Manuel
Kauers
JKU Linz
Search by speaker
Filter institution
Filter content
April 2025
March 2025
Jan
Pauls
UniversitĂ€t MĂŒnster
Christophe
Roux
ZIB
Marius
Zeinhofer
ETH, ZĂŒrich, Switzerland
First Optimize, Then Discretize for Scientific Machine Learning
Abstract.
This talk provides an infinite-dimensional viewpoint on optimization problems encountered in scientific machine learning and discusses the paradigm first optimize, then discretize for their solution. This amounts to first choosing an appropriate infinite-dimensional algorithm which is subsequently discretized in the tangent space of the neural network ansatz. To illustrate this point, we show that recently proposed state-of-the-art algorithms for scientific machine learning applications can be derived within this framework. Finally, we discuss the crucial aspect of scalability of the resulting algorithms.
February 2025
Giulio
Schimperna
University of Pavia, Italy
Some results on a modified Cahn-Hilliard model with chemotaxis
Abstract
Konrad
Mundiger
ZIB
Neural Networks for Unsupervised Discovery of Plane Colorings
Abstract.
We present a framework that transforms geometric and combinatorial problems into optimization tasks by designing loss functions that vanish precisely when the desired coloring properties are achieved. We employ neural networks trained through gradient descent to minimize these loss functions, allowing for efficient exploration of the solution space. We demonstrate the effectiveness of the method on variants of the Hadwiger-Nelson problem, which asks for plane colorings that avoid monochromatic unit-distance pairs and sketch how the approach can be applied to other problems.
Marc
Zimmermann
Uni Köln
Nikola
Sadovek
FU Berlin
Complex analogues of the Tverberg-VreÄica conjecture and central transversal theorems
Abstract.
The Tverberg-VreÄica conjecture is a broad generalization of Tverberg's classical theorem. One of its consequences, the central transversal theorem, extends both the centerpoint theorem and the ham sandwich theorem. In this talk, we will consider complex analogues of these results, where the corresponding transversals are complex affine spaces. The proofs of the complex Tverberg-VreÄica conjecture and its optimal colorful version rely on the non-vanishing of an equivariant Euler class. Furthermore, we obtain new Borsuk--Ulam-type theorems on complex Stiefel manifolds, which are interesting on their own. These theorems yield complex analogues of recent extensions of the ham sandwich theorem for mass assignments and provide a direct proof of the complex central transversal theorem. This talk is based on a joint work with Pablo SoberĂłn.
Fred Espen
Benth
U of Oslo
André
Schlichting
UniversitÀt Ulm
Breakdown of the mean-field description of interacting systems: Phase transitions, metastability and coarsening
Abstract
Konrad
Mundiger
ZIB
Neural Networks for Unsupervised Discovery of Plane Colorings
Abstract.
We present a framework that transforms geometric and combinatorial problems into optimization tasks by designing loss functions that vanish precisely when the desired coloring properties are achieved. We employ neural networks trained through gradient descent to minimize these loss functions, allowing for efficient exploration of the solution space. We demonstrate the effectiveness of the method on variants of the Hadwiger-Nelson problem, which asks for plane colorings that avoid monochromatic unit-distance pairs and sketch how the approach can be applied to other problems.
Mikhail
Gorskii
Hamburg UniversitÀt
Counting in Calabi-Yau categories
Abstract.
I will discuss a replacement of the notion of homotopy cardinality in the setting of even-dimensional Calabi--Yau categories and their relative generalizations. This includes cases where the usual definition does not apply, such as Z/2-graded dg categories. As a first application, this allows us to define a version of Hall algebras for odd-dimensional Calabi-Yau categories. I will explain its relation to some previously known constructions of Hall algebras. If time permits, I will also discuss another application in the context of invariants of smooth and graded Legendrian links, where we prove a conjecture of Ng-Rutherford-Shende-Sivek relating ruling polynomials with augmentation categories. The talk is based on joint work with Fabian Haiden, arxiv:2409.10154.
Stefan
Kober
Universite Libre de Bruxelles
Sebastian
Schwarzacher
Uppsala University, Sweden & Charles University Prague, Czech Republic
Time-periodic solutions for fluid-solid interactions
Abstract
Gioni
Mexi
ZIB
Demystifying Pseudo-Boolean Conflict Analysis through a MIP Lens
Abstract.
For almost two decades, mixed integer programming (MIP) solvers have used graph-based conflict analysis to learn from local infeasibilities during branch-and-bound search. In this talk, we discuss improvements for MIP conflict analysis by instead using reasoning based on cuts, inspired by the development of conflict-driven solvers for pseudo-Boolean optimization. Phrased in MIP terminology, this type of conflict analysis can be understood as a sequence of linear combinations, integer roundings, and cut generation. We leverage this MIP perspective to design a new conflict analysis algorithm based on mixed integer rounding cuts, which theoretically dominates the state-of-the-art method in pseudo-Boolean optimization using ChvĂĄtal-Gomory cuts. Furthermore, we discuss how to extend this cut-based conflict analysis from pure binary programs to mixed binary programs and-in limited form-to general MIP with also integer-valued variables. Our experimental results indicate that the new algorithm improves the default performance of SCIP in terms of running time, number of nodes in the search tree, and the number of instances solved.
Mattia
Cavicchi
Dijon
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 UniversitÀt 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 PlĂŒckebaum and Lutz Wilhelmy.
Alexandros
Saplaouras
ETH ZĂŒrich
The ItĂŽ Integral for Nonlinear LĂ©vy Processes: Insights into the G-LĂ©vy Framework
Abstract.
Nonlinear LĂ©vy processes, as established within the general framework by A. Neufeld and M. Nutz, offer a versatile foundation without restrictions on the characteristic triplets. Building on this foundational work, we focus specifically on G-LĂ©vy processes, a concept introduced by S. Peng. Adopting Peng's approach, we construct the ItĂŽ integral with respect to G-LĂ©vy processes and examine its associated properties. Alongside, we delve into results concerning the uniqueness of fully nonlinear integro-partial differential equations and briefly discuss the technical challenges.
Constantin
Ickstadt
Uni Frankfurt
MihĂĄly
KovĂĄcs
PPKE ITK Budapest
Helena
Kremp
TU/WIAS Berlin
Coherent Transport of Semiconductor Spin-Qubits: Modeling, Simulation and Optimal Control
Abstract
Laurent
Coté
Bonn
Fukaya categories of conical symplectic resolutions
Abstract.
