai
Hebrew University Jerusalem
Helly-type theorems
Amir
Dzambic
Univ. Frankfurt am Main
Hyperbolic 3-manifolds with maximal automorphism group
January 2009
Artem
Chernikov
HU Berlin
What is a Fraisse construction?
Abstract.
Take all finite subraphs of your infinite graph, now forget it and ask yourself whether you can recover the graph you started with just from this bunch of finite graphs or not? There is a precise and very simple answer, which is actually delivered by a general method of taking limits in certain categories called Fraisse construction. For example, the limit of all finite graphs is exactly the random graph, or say limit of all finite linear orders is the dense linear order (like in rationals). But you can apply the same procedure to groups, partial orders, metric spaces, fields and whatever else getting lots of fancy objects, sometimes well-known and sometimes totally new. I will show a couple of exotic species hopefully.
Stefan
Felsner
Sorting Pairs in Bins, aka 'Das Krawattenraetsel'
Jean-Benoit
Bost
Univ. Paris-Sud, Orsay
Unitary integrable connections defined over number fields
Fritz
Hörmann
HU Berlin
What is the Birch and Swinnerton-Dyer conjecture?
Abstract.
Elliptic curves — which can be given by equations of degree 3 (e.g. $X^3+Y^3=1$) — are the most interesting among all algebraic curves. It is an old question in number theory, called a Diophantine problem, to determine the set of rational points on such a curve. The elliptic case, again, is the most interesting and mysterious of all Diophantine problems of dimension 1. For example, there may or may not be infinitely many rational solutions. At present no known algorithm can determine this. However, already in the 60's, Birch and Swinnerton-Dyer experimentally found a deep and mysterious relation of this question to analytic properties of the zeta-function of the curve, which encodes the easily determined solutions of the congruences (e.g. $X^3+Y^3 \equiv 1 \pmod N$). It later became one of the most famous conjectures of mathematics, and is one of the millenium prize problems, for whose solution the Clay Mathematical Institute offers a reward of \$1,000,000.
Annette
Werner
Frankfurt
Bruhat-Tits buildings
Torsten
Ueckerdt
On the number of non-decreasing paths in grid graphs
Jay
Jorgenson
CCNY
Sup-norm bounds for weight k automorphic forms
Carsten
Hartmann
FU Berlin
What is an averaging principle?
Abstract.
The talk gives a high-level overview of problems that involve dynamical systems with slow and fast time scales. Prominent examples are the sun-earth-moon system in celestial mechanics or climate models in which the weather appears as a fast perturbation to the slowly varying climate. I will explain how the fast dynamics can be systematically eliminated from the equations of motion (averaging) thereby yielding closed-form equations that govern the motion the slow variables.
Mihyun
Kang
Yet Another Way of Counting Planar Graphs
Rolf-Peter
Holzapfel
HU Berlin
Picard modular del Pezzo surfaces, II
Matthias
Lenz
TU Berlin
What is the $\mathsf P$ versus $\mathsf{NP}$ problem?
Abstract.
Which boxes in the picture above should be chosen to maximize the amount of money while still keeping the overall weight under or equal to 15 kg? Problems of this type are called knapsack problems. They belong to the class of $\mathsf{NP}$ problems, which are in general very hard to solve. Let $\mathsf P$ be the class of problems that can be solved in polynomial time. $\mathsf P$ is a subset of $\mathsf{NP}$. One of the Millenium problems is to decide whether $\mathsf P$ equals $\mathsf{NP}$. Put differently, we want to know if there are problems whose answer can be quickly checked, but which require an impossibly long time to solve even on a supercomputer. We will define the classes $\mathsf P$ and $\mathsf{NP}$ using Turing machines, give plenty of examples and explain why the $\mathsf P$ vs. $\mathsf{NP}$ problem is important.
Susanne
Albers
Freiburg
Network Design Games
Paul
Bonsma
Avaloqix is strongly PSPACE-hard
K.
Künnemann
Univ. Regensburg
Line bundles with connections on projective varieties over function fields and number fields
Abstract.
