Sanaz
Pooya
University of Potsdam
What is Kazhdan Property (T)?
'What is...?' Seminar 📅
Homepage
Institute
Organizer
Fawzy Hegab,
Ji Hoon Chun,
Toluwani Okunola,
Alberto Richtsfeld,
Nicolas Alexander Weiss,
Aldo Kiem, and
Muhammed E. Guelen
Description
The "What is...?" Seminar is a cycle of talks organized each semester by students at the three universities of Berlin, the Freie Universität Berlin, the Humboldt-Universität zu Berlin, and the Technische Universität Berlin. Each talk of the "What is...?" Seminar is characterized by the following: It lasts around half an hour, introduces a mathematical idea or object answering a question of the form "What is...?", is directed to mathematics students or mathematicians unfamiliar with the exposed field, and happens in an open and relaxed environment where comments and questions are welcome.
Number of talks
296
December 2024
November 2024
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.
July 2024
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.
June 2024
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.
May 2024
Kanishka
Katipearachchi
HU Berlin
What is Hyperbolic Geometry?
Abstract.
In this talk I will give a gentle and non-rigourous introduction to hyperbolic geometry. Starting from a slight modification of the parallel postulate, we will build a model for the hyperbolic plane and use its transformation group to deduce a metric for it. We will then use this model to build hyperbolic surfaces and possibly define the Teichmüller space of hyperbolic structures for a topological surface.
Ji Hoon
Chun
TU Berlin
$\vec{w}h\alpha\mathfrak{t}\;\; \forall\mathbb{R}\varepsilon\ldots$sphere packing lower and upper bounds?
Abstract.
In Euclidean space, the densest sphere packings and their densities are only known in dimensions 1, 2 (Thue, Fejes Tóth), 3 (Hales), 8 (Viazovska), and 24 (Cohn et al.). However, several nontrivial lower and upper bounds for the density δ(d) of the densest packing in dimension d have been established. A simple "folklore" result states that δ(d) ≥ 1/2^d. In this talk we present the intuition and details of three other lower and upper bounds for δ(d): the Minkowski–Hlawka theorem for a lower bound, Blichfeldt's upper bound, and Rogers's upper bound. These results, among others, place δ(d) within a narrow strip of possible densities.
April 2024
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.
February 2024
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.
January 2024
Vittorio
di Fraia
FU Berlin
What is Algebraic K-theory?
Abstract.
Algebraic K-theory originated in the 1950s from Grothendieck's studies on algebraic varieties. Since then, it has proven to be a powerful tool in various fields such as algebraic geometry, algebraic topology, and number theory. We will focus on defining the first K-group, K₀, with some examples, and extend the discussion to higher K-groups.
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.
December 2023
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.
November 2023
Manuel
Staiger
FU Berlin
What is a sofic group?
Abstract.
In view of the MATH+ Friday lecture by Andreas Thom we introduce sofic groups and discuss some examples.
Jonas
Köppl
WIAS
What is a low-complexity coloring?
Abstract.
Informally, a tiling is a covering of the plane with tiles of various shapes, arranged to avoid any overlapping. Usually, these tiles have simple shapes (e.g. polygons), and one only allows a small number of different shapes to be used for a tiling. One particularly interesting class of tiles are the so-called Wang tiles which can alternatively be represented via finite colorings of \Z^d. Given a set of such tiles, one might ask whether one can actually use them to cover the plane and whether that is possible without ever repeating oneself, i.e., without becoming periodic. The goal of this talk is to introduce (finite) colorings of Z^d, discuss their relation to Wang tiles and the domino problem, and then speak about low complexity colorings and Nivat’s conjecture.
Ekin
Ergen
and
Lizaveta
Manzhulina
TU Berlin
What is (integer) linear programming?
Abstract.
Below is the abstract initially proposed by Lizaveta Manzhulina. Due to illness, however, Ekin Ergen did kindly and on our very short notice take over the talk. A plentitude of highly relevant real-world problems can be modeled by means of linear programs (LP): optimization problems with a linear objective and constraints. Luckily, there are efficient algorithms for solving them. However, once we additionally require the variables to be integers --- which unsurprisingly is oftentimes so in applications --- things go downhill. Integer linear programming (ILP) is NP-hard, and thus to deal with it in practice heuristic methods and approximation algorithms are used. After convincing ourselves that (I)LP arises everywhere around us, we will explore the intricate relationship between LP and ILP. In particular, we are going to discuss how LP can be of help in tackling NP-hard ILP. Along the way, we will be accompanied by classical examples coming from network problems --- and from combinatorial optimization in general.