Conical symplectic resolutions are a rather loosely defined class of hyperkÀhler varieties arising from canonical constructions in representation theory. Important examples include hypertoric varieties, Nakajima quiver varieties and Hitchin spaces. Assuming as little background as possible in symplectic geometry, I will introduce a categorical invariant called the Fukaya category, which is defined using methods of global analysis. When specialized to conical symplectic resolutions, the Fukaya category turns out to be intimately related to a category of longstanding interest in geometric representation theory, called Category O. This talk will report on joint work with (subsets of) Benjamin Gammage, Justin Hilburn, Christopher Kuo, David Nadler and Vivek Shende.
Daniel
Roggenkamp
MiS Leipzig
Rozansky-Witten models as extended defect TQFTs
Abstract.
Very few examples of extended topological quantum field theories are known explicitly. In this talk I will discuss a very explicit construction of the extended TQFTs associated to Rozanksy-Witten models with affine target spaces. I will furthermore explain how to incorporate defects into the extended TQFTs. This can be used for instance to derive the Hilbert spaces of affine Rozansky-Witten models associated to surfaces with arbitrary insertions of defect networks.
Carla
Cederbaum
U TĂŒbingen
Coordinates are messy in general (relativity)
Ekin
Ergen
Online Makespan Scheduling Under Scenarios
Julian
Marx
BeitrÀge zum ZÀhlen von BÀumen auf Punktmengen
Shelby
Cox
MPI MiS Leipzig
Weighted tropical Fermat-Weber points
Abstract.
Let $v_1,\ldots,v_m$ be points in a metric space $X$ with distance $d$, and let $w_1,\ldots,w_m$ be positive real weights. The weighted Fermat-Weber points are those points $x$ which minimize $\sum w_i d(v_i, x)$. When $X$ is the tropical projective torus, and $d$ is the asymmetric tropical distance, Comaneci and Joswig proved that the set of unweighted Fermat-Weber points agrees with the "central" covector cell of the tropical convex hull of $v_1,\ldots,v_m$. In a recent paper, Maize Curiel and I extend this result to the weighted setting using the combinatorics of tropical hyperplane arrangements and polyhedral geometry. Furthermore, we show that for any fixed data points $\bfv_1, \ldots, \bfv_m$, and any covector cell of the tropical convex hull of the data, there is a choice of weights that makes that cell the Fermat-Weber set. In the context of phylogenetics, the (weighted) Fermat-Weber points furnish a method for computing consensus trees.
Fabian
Lenzen
TU Berlin
Topological Data Analysis: Algebra and Computation
Abstract.
Topological Data Analysis (TDA) is an area that seeks to use methods from algebraic topology to develop new methods for data analysis. Arguably, persistent homology (PH) is the most prominent tool of TDA, which is a multi-scale approach estimating the homology of topological spaces from finite samples. From the PH of such a sample, one can then read off certain descriptors that contain information about the original space's topology. Computing these descriptors touches upon standard problems in computational algebra. In this talk, we will see some basic notions of PH, explore some of the algebraic properties of this structure, and touch upon some computational aspects crucial to the feasibility of PH in practice.
Jack
Allsop
Monash University
Subsquares of Latin squares
Abstract.
A Latin square of order \(n\) is an \(n \times n\) matrix of \(n\) symbols, each of which occurs exactly once in each row and column. A subsquare of a Latin square is a submatrix which is itself a Latin square. Any Latin square of order \(n\) has \(n^2\) subsquares of order \(1\) and a single subsquare of order \(n\). Any other subsquare is called proper. In this talk we discuss a number of problems regarding subsquares. Given integers \(n\) and \(m\), what is the maximum number of subsquares of order \(m\) in a Latin square of order \(n\)? What is the expected number of subsquares of order \(m\) in a uniformly random Latin square of order \(n\)? For what orders do there exist Latin squares without any proper subsquares?
Anja
Schlömerkemper
UniversitĂ€t WĂŒrzburg
A variational model for the evolution of magnetoelastic materials
Abstract
Jan-Frederik
Pietschmann
UniversitÀt Augsburg
Gradient flows on metric graphs with reservoirs
Abstract
Olivia
Röhrig
ZIB
Inference of Differential Privacy properties of julia code
Abstract.
Differential privacy is a concept that can be used to express the extent to which algorithms like database queries, statistics and machine learning procedures, preserve a certain notion of privacy of an input dataset. Prominent applications of the technique include the US census and user data aggregation procedures of multiple large tech companies. The correct implementation of such algorithms requires a substantial amount of care, which motivated the development of type systems tailored to verify the differential privacy properties of programs. We implemented a type checker for one such type system and integrated it with the julia programming language to enable not only verification but automatic inference of privacy parameters for a reasonable subset of julia code.
Vincent
Rivoirard
Université Daupine, Paris
PCA for point processes
Michael
Dettweiler
Bayreuth
Regular Galois Realizations of Groups of Lie Type
Abstract.
We survey the inverse Galois problem and state some new and old result on Galois realizations of groups of Lie type over the rational function field. Most of these can be obtained by a suitable combination of tensor products, convolutions on the affine line, and l-adic Fourier transformations.
Olivier
Marchal
Univ. St-Etienne
Isomonodromic deformations, quantization and exact WKB
Abstract.
In this talk, I will review the theory of isomonodromic deformations of meromorphic connections on gl2 and the underlying symplectic structure. In particular, I will explain how to obtain explicit formulas for the Hamiltonian systems and the Lax pairs. Next, I will explain how one can formally reconstruct these results using the quantization of the classical spectral curve using topological recursion. Finally, I will explain the current challenges and results to upgrade this formal reconstruction to an analytic one focusing on the genus zero case where one can use Borel resummation of WKB solutions. The talk is supposed not to require any knowledge in integrable systems, topological recursion of Borel resummation.
James
Davies
Frame matroids with a distinguished frame element
Dingding
Dong
Harvard University
Uncommon linear systems of two equations
Abstract.
A system of linear equations \(L\) is common over \(F_p\) if any 2-coloring of \(F_p^n\) gives at least as many monochromatic solutions to it as a random 2-coloring, asymptotically as \(n\to\infty\). It is an open question which linear systems are common. When \(L\) is a single equation, Fox, Pham and Zhao gave a complete characterization of common linear equations. When \(L\) consists of two equations, Kam\čev, Liebenau and Morrison showed that all irredundant 2*4 linear systems are uncommon. In joint work with Anqi Li and Yufei Zhao, we: (1) determine commonness of all 2*5 linear systems up to a small number of cases; (2) show that all \(2*k\) linear systems with \(k\) even and girth (length of the shortest equation) \(k-1\) are uncommon, answering a question of Kam\čev-Liebenau-Morrison.
Hana
Kourimska
UniversitÀt Potsdam
Amoebas: at the intersection of discrete, differential, and algebraic geometry
Abstract.