We report about joint work with with Jean-Benoit Bost (Orsay). Consider a hermitian line bundle on a smooth, projective variety over a number field. The arithmetic Atiyah class of the hermitian line vanishes by definition if and only if the unitary connection on the hermitian line bundle is already defined over the number field. We show that this can happen only if the class of the line bundle is torsion. This problem may be translated into a concrete problem of diophantine geometry, concerning rational points of the universal vector extension of the Picard variety. We investigate this problem, which was already considered and solved in some cases by Bertrand, by using a classical transcendence result of Schneider-Lang. We also consider a geometric analog of our arithmetic situation, namely a smooth, projective variety which is fibered on a curve defined over some field of characteristic zero. To any line bundle on the variety is attached its Atiyah class relative to the base curve. We describe precisely when this relative class vanishes. In particular, when the fixed part of the relative Picard variety is trivial, this holds only when the restriction of the line bundle to the generic fiber of the fibration is a torsion line bundle.
December 2008
Daria
Schymura
FU Berlin
What is similarity of shapes?
Abstract.
This talk is about shape matching, an interesting problem from Theoretical Computer Science. I will introduce the general matching problem, several classes of shapes and distance measures. I will also present results on how to compute the similarity of shapes.
Florent
Becker
Self-Assembling Tilings, Orders and Signal Systems
Anna
Pippich
HU Berlin
What is spectral expansion?
Abstract.
Starting from the Christmas guitar we will study a very basic example to answer the above question and we will indicate some applications to number theory. Check also the attached abstract.
Jay
Jorgenson
City College of New York
Identities in Euclidean and Hyperbolic Geometry which follow from Existence and Uniqueness Theorems
Britta
Peis
Covering Graphs by Colored Stable Sets
Philipp
Habegger
ETH Zürich
Points with multiplicative dependent coordinates on curves
Abstract.
The Mordell Conjecture states that a smooth projective curve of genus at least 2 has only finitely many points defined over a number field. This is now a theorem of Faltings; another proof using completely different ideas was found by Vojta. His approach can be encapsulated neatly in a single height inequality. Remond found a more uniform version of this inequality and also applications to abelian varieties in the context of the Zilber-Pink Conjecture governing the intersection of a variety with the union of certain algebraic subgroups. Maurin later built up on Remond's result in the toric case and showed that the curve contained in the algebraic torus of dimension 6 parametrized by (2,3,5,t,1-t,1+t) contains only finitely many points whose coordinates satisfy two independent multiplicative relations. Based ultimately on Vojta's method, Maurin's Theorem is ineffective: it does not allow us to determine the t's. We propose a different approach which circumvents Vojta's method and which is in principle effective in the toric setting.
Peter
Krautzberger
FU Berlin
What is Hindman's theorem?
Abstract.
In the talk I will introduce Hindman's theorem, a result in (infinite) Ramsey theory important for both its content and its historical role. If time allows it, I will try to sketch the proof due to Galvin and Glazer using ultrafilters.
Bruno
Benedetti
Bipartite Graphs and Basic k-Covers
Rolf-Peter
Holzapfel
HU Berlin
Picard modular del Pezzo surfaces
November 2008
Barbara
Jablonska
TU Berlin
What is geometric knot theory?
Abstract.
Using visualisation softare based on JReality I will introduce some basic notions of the classical (topologic) and geometric knot theory. I will also discuss some results from the latter field and if time allows I will present my recent result. The whole talk is ment to be very interactive and will teach you geometric knot theory mainly through graphics. No previous knowledge is required. Please note the attached PDF version.
Wolfgang
Lück
Münster
Topological Rigidity
Daniel
Heldt
A Reason Why Avaloqix Could Be 'Interesting' (i.e. NP-hard)
Anilatmaja
Aryasomayajula
HU Berlin
Hyperbolic and canonical metrics
Dror
Atariah
FU Berlin
What is loops and the fundamental group?
Abstract.
In this talk I will introduce an important tool for studying topological spaces. We will see how loops can characterize spaces, and define the fundamental group of a topological space.
Melanie
Win Myint
An Algebraic Characterization of Planar Graphs
Maria
Petkova
HU Berlin
Shimura curve computation
Michael
Rapoport
Bonn
On the zeta function of Shimura varieties
Mareike
Massow
The Boolean Lattice and its Linear Extension Diameter
Nathan
Ilten
FU Berlin
What is deformation?
Abstract.