July 2023
Dennis
Chemnitz
FU Berlin
What is the blow-up method?
Abstract.
Many dynamical systems have interesting dynamics at several time scales (think weather vs climate). The resulting phenomena can be difficult to capture using classical ODE-theory and thus new tools are required. The goal of this talk is to introduce fast/slow systems, geometric singular perturbation theory and the blow-up method. The Van-der-Pol oscillator will serve as a guiding example.
June 2023
Sanaz
Pooya
U Potsdam
What is a group C*-algebra?
Abstract.
In this talk I will start by introducing the notion of C*-algebras, focusing on some basic examples. Subsequently, I will introduce the important example of group C*-algebras. They form a bridge between group theory and operator algebras. For some concrete examples of groups I will describe their associated C*-algebras.
May 2023
Musxhu (Julian)
Kern
WIAS Berlin
What is an interacting particle system?
Abstract.
Most parts of the real world can be described via interacting particles. Whether you look at the microscopic level and see molecules bumping together or try to understand how electrical impulses in your brain form thoughts, one question is common to all particle systems: how can local interactions induce a macroscopic behaviour? In this week's "What is... ?" seminar, we will see how particle systems model natural phenomena, where randomness comes into play and what behaviour we might expect to see at different scales (=zoom levels).
Lukas
Abel
HU Berlin
What is important in mathematical modeling of materials?
Abstract.
Real world problems of material sciences are often mathematically modelled in the framework of calculus of variations. In this week’s „What is“ seminar starting with $\Gamma$-convergence we will go on a journey through different ideas, concepts and methods which are highly used and of great importance in this special area of applied analysis.
April 2023
Marco
Flores
HU Berlin
What is an Eisenstein series?
Abstract.
Modular forms are a certain kind of holomorphic maps defined on the upper half-plane whose Fourier expansions, very much surprisingly, mantain an intimate relationship with number theory. One of the most striking and intricate manifestations of this relationship is the proof by Andrew Wiles and others of Fermat's Last Theorem, a result about an integral equation with 4 variables which resisted proof for over 350 years. Eisenstein series are a particular kind of modular forms which can be written down explicitly, thus being ideal for experimentation in the theory of modular forms. In this talk I will give some examples of Eisenstein series, and we will witness exactly how number theoretic information can be extracted from their structure.
February 2023
Shpresim
Sadiku
TU Berlin, Zuse Institut Berlin
What is backpropagation?
Abstract.
Deep Neural Networks (DNNs) are a composition of several vector-valued functions. In order to train DNNs, it is necessary to calculate the gradient of the error function with respect to all parameters. As the error function of a DNN consists of several nonlinear functions, each with numerous parameters, this calculation is not trivial. We revisit the Backpropagation (BP) algorithm, widely used by practitioners to train DNNs. By leveraging the composite structure of the DNNs, we show that the BP algorithm is able to efficiently compute the gradient and that the number of layers in the network does not significantly impact the complexity of the calculation.
January 2023
Dingyu
Yang
HU Berlin
What is an infinity structure?
Abstract.
Producing a loop of unit length by joining two such ones is an operation which is almost associative. An infinity structure makes this precise. We explain how to think about such a higher structure, and illustrate using a couple of examples from linking of ancient Borromean rings to some glimpse of current research.
Andrea
Di Lorenzo
HU Berlin
What is a rational variety?
Abstract.
Starting from rational algebraic curves, I will discuss the notion of rational varieties, giving some concrete examples. Next, I will present a classical problem: how can we tell a rational variety from a non-rational (i.e. irrational) one? I will briefly sketch some strategies for tackling this problem and outline some open questions.
December 2022
Silas
Rathke
FU Berlin
What is a k-tensor?
Abstract.
Don't Panic! $k$-Tensors are quite different to tensor products and I also won't write down any commutative diagram, I promise! Instead, $k$-tensors are used to answer questions in extremal combinatorics using linear algebra. We are going look at a beautiful proof for a surprising fact in extremal combinatorics and then try to generalize the main idea, which will lead us to $k$-tensors and its slice rank.
Paul
Brommer-Wierig
FU Berlin
What is tame geometry?
Abstract.