Amoebas are mathematical objects introduced in algebraic geometry at the end of the last century. Formally, they are defined as an image of a zero set of a polynomial under the so-called log-absolute map: for each point in the zero set, you take the absolute value of each of its coordinates, and then the (real) logarithm of it, and you end up with a set in the Euclidean space. While this may sound discouraging, it turns out one can tell (and learn) a lot about amoebas also without an extensive knowledge in algebraic geometry. In my talk I will tell you about these types of properties, and also shed light on where amoebas can be applied, and how differential geometry can help in approximating the area of amoebas. This is joint work with Timo de Wolff.
Randolf
Altmeyer
Imperial College London
Ben
Hambly
U of Oxford
Patrik
Knopf
UniversitÀt Regensburg
The Cahn-Hilliard equation with dynamic boundary conditions and its application to two-phase flows
Abstract
Andreas
Kretschmer
HU Berlin
Symmetric Ideals and Invariant Hilbert Schemes
Abstract.
A symmetric ideal is an ideal in a polynomial ring which is stable under all permutations of the variables. Special classes of symmetric ideals (e.g. Specht ideals, Tanisaki ideals) feature prominently in algebraic combinatorics: For example, the non-negativity of the q-Kostka polynomials is eventually explained by its interpretation as the Hilbert series of the quotient of a Tanisaki ideal. In commutative algebra, chains of symmetric ideals in increasingly many variables stabilize "up to symmetry" and several of their invariants display a uniform behavior. This is closely related to the concepts of representation stability and FI-modules.<br>This project grew out of a desire to study symmetric ideals more systematically. In particular, we wanted to get a feeling for the naive question of "how many" (zero-dimensional) symmetric ideals there actually are, after fixing certain parameters. This leads to the study of the invariant Hilbert scheme for the action of the symmetric group on affine n-space. We prove irreducibility and smoothness results and compute dimensions in certain special cases. Moreover, we classify all homogeneous symmetric ideals of colength at most 2n and decide which of these define singular points. We also discuss a conjectural stabilization phenomenon and explain connections to the usual Hilbert schemes of points of affine space. This is joint work with Sebastian Debus.
Antonia
Chmiela
ZIB
Polyhedrality made easy: A generic cut framework with polyhedral closures
Abstract.
Cutting planes are a crucial tool in mixed-integer programming. When measuring the strength of a specific class of cutting planes, a key question is whether the corresponding cut closure is polyhedral, meaning that finitely many cuts are sufficient to obtain the closure. The vast amount of literature investigating polyhedrality for different kinds of cutting planes indicates that the field has not yet settled on a universal solution to this problem. In this talk, we will look at cutting plane generation through a unifying framework and establish conditions that ensure polyhedrality of cut closures. This enables us to prove polyhedrality for a broad range of cutting plane families more easily.
Xiaorui
Zuo
National University of Singapore
Cryptos have rough volatility and correlated jumps
Ning
Guo
Harbin Institute of Technology
Geometric presentations and applications
Abstract.
Various problems concerning cohomology sets and K-theory, including Gersten's injectivity and the Grothendieck-Serre conjecture, have affirmative answers in the equi-characteristic case. A helpful tool for dealing with mixed characteristic cases is geometric presentation theorems. Roughly speaking, these realize the local ring under consideration as a relatively smooth (or étale) object over a regular DVR or a field with a low relative dimension such that problems reduce to the DVR or field cases. In this talk, we will discuss several geometric presentation theorems (due to Gabber-Quillen, Colliot-ThélÚne, Ojanguren, etc) and their application to the Grothendieck-Serre conjecture in mixed characteristic. In particular, I will introduce the notion of weak elementary fibrations, which traces back to Artin's "bon voisinage" in SGA4. These are individual joint works with Ivan Panin, Fei Liu and Yisheng Tian.
Gaëtan
Borot
HU Berlin
Limits in topological recursion
Abstract.
I will discuss the subtle aspects of the behavior of topological recursion (= seen as the construction of W-algebra modules from branched covers of curves) along families of spectral curves. To do so, I will review basics of singularities of plane curves and discuss (maximal) equigeneric families.
TĂŒrkĂŒ ĂzlĂŒm
Ăelik
MPI of Molecular Cell Biology and Genetics, Dresden
Algebraic curves and Grassmannians via the KP Hierarchy
Abstract.
Algebraic curves are fundamental objects in the mathematical sciences. Integrable systems, particularly the Kadomtsev-Petviashvili hierarchy, provide an example of a such phenomenon, and also reaffirm the significance of the Grassmannians. In this talk we will examine connections between algebraic curves and Grassmannians guided by the hierarchy. We will explore these connections within the transcendental, real, and combinatorial algebraic geometry from a computational perspective.
Ilya
Chevyrev
U Edinburgh/TU Berlin
Rough Analysis
Ilya
Cheyrev
U Edinburgh
Luca
Pelizzari
WIAS
What is What is...rough analysis?
Abstract.
The field of rough analysis provides a powerful mathematical framework for understanding differential equations driven by irregular paths, which frequently arise in the modeling of complex real-world phenomena. Studying such equations requires an understanding of integration against paths, a problem that becomes increasingly challenging as the regularity of the path decreases. This motivates the fundamental notions of a rough path and the rough integral, pioneered by Terry Lyons in the 1990s, which are central for the theory of rough analysis. In this talk, we aim to give a gentle introduction to these concepts, and show how they lead to a robust theory for rough differential equations. We demonstrate consistency with ItĂŽ's stochastic calculus for Brownian paths, and illustrate how rough analysis addresses limitations in the theory of stochastic differential equations.
Felix
Schröder
The Beer Index and Related Convexity Measures
Mehmet Akif
Yildiz
University of Amsterdam
Path decompositions of oriented graphs
Abstract.
We consider the problem of decomposing the edges of a directed graph into as few paths as possible. There is a natural lower bound for the number of paths in any path decomposition dictated by vertex degree imbalances, and any directed graph that meets this bound is called consistent. In 1976, Alspach, Mason, and Pullman conjectured that (as a generalization of Kelly's conjecture on Hamilton decompositions of regular tournaments) every tournament of even order is consistent. This conjecture was recently verified for large tournaments by GirĂŁo, Granet, KĂŒhn, Lo, and Osthus. A stronger conjecture, proposed by Pullman in 1980, states that every orientation of every regular graph with odd degree is consistent. In this talk, I will present our recent work (joint with Viresh Patel) that establishes Pullmanâs conjecture for random regular graphs and regular graphs without short cycles.
Roxana
Radulescu
Utrecht University
Decision-making in Multi-Objective Multi-Agent Systems
Abstract.