I plan to give a concise introduction to deformations of singularities. After showing some very pretty pictures, I will define what a deformation is. Additionally, I hope to make remarks concerning concepts such as flatness, induced deformations, and versality. Time permitting, I will present Pinkham's famous example of the cone over the rational normal curve of degree 4.
Sarah
Li
Characterization of Maps with Order Dimension at most 2
Anna
von Pippich
HU Berlin
Elliptic and hyperbolic degeneration
October 2008
Martin
Skutella
TU Berlin
Flows over time - classical and more recent results
Anton
Dochtermann
TU Berlin
What is homotopical algebra?
Abstract.
Homotopy theory of topological spaces represents a rich and interesting interplay between relatively easy-to-define notions such as homotopy of maps, homotopy groups of spaces, fibrations, etc. In the sixties Quillen realized that topological structures like these could be encoded in a set of axioms which, if satisfied, allow one to talk about 'homotopy theory' in a more abstract setting. Any category that satisfies these axioms is called a '(closed) model category'. Although many instances arise in a geometric context, a perhaps surprising application of model categories is in a more algebraic setting: one of the early success of model categories was the proof that the combinatorial notion of 'simplicial sets' sufficiently 'models' the homotopy category of topological spaces. It turns out that chain complexes of modules also satisfy the axioms (this lead Quillen to the notion of 'homotopical algebra') and hence we can talk about such things as the 'suspension of a chain complex', etc. More recently model categories have been introduced in algebraic geometry in the context of '$A^1$ homotopy' of schemes. In this talk we will introduce the axioms for a model category and discuss a couple of examples and applications (among those mentioned above).
Stefan
Felsner
Partitioning Boolean Lattices into Intervals
Jürg
Kramer
HU Berlin
Estimating Green's Functions I
Stefanie
Frick
FU Berlin
What is the random graph?
Abstract.
The random graph can be defined on the set of natural numbers: For each pair $(i,j)$ a coin flip decides whether or not the two numbers are connected with an edge. The resulting graph is universal in the sense that every countable graph is contained as an induced subgraph. We will define the graph and use it to show the so called 0-1-laws. If the probability that a graph on $n$ nodes has a given property converges to 1 (for increasing $n$), we say that the property applies 'almost for all'. If the probability converges to 0, we say the property applies 'almost never'. The 0-1-law states the rather surprising fact that for all first order order statements always one of the two cases applies. The elegant and for a general audience accessible proof will be presented. It combines logical arguments (Gödelian completeness, compactness) with probabilistic considerations. The random graph has been studied in different setups, under different angles and in different disciplines. If there is time left, I will present some results in combinatorics. That is, we will consider colorings of edges and vertices of the random graph.
Hal
Kierstead
Efficient packing via game coloring
Norbert
Schappacher
Strasbourg
Political Space Curves
Torsten
Ueckerdt
How to Eat 4/9 of a Pizza
Daniel
Heldt
and
Jannik
Matuschke
and
Madeleine
Theile
Avaloqix - A Max Flow Game
September 2008
Bartłomiej
Bosek
and
Kamil
Kloch
Online Dimension of Semi-Orders with Representation
Kamil
Kloch
and
Piotr
Micek
Some Open Problems We Like
July 2008
Uwe
Schauz
Tutte's Flow Conjectures and Berlekamp's Switching Game
Michael
Payne
FU Berlin
What is the chromatic number of the plane?
Bernd
Sturmfels
Evolution on distributive lattices
Andrea
Miller
Harvard Univ.
Chow-Künneth decompositions for some mixed Shimura varieties
Torsten
Ueckerdt
Distributive polyhedra and related problems on weighted parameterized graphs
Laura
Hinsch
Univ. Göttingen
Projective spaces and sheaves of modules on F_1
Gido
Scharfenberger-Fabian
FU Berlin
Was ist eigentlich das Kontinuum und was ist sein Problem?
Melanie
Win Myint
Cycles and Bicycles in Locally Finite Graphs II
Riccardo
Salvati Manni
Universita La Sapienza, Roma I
The loci of abelian varieties with singular points of order two on the theta divisor
June 2008
Peter
Krautzberger
FU Berlin
Was ist eigentlich eine sinnvolle Erweiterung der Halbgruppe der natürlichen Zahlen?