In the 80's Grothendieck claimed that general topology was unfit for the study of the shape of geometric objects since it is possible to realise arbitrary wild pathologies within it. He proposed a tame geometry which should not exhibit such phenomena. Around the same time, model theorists interested in a seemingly unrelated question developed the now widely accepted framework of o-minimal structures. In this talk, I will explain the notion of an o-minimal structure and discuss basic examples of it. I will then continue explaining why they lead to the correct notion of tame geometry. I will finish the talk by sketching applications of o-minimal structures in algebraic geometry.
November 2022
Gari Yamel
Peralta Alvarez
HU Berlin
What is a Grothendieck topology?
Abstract.
A Grothendieck topology is a structure on a category that mimics the properties of open covers of topological spaces. These properties are particularly useful to define geometric structures in terms of local information, for example, an atlas on a manifold or a sheaf on an algebraic variety. The goal of this talk is to explain in simple terms how Grothendieck topologies and sheaves generalize familiar geometric objects.
October 2022
Kyle
Huang
FU Berlin
What is a Semialgebraic Set?
Abstract.
Semialgebraic sets are a generalization of polyhedra, where instead of linear constraints we take polynomial inequalities. As the fundamental object of real algebraic geometry and capable of modeling diverse real-world phenomena, they are of interest to both pure and applied mathematicians. We'll begin with an introduction to the motivations and definitions surrounding real algebraic geometry, after which we will discuss the Tarski-Seidenberg Principle and see an illustrative application to robotics. With computer algebra!
July 2022
Ekin
Ergen
TU Berlin
What is the generalized Poincaré conjecture?
Abstract.
We are going to explore a higher-dimension version of the Poincaré conjecture. In dimension three, this corresponds to the only Millennium Prize problem that is solved as of today. Roughly, the conjecture tells us that, if a manifold is homotopy equivalent to a sphere of its dimension, it is a sphere. We are going to discuss the history of this conjecture, and sketch a proof of the higher-dimension version via the $h$-cobordism theorem, due to Smale (1960). We are also going to introduce handle decompositions and the so-called Whitney trick, due to Whitney, which helps us tidy up handles. Prepare to see a lot of great achievements and of course, a lot of pictures!
Mahmut Levent
Doğan
TU Berlin
What is a probabilistically checkable proof?
Abstract.
The complexity class NP consists of search problems where the sought object may be hard to find, but once the object is provided by an all-knowing oracle, it is possible to verify it efficiently and deterministically. However, if we allow ourselves interaction and small probability of error in our verification processes, we can reach higher complexity classes than NP. These “probabilistic proof systems” consist of interactive proofs, zero-knowledge proofs and the main object of the talk, the probabilistically checkable proofs. In this talk, we will provide a humble introduction to probabilistic proof systems and talk about one of the most important theorems of complexity theory, the PCP theorem. If time allows, we will also show how the PCP theorem implies inapproximability of certain combinatoric optimizatiton problems.
June 2022
Martha
Nansubuga
HU Berlin
What is a McKean-Vlasov process?
Abstract.
A McKean-Vlasov process is a stochastic process described by a stochastic differential equation where the coefficients depend on the distribution of the process itself. The story of these processes started with a stochastic toy model for the Vlasov equation of plasma proposed by Mark Kac in his paper "Foundations of kinetic theory (1956)". They have been applied in several areas like physics, finance, social interactions and extra. In order to understand McKean-Vlasov processes, I will give a brief introduction to stochastic differential equations, too.
Angela
Ortega
HU Berlin
What is a linear system of plane curves?
Abstract.
The interpolation problem can be roughly stated as follows. Given a set of points in the plane find a polynomial $f(x, y)$ with these points as roots. More generally, one can also ask for the vector space of all interpolating polynomials with bounded degree and with specified vanishing order at each one of the points. In this talk we will look at the problem of estimating the dimension of such space and will explain why does it become hard to give a general answer to this question. We will spice the exposition with a couple of examples.
Khai Van
Tran
TU Berlin
What is submodularity?
Abstract.
Submodularity is a property of functions that assign values to subsets of a ground set. It can be characterized by the inequality $f(X) + f(Y) \geq f(X \cap Y) + f(X \cup Y)$. In this talk, we will explain why submodularity is regarded as a discrete equivalent to both convexity and concavity. Furthermore, we will demonstrate some common techniques making use of submodular functions.
May 2022
Nicholas
Schmidt
HU Berlin
What is a zeta function?
Abstract.