Many challenging tasks such as managing traffic systems, taxation policy design, or supply chains involve complex decision-making processes that must balance multiple conflicting objectives and coordinate the actions of various independent decision-makers. In a nutshell, the majority of real world problems are multi-agent and multi-objective in nature. Multi-objective multi-agent systems are more holistic and realistic models that capture the complexity of interactions among agents and the multiple dimensions of their objectives. Multi-objective multi-agent decision making (MOMADM) can potentially provide solutions that explore the possible trade-offs among the objectives, as well as the dependencies between the agents. This talk presents an overview of current developments in MOMADM, with a focus on how to structure the field, the current theoretical results and potential future directions.
Johannes
Schmidt-Hieber
University of Twente
Statistical estimation using zeroth-order optimization
Abstract.
In this talk, we study statistical properties of zeroth-order optimization schemes, which do not have access to the gradient of the loss and rely solely on evaluating the loss function. Such methods are often considered to be suboptimal for high-dimensional problems, as their convergence rates to the minimizer of the objective function are typically slower than those of gradient-based methods. This performance gap becomes more pronounced as the number of parameters increases. Considering the linear model, we show that reusing the same data point for multiple zeroth-order updates can overcome the gap in the estimation rates. Additionally, we demonstrate that zeroth-order optimization methods can achieve the optimal estimation rate when only queries from the linear regression model are available. Special attention will be given to the non-standard minimax lower bound in the query model. This is joint work with Thijs Bos, Niklas Dexheimer and Wouter Koolen.
TĂŒrkĂŒ ĂzlĂŒm
Ăelik
MPI of Molecular Cell Biology and Genetics, Dresden
December 2024
Mahrud
Sayrafi
MPI MiS Leipzig
Splitting of Vector Bundles on Toric Varieties
Abstract.
In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Generalizations of this criterion, under additional hypotheses, have been proven for other toric varieties, for instance by Eisenbud-Erman-Schreyer for products of projective spaces, by Schreyer for Segre-Veronese varieties, and Ottaviani for Grassmannians and quadrics. This talk is about a splitting criterion for arbitrary smooth projective toric varieties, as well as an algorithm for finding indecomposable summands of sheaves and modules in the more general setting of Mori dream spaces.
Christoph
Spiegel
Zuse-Institut Berlin
Coloring the plane with neural networks
Abstract.
We present two novel six-colorings of the Euclidean plane that avoid monochromatic pairs of points at unit distance in five colors and monochromatic pairs at another specified distance din the sixth color. Such colorings have previously been known to exist for \(0.41 < \sqrt 2â1 \le d \le 1/\sqrt 5 < 0.45\). Our results significantly expand that range to \(0.354 \le d \le0.657\), the first improvement in 30 years. The constructions underlying this notably were derived by formalizing colorings suggested by a custom machine learning approach. This is joint work with Sebastian Pokutta, Konrad Mundiger, and Max Zimmer.
Zhen-Qing
Chen
U of Washington
Pascal
Maillard
CNRS
Computational Aspects of Quadratic Forms in Determining the Representation Type of Quiver Algebras
Abstract
Gaëtan
Borot
HU Berlin
Limits in topological recursion
Abstract.
I will discuss the subtle aspects of the behavior of topological recursion (= seen as the construction of W-algebra modules from branched covers of curves) along families of spectral curves. To do so, I will review basics of singularities of plane curves
Niklas
Martensen
HU Berlin
and
Jan
Pulmann
HU Berlin
Moduli spaces of flat connections as KĂ€hler spaces, part III
Abstract.
Building on earlier sessions, we will propose the existence of an approach to obtain KĂ€hler structures on the moduli spaces analogous to the quasi-Hamiltonian reduction.
Jan
Pulmann
HU Berlin
Moduli spaces of flat connections as KĂ€hler spaces, part III
Abstract.
Building on earlier sessions, we will propose the existence of an approach to obtain KĂ€hler structures on the moduli spaces analogous to the quasi-Hamiltonian reduction.
Silvia
Ragni
University of Padua/HU Berlin
Teleman reconstruction of CohFTs
Abstract.
Given a CohFT, it is possible to define the action of the so called Givental group. Teleman in 2010 managed to prove that, given certain conditions, it is possible to reconstruct a whole CohFT from its 0-degree part. In this talk we are going to deal with the main ideas behind Teleman's reconstruction proof.
Multi-objective optimization with linear hyperbolic PDE constraints: generalized Nash equilibrium problems and gas market applications
Abstract.
The concept of Nash equilibrium is fundamental to a wide range of applications, spanning fields from particle mechanics to micro and macroeconomics. However, much of the existing literature focuses on finite-dimensional settings. In this seminar, we draw on energy markets coupled with transport dynamics to motivate the study of multi-objective optimization problems with hyperbolic PDE constraints. We will explore the core ideas and challenges posed by generalized Nash equilibrium problems, particularly those related to dimensionality and regularity. Finally, we present some recent results on the existence and characterization of equilibria, emphasizing optimality conditions as a framework for understanding such solutions.
Roman
Sauer
KIT
From Kazhdanâs property (T) to higher-dimensional expansion
Sanaz
Pooya
University of Potsdam
What is Kazhdan Property (T)?
Abstract.
Property (T), a rigidity property for groups, was introduced by Kazhdan in 1967. It is connected to the theory of expander graphs, which are sequences of graphs that are sparse and well connected at the same time. Notably, Margulis in 1973 exploited property (T) and constructed the first explicit example of an expander. While existence of expanders was known thanks to a probabilistic proof, an explicit construction was a highly non-trivial problem at the time. In this talk, I will define both property (T) and expander graphs. The example of special linear groups will exemplify Margulis' construction. In the end, I will briefly mention a higher dimensional analogue of property (T) that has attracted interest over the past decade.
Dante
Luber
Thesis defense
Abstract.
Thesis defense Dante Luber
MartĂn
HernĂĄndez
FAU Erlangen-NĂŒrnberg
Random Batch Method for PDEs on Graphs
Hana
Kourimska
UniversitÀt Potsdam
Amoebas: at the intersection of discrete, differential, and algebraic geometry
Bernold
Fiedler
Freie UniversitÀt Berlin
 Ist reale Zeit wirklich reell?
Abstract.