Felix
Fischer
Voting Caterpillars
Eva
Bayer-Fluckiger
EPFL
The Euclidean division
Felix
Breuer
FU Berlin
Was ist eigentlich eine sinnvolle Verallgemeinerung des Zwischenwertsatz?
Melanie
Win Myint
Cycles and Bicycles in Locally Finite Graphs I
Nathan
Ilten
FU Berlin
Was ist eigentlich eine Garbe?
Peter
Allen
A connection between Ramsey number and chromatic number
Hans
Föllmer
HU Berlin
Bright spotlights, dark shadows: probabilistic aspects of financial risk
Frederik
von Heymann
FU Berlin
Was ist eigentlich die Schälung eines Polytops?
Daniel
Garbin
CUNY Graduate Center, New York
Spectral convergence of elliptically degenerated Riemann surfaces
Mareike
Massow
LINEAR EXTENSION DIAMETER is NP-complete
Tobias
Hahn
ETH Zürich
Arithmetic Riemann Roch theorem for modular curves
May 2008
Tobias
Marxen
FU Berlin
Was ist eigentlich der Ricci-Fluss?
Sarah
Li
Random Walks and Search Problems
Bernhard
Heim
MPI Bonn
On the Modularity of the GL_2-twisted Spinor L-function
Abstract.
There are famous theorems on the modularity of Dirichlet series with Euler product attached to geometric or arithmetic objects. There is Hecke's converse theorem, Wiles proof of the Taniyama-Shimura conjecture, and Fermat's Last Theorem to name a few. In this talk in the spirit of the Langlands philosophy we raise the question on the modularity of the GL_2-twisted spinor L-function L(s,G imes h) related to automorphic forms G, h on the symplectic group with similitudes GSp(4) of degree 2 and GL_2. This leads to several promising results and finally culminates into a precise very general conjecture. This gives new insides into the Miyawaki conjecture on spinor L-functions of modular forms. We indicate how this topic is related to Ramakrishnan's work on the modularity of the Rankin-Selberg L-series.
Michael
Hopkins
Harvard University, Cambridge, MA, USA
How topologists count things
Peter
Krautzberger
FU Berlin
Was ist eigentlich ein Ultrafilter
Ludmila
Scharf
Inducing Polygons of Line Arrangements
Vladimir
Berkovich
Weizmann Institute/MPI Bonn
Analytic geometry over the field of one element
Mareike
Massow
Linear Extensions in Diametral Pairs Don't Have to Be Reversing
Elmar
Große-Klönne
HU Berlin
On the crystalline monodromy-weight conjecture for p-adically uniformized varieties
Wolfgang
Dahmen
Aachen
Compressed Sensing
Anna
von Pippich
HU Berlin
Elliptic Eisenstein series for PSL_2(Z)
April 2008
Paul
Bonsma
A 3/2-Approximation Algorithm for Finding Spanning Trees with Many Leaves in Cubic Graphs
Jürg
Kramer
HU Berlin
Spectral expansion of hyperbolic Eisenstein series
Gerard
van der Geer
Amsterdam
Modular forms and counting curves over finite fields
Uwe
Schauz
Mr. Paint and Mrs. Sandpaper
Jorge A.
Pineiro
City Univ. of New York
Heights and algebraic dynamics
Axel
Werner
Linkages in Polytope Graphs
Bernd
Sturmfels
Permutohedra, Semigraphoids, and Mice
March 2008
Stefan
Felsner
Hamiltonicity Properties of Arrangement Graphs
Kolja
Knauer
On Frankl's Conjecture
Cornelia
Dangelmayr
k-Segments in Arrangements of Pseudolines
February 2008
Patrick
Baier
Distributed Computation of Virtual Coordinates
Stefan
Felsner
Markov Chains and the Second Eigenvalue
Moritz
Schmitt
The Spectrum of a Graph
Emmanuel
Ullmo
Universite Paris-Sud, Orsay
Rational points on Shimura varieties
Shmuel
Friedland
Chicago/Berlin
Counting matchings in graphs, with applications to the monomer-dimer models
Anton
Dochtermann
Foldings and the Topology of Graph Homomorphisms
Jose Ignacio
Burgos Gil
Univ. of Barcelona
The arithmetic Riemann Roch theorem for closed immersions
January 2008
Mareike
Massow
Reconstructing Posets from Linear Extension Graphs
Sascha
Orlik
Univ. Leipzig
Äquivariante Vektorbündel auf Drinfelds Halbraum
Ruth
Kellerhals
Fribourg
Hyperbolic isometry groups and small volume
Kolja
Knauer
A Distributive Lattice on Pseudo-Flows
Giancarlo
Urzua
Univ. of Michigan
Arrangements of curves and algebraic surfaces
Abstract.