Riemann, Dedekind, Hecke, Artin, Selberg, Ruelle,... the list of names associated to "zeta functions" is long, nearly as long the list of objects to which zeta functions have been attached, ranging from number fields over riemannian manifolds to partially ordered sets, dynamical systems and — finally — groups. This talk will try to give a conceptual answer to the question "what is a zeta function" by "categorifying" Dirichlet series, and to look at some recurring themes connecting various kinds of zeta functions.
Ji Hoon
Chun
TU Berlin
What is a sphere packing?
Abstract.
The problem of finding the densest way to pack equally sized spheres in Euclidean space has been studied for hundreds of years. The densest packing is only known in dimensions 1, 2, 3, 8, and 24, with the last two being recently solved by Viazovska (2016) and Cohn et al. (2016). This talk will give an overview of several lower and upper bounds for the maximum packing density. We will also show examples of dense sphere packings in small and high dimensions and explain their properties and connections to other areas of mathematics.
April 2022
Johannes
Zonker
FU Berlin
What is epidemic modeling?
Abstract.
While the spreading of infectious diseases has already been a popular research area for a long time, since the beginning of the COVID19 pandemic almost everyone (including non-mathematicians) has at least heard about the term epidemic modeling. The goal of this talk is to provide an introduction to the basic concepts and core assumptions of epidimic models as well as an overview of common model types. Among the different model types we will focus on the widely used compartmental ODE models and related network and metapopulation approaches. Slides are available here.
February 2022
Anastasija
Pesic
HU Berlin
What is direct method of calculus of variations?
Abstract.
When looking to show that a minimum of a functional is attained, one usually first turns to the Direct Method of Calculus of Variations. In this lecture, following one concrete example from electrostatics, we will first explore the necessary condition that a minimizer must satisfy — the Euler-Lagrange equation. Then we will move on to the problem of existence of minimizers and present the Direct Method. We will discuss the assumptions involved and explore the method's limits.
January 2022
Henriette
Lipschütz
FU Berlin
What is a regular polytope?
Abstract.
Polytopes in three dimensions are present in several objects used on daily basis: everything having planar sides without dents or holes belongs to this class of objects. Mathematically, they can be described as intersection of half spaces. Next to this definition, we will consider some of their basic properties and special classes of polytopes in 3D such as the Platonic and Archimedean solids.
December 2021
Pierre
Clavier
IRIMAS, Université de Haute Alsace
What is resurgence theory?
Abstract.
Resurgence theory was invented over four decades ago by J. Ecalle, but has recently been gaining a lot of interest, in particular because of its applications to Physics. In this introductory talk, I will start to explain why is resugence interesting for physicists, then move on to detail how trans-series arise and can be worked out within the resurgent framework. I will also mention some interesting properties of (accelero-)summation a la Ecalle when applied to quantum field theory.
November 2021
Jannik
Peters
TU Berlin
What is fair division?
Abstract.
In this seminar talk, I will give a short introduction to the theory of fair division and present the basic solution concepts and models in fair division and how/if they are achievable. If you are interested in cutting cakes, fairness, and interesting computational problems come join the talk!
October 2021
Simon
Breneis
WIAS Berlin
What is rough volatility?
Abstract.
Starting with an introduction of Brownian motion, we discuss the general goals of mathematical finance and explain the intuition behind the Black-Scholes model. After discussing option pricing in this standard framework, we observe some of the shortcomings of the Black-Scholes model. Finally, to overcome these deficiencies, we introduce stochastic volatility and rough volatility models.
July 2021
Paramjit
Singh
HU Berlin
What is Floer homology?
Abstract.
We shall introduce Morse theory and the notion of Floer homology for a finite dimensional orientable closed manifold with an example. Then we will outline how this is generalized for infinite dimensional manifolds (like loop spaces, curves in a manifold, etc) and in the symplectic/contact settings.
Clemens
Sirotenko
WIAS Berlin
What is variational image processing?
Abstract.
We will briefly introduce the concept of inverse problems in the context of image reconstruction tasks and take a look on practical examples. Subsequently, we discuss the variational approach to solve these problems and point out existing advantages, disadvantages and potential challenges. After a small glimpse at some theoretical results, we turn to more modern methods that aim to integrate information from existing data into the solution process.
Ivan
Spirandelli
U Potsdam
What is persistent homology?
Abstract.