 FĂŒr Augustinus von Hippo ist die reale Zeit reell: er argumentiert mit dem Dedekindschen Schnitt. Die islamische Philosophie des KalÄm betrachtet alle Schöpfung als endlich: Raum, Zeit, Materie. Nietzsche bekĂŒmmert die ewige Wiederkehr: "Die Zeit selber ist ein Kreis". SpĂ€testens seit Weierstrass kennt allerdings selbst das physikalische Pendel neben der sichtbaren Periode in â noch eine unsichtbare Periode in â \ â. Der philosophische Determinismus sagt, dass perfekte Kenntnis des "Jetzt" alle Zukunft exakt vorherbestimmt. Die Chaostheorie meckert, dass das wirklich nicht so einfach ist. Und manche meinen ja, es sei eh' alles Zufall. Wir wollen mal schauen, wie das alles womöglich zusammenhĂ€ngt.
Sebastian
Bechtel
Université Paris-Saclay, France
What we know about square roots of elliptic systems -- and a bit more!
Abstract
Thomas
Kerdreux
Galeio
Some Challenges for Satellite Analysis with AI
Abstract.
Over the past decade, an enormous amount of satellite data has become freely available, unlocking countless opportunities for researchers to address a wide range of monitoring challenges. These include tasks like greenhouse gas estimation, ground displacement detection, soil moisture analysis, and many more. Despite this abundance of data, significant challenges remain when it comes to developing truly operational satellite analytics productsâespecially when scaling to create diverse solutions for various use cases. In this talk, I will explore these challenges from both a research and commercial perspective, highlighting the practical hurdles faced in turning satellite data into actionable products. I will also discuss how recent advancements in geospatial foundation models are beginning to emerge as a versatile, "Swiss Army knife" solution for building scalable and efficient satellite analytics tools.
Chloé
Rouyer
UniversitÀt Potsdam
Foundations of online learning for easy and worst-case data.
Abstract.
Online learning is a well-studied framework used to represent learning problems where the learner only has access to one data-point at the time and has to learn sequentially. This problem is particularly challenging in the bandit framework, which is a repeated game between the learner and the environment. In this game, the learner is faced with a list of actions and the environment generates losses associated with these actions. Then, the learner repeadly needs to play an action within this list in order to minimize their cumulative loss, but they can only observe the loss associated with the action they played. This means that at each round, the learner has to balance exploration (gathering information on less studied actions) and exploitation (using the already gathered information to play an action with a supposed small loss). Developing learner strategies for this problem depends on the assumptions made on the environment. There have been two major lines of research in this field, one assuming that these losses follow some unknown stochastic distributions and the other only assuming that these losses are bounded and independent of the learner's actions. In this talk, we introduce the recent field of best-of-both worlds sequential learning, which aims to develop algorithms that are optimal for both types of losses simultaneously.
Niklas
Martensen
HU Berlin
and
Jan
Pulmann
HU Berlin
Moduli spaces of flat connections as KĂ€hler spaces, part II
Abstract.
Building on the introduction of the various moduli spaces in part I, we will sketch the proof of the non-abelian Hodge correspondence for flat bundles on compact KĂ€hler manifolds, which provides a homeomorphism between the moduli space of (semisimple) flat connections and of (certain) Higgs bundles. Moreover, we will propose the existence of an approach to obtain KĂ€hler structures on the moduli spaces analogous to the quasi-Hamiltonian reduction.
Serhii
Stasyuk
Institute of Mathematics, Ukraine and TU Chemnitz
Sparse trigonometric approximation of Besov classes of periodic multivariate functions
Patrick
Cheridito
ETH ZĂŒrich
Sentiment-based asset pricing
Abstract.
We propose a continuous-time equilibrium model with a representative agent that is subject to stochastically fluctuating sentiments. Sentiments dynamically respond to past price movements and exhibit jumps, which occur more frequently when sentiments are disconnected from underlying fundamentals. We model feedback effects between asset prices and sentiment in both directions. Our analysis shows that in equilibrium, sentiments affect prices even though they have no direct impact on the assetâs fundamentals. Empirically, the equilibrium risk premia and risk-free rate respond to measurable shifts in sentiment in the direction predicted by the model.
Stefanos
Theodorakopoulous
TU Berlin
Topics on mean-field and McKeanâVlasov BSDEs, and the backward propagation of chaos
Abstract.
We shall present different versions of McKean-Vlasov and mean-field BSDEs of increasing generality, and the notion of backward propagation of chaos. We will then discuss some of the technical difficulties associated with the corresponding limit theorems and see some of their immediate corollaries and rates of convergence. Finally, we will introduce the concept of stability with respect to data sets for the backward propagation of chaos, and state the intermediate results that allowed us to prove its validity under a natural framework.
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.
Milos
Stojakovic
University of Novi Sad
Maker-Breaker games on random boards
Abstract.
In Maker-Breaker games played on edge sets of graphs, two players, Maker and Breaker, alternately claim unclaimed edges of a given graph until all of its edges are claimed. Maker wins the game if he claims all edges of one representative of a prescribed graph-theoretic structure (e.g. a Hamiltonian cycle, or a fixed graph \(H\)). Breaker wins otherwise. We take a closer look at various Maker-Breaker games played on the edge sets of random graphs.
Patrick
Cheridito
ETH Zurich
GĂŒnter
Last
KIT
Georg
Heinze
WIAS
Discrete-to-continuum limit for reaction-diffusion systems via variational convergence of gradient systems
Abstract
Orr
Paradise
Theory of Computation at UC Berkeley
Models That Prove Their Own Correctness
Abstract.
How can we trust the correctness of a learned model on a specific input of interest? This work introduces Self-Proving Modelsâmodels that can prove the correctness of their outputs via an interactive proof system. These models offer high-probability correctness for most outputs, with a verification algorithm that reliably detects any incorrect ones. The framework, rooted in theoretical guarantees, is demonstrated through experiments using transformers trained for arithmetic tasks such as computing the greatest common divisor (GCD).
Niklas
Martensen
HU Berlin
and
Jan
Pulmann
HU Berlin
Moduli spaces of flat connections as KĂ€hler spaces, part I
Abstract.
Moduli spaces of flat connections on surfaces appear in physics and geometry. Consequently, they themselves carry various geometric structures. We will recall the constructions of Poisson and complex structures on these moduli spaces, as well as the non-abelian Hodge correspondence. This talk will be mainly a literature review; our goal for the second part is to discuss the problem of inventing 'quasi' generalizations of KĂ€hler structures.
November 2024
Laura
Ciobanu
HWU
Undecidable problems in group theory
Christoph
Spiegel
Zuse-Institut Berlin
An Unsure Talk on an Un-Schur Problem
Abstract.