We show a close relation between Chern and log Chern numbers of complex algebraic surfaces. In few words, given a log surface (Y,D) of a certain type, we prove that there exist smooth projective surfaces X with Chern ratio arbitrarily close to the log Chern ratio of (Y,D). The method is a "random" p-th root cover which exploits a large scale behavior of Dedekind sums and negative-regular continued fractions. We emphasize that the "random" hypothesis is necessary for this limit result. For certain divisors D, this construction controls the irregularity and/or the topological fundamental group of the new surfaces X. For example, we show how to obtain simply connected smooth projective surfaces, which come from the dual Hesse arrangement, with Chern ratio arbitrarily close to 8/3. In addition, by means of the Hirzebruch inequality for complex line arrangements, we show that this is the (unique) best result (closer to the Miyaoka-Yau bound 3) for complex line arrangements.
Gesine
Koch
Counting Paths in Cube Configurations
Amir
Dzambic
Univ. Frankfurt a. Main
Geometrie und Arithmetik falscher projektiver Ebenen
Rupert
Klein
FU Berlin
Multiple scales in Weather and Climate
Paul
Bonsma
A Polynomial Time Algorithm for Finding Fullerene Patches
Jay
Jorgenson
CCNY
Zeta functions and determinants on discrete tori
Abstract.
For any integer m, we let mZ\Z denote a discrete circle, and we define a discrete torus to be a product of a finite number of discrete circles. Associated to the combinatorial Laplacian, one has a finite set of eigenvalues from which one can form the determinant, namely the product of the eigenvalues. We prove, under reasonably general assumptions, an asymptotic expansion of the determinant as the parameters m in the discrete circles tend to infinity. Specifically, we establish a "lead term" which solely involves the parameters of the discrete circles, and a "second order term" which is a modular form associated to the Kronecker limit problem for Epstein zeta functions. We show that the modular form is that which is obtained when defining the regularized determinant of the Laplacian on real tori, thus establishing a new connection between determinants and zeta regularized determinants, as well as discrete and real tori.
December 2007
Graham
Brightwell
Polyhedral Methods for Linear Extensions of Partial Orders
Ilka
Agricola
HU Berlin
The E8 challenge
Sarah
Li
The Dictator Paradox
Stefan
Felsner
Sorting Using Networks of Stacks and Queues
Maria
Petkova
HU Berlin
Ball quotients curves with application in the coding theory
November 2007
Florian
Zickfeld
Leafy Trees in Planar Graphs
Walter
Gubler
Univ. Dortmund, z.Z. HU Berlin
Tropische analytische Geometrie und die Bogomolov-Vermutung
Claus
Michael Ringel
Bielefeld
Invariant subspaces of nilpotent operators
Nina
Siebold
How to Draw an Order
Aleksander
Momot
ETH Zürich
Toroidally compactified ball quotient surfaces in small Kodaira dimension
Abstract.
As well as Hilbert modular surfaces, compact and compactified quotients $X = ar{\Gamma \\setminus \\mathbf{B}}$ of the open complex unit-ball $\\mathbf{B} \\subset \\mathbb{C}^2$ serve as the two-dimensional analog of modular curves. They are thus intimately connected with modular problems, but also provide interesting and extremal examples in the geography of surfaces. While Holzapfel has classified such surfaces for the case that they are defined by Picard modular ball-lattices $\\Gamma = \\mathbf{PU}(2,1; \\mathfrak{a})$, $\\mathfrak{a}$ an order in an imaginary quadratic number field, I aim a classification avoiding arithmetic conditions on Gamma. In the main part of the talk I will sketch the classification for kod(X)<= 0, q(X)> 0.