In the talk I give a practical introduction to alpha complexes and persistent homology. Persistent homology is a formalism that enables us to compute topological features of a space at different spatial resolutions. It is often used on simplicial complexes that were constructed on point clouds. Alpha complexes are one popular way of constructing these simplicial complexes.
May 2021
Kıvanç
Ersoy
FU Berlin
What is a finite simple group?
Abstract.
In this introductory talk, finite simple groups and their classification will be introduced. In particular, we will talk about some examples of simple groups of Lie type.
Elias
Wirth
TU Berlin/Zuse-Institut Berlin
What is conditional gradients?
Abstract.
We introduce the Frank-Wolfe Algorithm, also known as Conditional Gradients. Recently, the algorithm has been revisited for various applications in machine learning. We motivate the Frank-Wolfe Algorithm and discuss the properties that make it appealing for practitioners.
April 2021
Raphael
Steiner
TU Berlin
What is a tree-decomposition?
Abstract.
In the talk I explain the definition of a tree decomposition of a graph, and using simple examples I illustrate some of the intuition behind this important combinatorial concept, which is useful in computer science for designing efficient algorithms as well as in structural graph theory as a tool for proving theoretical results.
February 2021
Gaëtan
Borot
HU Berlin
What is a Coulomb gas?
Abstract.
I will explain some physical and mathematical motivations underlying the study of Coulomb gases (and more generally, of repulsive particle systems) and set up natural questions about the macroscopic and microscopic properties of such models. In the way we will introduce the notion of equilibrium measure, large deviation principles, and Gaussian free field. Prof. Borot allowed us to share his slides; you can find them here.
January 2021
Felix
Baumann
Zuse-Institut Berlin
What is a multigrid method?
Abstract.
The discretization of PDEs leads to linear systems with a very large number of unknowns. While direct solvers fail due to the large scale, Multigrid Methods provide a powerful solution technique. In this talk we present the core ideas behind the Multigrid Method and discuss mesh-independent convergence and its optimal complexity.
Alp
Müyesser
FU Berlin
What is an expander graph?
Abstract.
A graph has good expansion if it is sparse yet very well-connected. In this talk, we will discuss several equivalent ways to quantify the expansion of a given graph. Further, we will do a couple of case studies demonstrating why graphs with good expansion are useful in various areas of mathematics.
December 2020
Niklas
Martensen
HU Berlin
What is Noether's theorem?
Abstract.
Noether's theorem is one of the most important results in classical mechanics. It shows that any continuous symmetry of a mechanical system leads to a conserved quantity such as energy or momentum. In order to understand this result we give a brief introduction into Lagrangian mechanics and show applications of the theorem.
June 2020
Shah
Faisal
HU Berlin
What is an ergodic decomposition of invariant measures?
Abstract.
Ergodic systems, being indecomposable, are from the main objects of study in dynamical systems. If a system is not ergodic, it is natural to ask the following question: Is it possible to split it into ergodic systems in such a way that the study of the former reduces to the study of latter ones? In this talk, we will answer this question for measurable maps defined on complete separable metric spaces with Borel probability measure. No background in ergodic theory is assumed! The contents of the talk are taken from our preprint Ergodic Decomposition, to appear in Indagationes Mathematicae. For technical reasons, the talk could not be recorded. However, its slides can be found here.
February 2020
Felix
Schröder
TU Berlin
What is the dimension of a partial order?
Abstract.
Partial orders are among the most basic combinatorial structures, related to foundational mathematics such as properties of set systems as well as applied mathematics such as scheduling. We will look at multiple ways to visualize these posets, yielding a combinatorial definition of their dimension. We will then investigate the dimension of some special posets.
January 2020
Paul
Hager
TU Berlin
What is gaussian multiplicative chaos?
Abstract.
We will talk about the random measure which was originally introduced by Kahane in 1985 with the motivation of giving a rigorous construction to the Kolmogorov-Obukhov model of fully developed turbulences. In particular we are going to discuss the applications, properties and different approaches to its construction.
Ander
Lamaison
FU Berlin
What is the behavior of random graphs?
Abstract.
We construct a graph randomly by taking $n$ vertices and, for every pair of them, we draw an edge between them with probability $p$. As it turns out, if we slowly increase $p$, the properties of the graph that we obtain change drastically at certain values of $p$. In this talk we introduce the concept of threshold probability, and discuss what properties we expect to see in the random graph for different ranges of $p$.
December 2019
Josué
Tonelli-Cueto
TU Berlin
What is best voting system?