Graham, Rödl, and RuciĆski originally posed the problem of determining the minimum number of monochromatic Schur triples that must appear in any 2-coloring of the first \(n\) integers. This question was subsequently resolved independently by Datskovsky, Schoen, and Robertson and Zeilberger. In this talk we will study a natural anti-Ramsey variant of this question and establish the first non-trivial bounds by proving that the maximum fraction of Schur triples that can be rainbow in a given \(3\)-coloring of the first \(n\) integers is at least \(0.4\) and at most \(0.66656\). We conjecture the lower bound to be tight. This question is also motivated by a famous analogous problem in graph theory due to ErdĆs and SĂłs regarding the maximum number of rainbow triangles in any \(3\)-coloring of \(K_n\), which was settled by Balogh et al. The contents of this talk are joint work with Olaf Parczyk.
Michael
Jowsig
TU Berlin
Using OSCAR for experiments on Bergman's compact amalgamation problem
Abstract.
We study the question whether copies of S^1 in SU(3) can be amalgamated in a compact group. This is the simplest instance of a fundamental open problem in the theory of compact groups raised by George Bergman in 1987. This talk sets a focus on our considerable computational experiments with the new computer algebra system OSCAR. These computations suggest that the answer is positive in this case. Joint work with Mario Kummer, Andreas Thom and Claudia He Yun.
Julien
Chhor
Toulouse School of Economics
Locally sharp goodness-of-fit testing in sup norm for high-dimensional counts
Sandro
Roch
Technische UniversitÀt 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 vielfĂ€ltigen BezĂŒgen zu verwandten Strukturen aus Kombinatorik, Geometrie und Informatik, wie zum Beispiel Sortiernetzwerke, Rhombenpflasterungen oder Young-Tableaux. Uns beschĂ€ftigt unter anderem die Frage, wie sich diese zufĂ€llig 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 Boalchâs 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-Zhuâs 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.
Stefan
Felsner
Fast and Simple Sorting Using Partial Information
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 Itoâs 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 Itoâs 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.
Aldo
Kiem
Zuse-Institut Berlin
Forcing Graphs and Graph Algebra Operators
Abstract.
We report on recent progress on Sidorenko's conjecture and the forcing conjecture. Our main observation is that graph operators such as subdivisions, blow-ups or adding disjoint vertices correspond to order-preserving operators on Graph Algebras. This way we easily obtain some new results, in particular that the proper balanced blow-ups of Sidorenko graphs are forcing. This is joint work with Olaf Parczyk and Christoph Spiegel. The talk will survey our results and assume no prerequisites with graph algebras and categories.
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
UniversitÀt Potsdam
Contraction rates for conjugate gradient
and Lanczos approximate posteriors in Gaussian process regression
Sonja
Steffensen
RWTH Aachen
Solving Potential Multi-Leader-Follower Games via Parametric Optimization
Addis Belete
Zewde
Dire Dawa University
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, KollĂĄr 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-BenoĂźt
Bost
U Paris XI
and
Maryna
Viazovska
EPFL
A Day of Arithmetic Geometry (on the occasion of the retirement of JĂŒrg Kramer)
Abstract
Robert
Lauff
On triangular separation of bichromatic point sets
Aldo
Kiem
Zuse-Institut Berlin
Forcing Graphs and Graph Algebra Operators
Abstract.
We report on recent progress on Sidorenko's conjecture and the forcing conjecture. Our main observation is that graph operators such as subdivisions, blow-ups or adding disjoint vertices correspond to order-preserving operators on Graph Algebras. This way we easily obtain some new results, in particular that the proper balanced blow-ups of Sidorenko graphs are forcing. This is joint work with Olaf Parczyk and Christoph Spiegel. The talk will survey our results and assume no prerequisites with graph algebras and categories.
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
UniversitĂ 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.
Laura
Ghilardi
Politechnico Milano
Optimization of high pressure gas networks
Gernot
Akemann
Bielefeld UniversitÀt
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 AygĂŒn, Mario Kieburg and Patricia PĂ€Ăler in arXiv/2410.21032
Vesna
IrĆĄiÄ 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 âsquare-root lawâ 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.
Benjamin
Gess
TU Berlin
David
Hobson
U of Warwick
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 UniversitÀt
1D Landau-Ginzburg superpotential of big quantum cohomology of CP2
Abstract.
Using the inverse period map of the Gauss-Manin connection associated with QHâ(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 QHâ(CP2) in closed form as the composition of the Weierstrass â-function and the universal coverings of C \ (Z â jZ) and C \ (Z â 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.
Yan
Alves Radtke
Signotopes and Pseudoconfigurations
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-NeĆĄetĆil-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.
TomĂĄĆĄ
RoubĂÄek
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 UniversitÀt
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 LĂ©on 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 PlĂŒcker 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.
Alexandra
Wesolek
TurĂĄn-good graphs
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).
Ioannis
Gasteratos
TU Berlin
Arno
Siri-Jegousse
UNAM
Jean Baptiste
Teyssier
Paris
Pedro
Montero
Universidad TĂ©cnica ValparaĂso
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" \(\mathcal{G}_a^nâ\)), 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 Ă©tale 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.
Victor
Falgas-Ravry
Umea University
1-independent percolation on high dimensional lattices
Abstract.
The celebrated Harris-Kesten theorem states that if each edge of the square integer lattice is open with probability \(p\), independently of all other edges, then for \(p\) at most \(1/2\) almost surely all connected components of open edges are finite, while if \(p>1/2\) almost surely there exists a unique infinite connected component of open edges. What if one allows the state of an edge (open or closed) to depend on that of neighbouring edges? Could one use such local dependencies to delay the emergence of an infinite component? Such questions lead to fascinating problems at the interface between extremal combinatorics and percolation theory. In this talk, I will present recent work related to one such decade-old and still open problem of Balister and BollobĂĄs on 1-independent percolation on high-dimensional integer lattices. Joint work with Vincent Pfenninger (TU Graz)
Rolf
Krause
UniversitĂ della Svizzera italiana
Multilevel methods for non-smooth minimization and variational inequalities
Olexandr
Burylko
Institute of Mathematics, Ukraine
Reversible saddle-node separatrix-loop bifurcation
Joseph
Fehribach
Worcester Polytechnic Institute
Families of Kirchhoff Graphs
Abstract.
Given a set of \(n\) vectors in any vector space over the rationals, suppose that \(k<n\) are linear independent. Kirchhoff graphs are vector graphs (graphs whose edges are these vectors), whose cycles represent the dependencies of these vectors and whose vertex cuts are orthogonal to these cycles. This presentation discusses the uniformity of vector 2-connected Kirchhoff graphs, and how graph tiling can generate families of Kirchhoff graphs. These families are composed of prime graphs (those having no Kirchhoff subgraphs), and composite graphs (not prime), all generated by a set of fundamental Kirchhoff graphs (smallest prime Kirchhoff graphs that can generate other prime and composite graphs for our set of vectors).