Leonid
Ioffe
Dimension Of Orders via Constraint Programming
Rainer
Weissauer
Heidelberg
Coverings of Riemann surfaces and a new kind of Galois groups
Patrick
Baier
Greedy Drawings of Planar Triangulations
Gabor
Wiese
Univ. Duisburg-Essen, Campus Essen
Zur Erzeugung von Koeffizientenkörpern von Neuformen durch einen einzigen Hecke-Eigenwert
Wendelin
Werner
Université Paris Sud
Coloring at random
October 2007
Cornelia
Dangelmayr
k-Level Complexity of Arrangements of (Pseudo-)Segments
Christian
Christensen
Univ. Dortmund, z.Z. HU Berlin
Der relative Satz von Schanuel
Herbert
Edelsbrunner
Duke/Berlin
Measuring scale before simplification
Kolja
Knauer
Construction of Steiner Triple Systems
Martin
Pergel
Recognition and Optimization Problems on Geometric Intersection Graphs
Balazs
Keszegh
Weak Conflict-Free Colorings of Point Sets and Simple Regions
September 2007
Florian
Pfender
Turan's Theorem for Multipartite Graphs
Stefan
Felsner
Compact Drawings with Bends
Mareike
Massow
Partitioning Posets
August 2007
Stefan
Felsner
Matchings and Edge Colorings in Regular Graphs
Carolyn
Chun
Unavoidable Minors of c-connected Graphs
Bertrand
Marc
2-Arrangements of Pseudolines
July 2007
Florian
Zickfeld
Leafy Trees
Jan
Kristensen
Oxford/HU Berlin
The Problem of Regularity in the Calculus of Variations
Eric
Fusy
A bijection between loopless maps and triangulations
Bernd
Sturmfels
Berkeley/Berlin
Algebraic Degree of Semidefinite Programming
Patrick
Baier
Digital Geometry
Stefan
Felsner
Infeasibility of Arrangements
June 2007
Albert
Cohen
Harmonic Analysis and Image Processing
Kamil
Kloch
and
Tomasz
Krawczyk
and
Piotr
Micek
On-line Dimension of Intervals
Andrei
Okounkov
Princeton
Box counting
Kamil
Kloch
On-line Colouring of Intervals
Bill
Casselman
UBC, Vancouver
An illustrated history of geometry
Olivier
Bernardi
Bijective Countings of Tree-Rooted Maps
Alexander
Barvinok
U Michigan, Ann Arbor
The asymptotic equivalence of counting and optimization
Cornelia
Dangelmayr
Planar Graphs are in 1-STRING
May 2007
Stefan
Hildebrandt
Euler und die Analysis
Stefan
Felsner
On-line chain partitions of up-growing 2-dimensional orders
Felix
Breuer
Perlenketten mit Quoten
Olga
Holtz
TU Berlin
How to multiply matrices fast (and why bother)?
Mareike
Massow
Diametral Pairs of Linear Extensions of a Poset
George
Francis
U Illinois, Urbana-Champaign
Designing computer animations with chalk and blackboard
Florian
Zickfeld
Hardness of Counting Orientations with Fixed Out-Degrees
April 2007
Sarah
Kappes
Was ist ein semidefinites Programm?
Patrick
Baier
Compass Routing
Cornelia
Dangelmayr
Pseudo-Connections in the Plane
Stefan
Felsner
Triangle Contact Representations of Graphs
March 2007
Andreas
Hoffmeister
Random sampling in distributive lattices
Stefan
Felsner
Online Ramsey Theory
February 2007
Eberhard
Knobloch
Euler and Leibniz on the infinite (celebrating Euler year 2007)
January 2007
Mareike
Massow
2-Dimensional Partial Orders Revisited
Florian
Zickfeld
Counting Planar Eulerian Orientations is #P-complete
Martin
Aigner
FU Berlin
Volumes, polyhedra, and prime numbers: A common combinatorial look
Cornelia
Dangelmayr
How Many 3-Pseudosegments can a Pseudoline Arrangements have?
Patrick
Baier
The Crossing Number of Geometric Complete Graphs
December 2006
Stefan
Felsner
Arrangements of pseudolines; news, problems, and goodies.