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^Î(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 CortĂ©s KĂŒhnast, 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 subcontinentâmathematics, astronomy, and related disciplinesâhave 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 Indiaâs 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
Matthias
GrĂŒtzner
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
Zuse Institute Berlin and Technische UniversitÀt 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, BollobĂĄs, Janson, and Riordan introduced a class of branching processes parametrized by an LÂč-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 GrebĂk and Rocha.
Michael
Anastos
Institute of Science and Technology Austria
Climbing up a random subgraph of the hypercube
Shagnik
Das
National Taiwan University
Fractionally Intersecting Families
Anita
Liebenau
UNSW Sydney
Ramsey with purple edges
Gregory
Sorkin
London School of Economics and Political Science
Matchings and Loose Cycles in the Semirandom Hypergraph Model
Hugo
Abreu
Zuse Institute Berlin
Bell inequalities with classical communication via Frank-Wolfe algorithms
Abstract.
Bellâs 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 MĂŒnchen
Covariance modulated optimal transport: geometry and gradient flows
Stefan
Sauter
Uni ZĂŒrich
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.
JĂŒrg
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 algorithmâs 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 â [0, T] Ă 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] Ă 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 OsnabrĂŒck
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.
Vittorio
Di Fraia
Localizing Invariants
Chris
Huggle
Sheaves and Cosheaves on continuous posets
Dan
Crisan
Imperial College London
Jiniao
Qiu
U of Calgary
Caroline
Lasser
FakultĂ€t fĂŒr Mathematik, TU MĂŒnchen
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 ĂŒber nichtglatte Nash Equilibrium Probleme
Eleonore
Bach
EPFL
Vittorio
Di Fraia
Localizing Invariants
Georg
Lehner
Limits of dualizable infinity-categories
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 HamiltonâJacobiâBellman 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 Ă©tale 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 Ă©tale and pro-Ă©tale 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 PagĂšs.
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 â-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 KĂŒhn, Arne Kuhrs, and Alejandro Vargas; as well as with Desmond Coles.
Tibor
SzabĂł
Freie UniversitÀt Berlin
Global rigidity of random graphs in R
Benjamin
Jourdain
Ăcole des Ponts ParisTech
Noemi
Kurt
Goethe University Frankfurt
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 'equitableâ or 'altruisticâ is possible to arise from pure survival-of-the-fittest objective functions
Clément
Berenfeld
UniversitÀt 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.
LĂdia
Rossell RodrĂguez
Zuse Institute Berlin and Universitat PolitĂšcnica de Catalunya
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
Ramziâs Theorem and limits of compactly assembled categories
Deborah
Hendrych
ZIB
Solving the Optimal Experiment Design Problem with mixed-integer convex methods
Simon
Schmidt
Ruhr-UniversitÀt 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
Ăcole 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, HuyĂȘn Pham, Sturmius Tuschmann, and Moritz Voss.
Vladimir
Shikmann
Technische UniversitÀt 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.
BĂĄlint
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 Igusaâs 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 UniversitÀt Berlin
 Zauberhafte Mathematik
Abstract.
 Wenn KunststĂŒcke 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 aufwĂ€ndigen elektronischen Schaltungen usw. Man kann sich aber auch Tatsachen aus verschiedenen Wissenschaftsgebieten zunutze machen: Physik, Chemie und natĂŒrlich auch Mathematik. In dem Vortrag soll gezeigt werden, wie man verschiedene Aspekte unseres Faches fĂŒr 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 verstĂ€ndlich" bis "ein bisschen anspruchsvoll". FĂŒr Bachelorstudierende sollte es keine Probleme geben. Es empfiehlt sich, etwas zum Schreiben und ein Kartenspiel dabei zu haben. Dann kann man einige der KunststĂŒcke gleich ausprobieren.
Georg
Lehner
Compactly assembled infinity-categories II
Elisabeth
a.d. Siepen
and
Georg
Lehner
Compactly assembled infinity-categories II + Examples
Andre
Zepernick
Quasi-Monte Carlo Methods for PDEs on Random Domains
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.
Rene
Brandenberg
TU MĂŒnchen
Dhruv
Mubayi
University of Illinois
New bounds for the ErdĆs-Rogers problem
Benedikt
GrĂ€Ăle
HU Berlin
Analysing WOPSIP by supercloseness
Stefan
Sauter
Uni ZĂŒrich
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.
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 TeichmĂŒller 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 ânon-symmetricâ 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 TamĂĄs KĂĄlmĂĄn.
Geometric Machine Learning and Graph Machine Learning
Elisabeth
a.d. Siepen
Compactly assembled infinity-categories
Konrad
Mundinger
ZIB
Learning Operators via Hypernetworks
Nikolay
Barashkov
MPI Leipzig
Shige
Peng
Shandong University
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 ZĂŒrich
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
MĂŒcksch
Leibniz UniversitÀt 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).
JĂŒrgen
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.
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.
Alexander
Suciu
Northeastern University
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 ErdĆs-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 worldâs 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
FlaĂkamp
UniversitÀt 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.<br>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.<br>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 LipschitzâKilling 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.
Lukas
Abel
A Scaling Law for a Model of Epitaxial Growth with Dislocations
Melanie
Koser
Energy Driven Pattern Formation in a Model for Two-Dimensional Frustrated Spin Systems
Anastasija
Pesic
Variational Models for Pattern Formation in Biomembranes
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.
Mate
Gerencser
TU Vienna
Hendrik
Weber
U of MĂŒnster
Markus
Schmidtchen
Institut fĂŒr Wissenschaftliches Rechnen, TU Dresden
Gradient flow solutions for porous medium equations with nonlocal LĂ©vy-type pressure
Abstract
Benedikt
GrĂ€Ăle
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 ServĂĄn 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 TĂłth), 3 (Hales), 8 (Viazovska), and 24 (Cohn et al.). However, several nontrivial lower and upper bounds for the density ÎŽ(d) of the densest packing in dimension d have been established. A simple "folklore" result states that ÎŽ(d) â„ 1/2^d. In this talk we present the intuition and details of three other lower and upper bounds for ÎŽ(d): the MinkowskiâHlawka theorem for a lower bound, Blichfeldt's upper bound, and Rogers's upper bound. These results, among others, place ÎŽ(d) within a narrow strip of possible densities.
Robert
Lauff
Trees in Planar Graphs
Katharina
Oberpriller
UniversitĂ€t MĂŒnchen
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 ReprÀsentationen dreiecksfreier Graphen mit hoher chromatischer Zahl (Wiederholung)
Markus
Herrmann-Wicklmayr
UniversitÀt 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.