Karl
Sigmund
University of Vienna
Gödel's Vienna (celebrating the end of the Gödel year)
Kolja
Knauer
α-Orientations and Regular Oriented Matroids
Kolja
Knauer
α-Orientations and FlipFlops
November 2006
Sarah
Kappes
Kombinatorik orthogonaler Flächen
Robert
Bryant
Duke University
Aufwiedersehen Surfaces, Revisited
Florian
Zickfeld
α-Orientations and Bipartite Perfect Matchings
Hélène
Esnault
Congruences for the number of rational points of varieties defined over finite fields
Leonid
Ioffe
Constraint Programming
Peter
Schröder
CalTech
Discrete Exterior Calculus with an application to circulation preserving simplicial fluid simulation
Martin
Pergel
On Complicacy of Graphs Representable by Polygons
October 2006
Dominik
Piesker
Spezialfälle der 'list edge coloring conjecture'
Stefan
Felsner
Baxter and Schröder Families
August 2006
Patrick
Baier
Adaptive colouring of upgrowing posets
July 2006
Cornelia
Dangelmayr
Dominating pairs in AT-free graphs
Stefan
Felsner
Counting Bipolar Orientations on the Grid
June 2006
Sarah
Kappes
Convex-Pseudo Decompositions
Maria del Pilar
Sabariego Arenas
3-Loop Networks with many Minimum Distance Diagrams
Eric
Fusy
Enumeration of Bipolar Orientations
Johan
Nilsson
The Gupta-Newman-Rabinovich-Sinclair Conjecture
May 2006
Florian
Zickfeld
Integer Realizations of 3-Polytopes
Stefan
Felsner
Proofs of Untileability
Noga
Alon
Graphs, Euler's theorem, Grothendieck's inequality and Szemerédi's regularity lemma
Cornelia
Dangelmayr
Cocomparability graphs of posets with (interval) dimension at most two
Patrick
Baier
Online Chain Partitioning of Upgrowing Interval Posets - The Upper Bound
Mareike
Massow
Semi Bar 1-Visibility Graphs
April 2006
Harald
Schülzke
The Normal Graph Conjecture for line graphs
Steffen
Melang
Bipolar Orientations
Maria del Pilar
Sabariego Arenas
Circulant Graphs and Binomial Ideals
March 2006
Florian
Zickfeld
On the Number of 3-Orientations of a Triangulation
Sarah
Kappes
A binary labeling for the angles of a plane quadrangulation
Johan
Nilsson
Reachability substitutes
February 2006
Stefan
Felsner
Two Algorithms for Bipartite Cardinality Matching
Mareike
Massow
Bar 1-Visibility Graphs - Introduction and First Results
Stefan
Felsner
Counting Linear Extensions
January 2006
Georg
Melamed
Skelette 3-dimensionaler Ordnungen
Cornelia
Dangelmayr
A Representation of the Subset 'VPT' of Chordal Graphs as Intersection Graphs of Pseudosegments
Patrick
Baier
The Upper Bound for Online Chain Covering of Upgrowing Interval Posets
December 2005
Florian
Zickfeld
Schnyder Woods and Pairs of Non-Crossing Dyck Paths
Stefan
Felsner
Box Graphs and Graph Dimension
November 2005
Sarah
Kappes
Realizers for d-polytopes with d+2 vertices
Viktor
Blåsjö
On the Braid Page of Gauss' Handbook 7
Cornelia
Dangelmayr
Relations within the class of intersection graphs
Patrick
Baier
A special case of online chain covering of posets
Stefan
Felsner
Dualization in Products of Chains
October 2005
Florian
Pfender
Visibility Graphs on Point Sets
Mareike
Massow
What makes the treewidth large?
September 2005
Viktor
Blåsjö
Enumeration of Reduced Words in Combinatorial Group Theory
Krisztián
Tichler
Lies and Hypercubes
August 2005
Florian
Zickfeld
A Schnyder Wood with no rigid AND coplanar embedding
Sarah
Kappes
Orthogonal Surfaces - A combinatorial definition for characteristic points
Eric
Fusy
Enumeration of d-polytopes with d+3 vertices
Stefan
Felsner
Poset Polytopes and Linear Extensions
July 2005
Cornelia
Dangelmayr
Intersection Graphs
Patrick
Baier
An upper bound for online chain covering
June 2005
Florian
Pfender
Closure and Hamiltonian Properties in Claw-free Graphs
Kamil
Kloch
Partial Orders - fooling Alice
Sarah
Kappes
What is an orthogonal complex?
Krisztián
Tichler
A Combinatorial Property of Databases
May 2005
Dan
Kral
Claws and Paws