Todor
AntiÄ
Star-Forest Decompositions of Complete (Geometric) Graphs
Hannes
Tarun
Konstruktion und Geometrische ReprÀsentationen dreiecksfreier Graphen mit hoher chromatischer Zahl (Wiederholung)
Nils
Detering
UniversitĂ€t DĂŒsseldorf
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
SantalĂł Geometry of Convex Polytopes
Abstract.
The SantalĂł 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 SantalĂł 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
UniversitĂ 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 humansâ 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 âcollectiveâ 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 ZĂŒrich
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 ReprÀsentationen dreiecksfreier Graphen mit hoher chromatischer Zahl
George
Balla
and
Marcel
Wack
and
Lena
Weis
Arxiv Seminar
Yassine
El Maazouz
Marina
Garrote
MPI MIS
Marina
Garrote-LĂłpez
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.
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 BlagojeviÄ & 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 âiterated' 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 BlagojeviÄ.
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, BĂžgvad'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 Hensleyâs 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 ZĂŒrich
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 UniversitÀt 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
UniversitÀt 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. GonzĂĄlez DâLeĂłn, Alejandro H. Morales, Daniel Tamayo JimĂ©nez, 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)
Yves Etienne
JĂ€ckle
Zuse Institute Berlin and Technische UniversitÀt 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 âsymbol bootstrapâ 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
HĂ©lĂšne
Esnault
UniversitÀt Kopenhagen
Lokale Systeme in der Algebraischen und Arithmetischen Geometrie
Abstract.
 Von Galois, zu Riemann, zu PoincarĂ©, zu⊠Grothendieck, zu⊠Simpson, zu⊠Langlands⊠Die Fundamentalgruppen (die Galoisgruppen) sind zwar sehr klar definiert, ĂŒber deren Eigenschaften weiĂ 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
UniversitÀt 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 algorithmâs 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
UniversitÀt 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 LÂČ-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-UniversitÀt 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
GrĂŒtzner
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 âartificial intelligenceâ (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.
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.<br>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.<br>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 orthantââ 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).
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, Kâ, with some examples, and extend the discussion to higher K-groups.
Malin
Heinacher
Trees and co-trees in planar graphs
Adam
Afandi
University of MĂŒnster
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.
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 Dantzigâs rule, with Blandâs 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 Dantzigâs 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
UniversitĂ€t TĂŒbingen
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 BorzĂŹ 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
UniversitĂ 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.
JĂŒrg
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 GauĂ 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 â generisch gesehen â solche Polynome (unendlich) viele rationale Nullstellen haben, wenn deren Grad âkleinâ ist, wogegen sie höchstens endlich viele rationale Nullstellen besitzen, falls deren Grad âgroĂâ ist. Um diese qualitativen Sachverhalte quantitativ genauer zu verstehen, werden wir uns vom IrrationalitĂ€tsbeweis von  â2  mit Hilfe von Fermats unendlichem Abstieg inspirieren lassen. Diese Ăberlegungen werden uns auch zu interessanten zahlentheoretischen Fragestellungen fĂŒhren, wie beispielsweise dem Kongruenzzahlproblem, die Gegenstand aktueller mathematischer Forschung sind.
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.
Bennet
Gebken
UniversitÀt 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 BlagojeviÄ & 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 âiterated' 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 BlagojeviÄ.
Michael
Krivelevich
Tel Aviv University
Percolation through isoperimetry
Jona
Rörig
HU Berlin
Gromov's Compactness Theorem for Alexandrov Spaces with Lower Curvature Bounds
RDM â More than a Chore!
Matteo
Giordano
UniversitĂ 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 ZĂŒrich
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
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.
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.
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 GopakumarâVafa 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 GromovâWitten theory arising from physics. Very roughly speaking, the GromovâWitten 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 GromovâWitten invariants are quite complicated (or âhave a rich internal structureâ). This is true especially for if \((X,\omega)\) is a symplectic CalabiâYau 3âfold (that is: \(\mathrm{dim}X=6\), \(c_1(X,\omega) = 0\)). In 1998, using arguments from Mâtheory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic CalabiâYau 3âfolds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the GromovâWitten invariants by a transformation of the generating series. The GopakumarâVafa 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 Gromovâs compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that Gromovâs compactness theorem can (and should!) be replaced with the work of FedererâFlemming, Allard, and De LellisâSpadaroâSpolaor. This upgrade of Ionel and Parkerâs cluster formalism proves both the integrality and finiteness conjecture. This talk is based on joint work with Eleny Ionel and Aleksander Doan.
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.
Nonlinear Electrokinetics in Anisotropic Microfluids â Analysis, Simulation, and Optimal Control
Abstract
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 Castelnuovoâs 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 ZĂŒrich
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.
HĂ©lĂšne
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 Lâ 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 AndrĂ©âs 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
PreiĂ
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)
LĂĄszlĂł
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
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, âŠ),  or agents (individual humans, media agents, âŠ). We are interested in systems with at least two types of such units, one type of which just a âfewâ individual units are present and another type of which there are âmanyâ. 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 âmanyâ 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 VĂ©gh (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 LĂĄszlĂł VĂ©gh.
Igor
Burban
UniversitÀt 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 lot of functionality at a decent speed, is called Singular:Letterplace and it is OSCAR-aware.
Deborah
Hendrych
Zuse Institute Berlin and Technische UniversitÀt 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, ErdĆs 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
UniversitÀt 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-UniversitÀt 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 ÎČ-model. Using Pfaffian formulae for averages of characteristic polynomials when ÎČ = 1 or ÎČ =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 GaĂ«tan Borot.
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.
Equations over groups       Â
Carsten
Carstensen
HU Berlin
Hs(Ω) for 0<s<1
Jeremie
Houssineau
University of Warwick
Ensemble Kalman filtering for epistemic uncertainty
Manfred
Scheucher
On the empty hexagon theorem
November 2023
Sonja
Steffensen
Institut fĂŒr 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 TSOâs 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
UrzĂșa
Pontificia Universidad CatĂłlica 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 BÄdescu 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 ZĂșñiga (Ph.D. student at UC Chile), the pre-print can be found here.
Martin
Spindler
UniversitÀt Hamburg
High-dimensional L2-boosting: Rate of convergence (hybrid talk)
Julia
Schaeffer
HU Berlin
Contour integration methods for nonlinear eigenvalue problems in nanooptics
Yannik
SchĂŒler
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.
Jonas
LautenschlÀger
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 âthe bestâ, 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.
Lukas
SchĂŒrmann
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 UniversitÀt
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, Ivån 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 Lâ 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
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 Nivatâs conjecture.
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.
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