Abstract.
Democracy might look like a straightforward process: whatever is the option preferred by the group should be the chosen option. But what does it mean to be the preferred option by the group? To answer this question, several voting systems (Condorcet's method, plurality voting, Borda count...), each with its advantages and disadvantages, have been proposed. As it has become clear in current times, how we vote influences radically the outcome of the election process. Can't we just choose the objectively best possible voting system? In this talk, we will see what mathematics has to say about this.
November 2019
Janusz
Ginster
HU Berlin
What is the role of
convexity in variational problems?
Abstract.
A classical problem in the Calculus of Variations is to minimize an integral functional in a certain class of functions. In this talk we present several simple examples which illustrate why and how non-convexity of the integrand can result in the non-existence of minimizers or the formation of small-scale oscillations.
Georg
Lehner
FU Berlin
What is a topological surface?
Abstract.
Everyone knows what a two dimensional surface is. If we allow surfaces to bend or stretch, but not rip and tear, how many fundamentally different surfaces are there? The answer is known since the 1860's but it hasn't lost its simple elegance since: Everything can be obtained by cutting a number of holes into a sphere and glueing into some of them either handles or Möbius strips. We present the simple ZIP proof due to Conway and we will draw lots and lots of pictures. No formulas needed!
Hannah
Sjöberg
FU Berlin
What is the Euler characteristic of a polytope?
Abstract.
The Euler characteristic of a non-empty (solid) polytope $P$ is the alternating sum of the number of non-empty $i$-dimensional faces of $P$. The Euler-Poincaré formula asserts that this alternating sum is equal to $1$. In this talk we discuss how the Euler characteristic can be constructed as a valuation. The construction has nice applications: we will see that it gives us simple proofs of the Euler-Poincaré formula and of a theorem on the number of regions in a hyperplane arrangement.
Dominic
Bunnett
TU Berlin
and
Marta
Panizzut
TU Berlin
What is algebraic curves and their tropical friends?
Abstract.
We will begin by illustrating stable tropical curves of genus $g$. We will then give a pictorial introduction to smooth algebraic curves of genus $g$ and their degenerations. We will define the moduli space of such curves and show the appearance of tropical curves in its study.
October 2019
Moritz
Schmitt
TU Berlin
What is Hilbert's 17th problem?
Abstract.
At the International Congress of Mathematicians in 1900 in Paris, Hilbert presented his now famous list of 23 problems. Many of these were rather influential for 20th century mathematics. One of these is Hilbert's 17th problem: Given a real polynomial in several variables that is non-negative, can it be represented as a sum of squares of rational functions? Artin answered this questions in 1927 in the affirmative. In this talk we will take a closer look at both Hilbert's problem and Artin's solution, and try to understand how all this relates to the upcoming lecture of M. F. Roy.
Georg
Lehner
FU Berlin
What is the point of pointless topology?
Abstract.
The open sets of a topological space together with union and intersection form a special kind of partially ordered set — a complete lattice in which finite meets distribute over arbitrary joins. These algebraic gadgets are called locales and as it turns out they behave very similar to the classical spaces we know from point set topology — with one major exception: There are interesting locales which have no points at all! We will give a brief glimpse into this pointfree world and hopefully convince you that giving up points is not as bad an idea as one might think.
Jean-Philippe
Labbé
FU Berlin
What is counting?
Abstract.
Symmetry is a central concept in many areas of mathematics, if not all of them. In this talk, I will present an intriguing numerical coincidence bridging two seemingly unrelated objects — the first one coming from representation theory, the second from discrete geometry — where symmetry could be the culprit. Confirming the guilt of symmetry would shed new light on both objects, and perhaps help to solve some conjecture in discrete geometry.
Dominic
Bunnett
TU Berlin
What is $M_{g,n}$?
Abstract.
$M_{g,n}$ is an algebraic variety which parameterises isomorphism classes of smooth curves of genus g and n marked points. The study of $M_{g,n}$ goes back to Riemann in 1857 and has been an object of study ever since, although the first rigorous construction is due to Mumford in 1965. We give a gentle introduction to the construction of $M_{g,n}$ and the techniques used to study its geometry.
Carlos
Améndola
Technische Universität München
What is estimation for Gaussian models?
Abstract.
The multivariate Gaussian distribution is fundamental in statistics. In this talk I will introduce two methods for estimating parameters: maximum likelihood and method of moments. Then I will present examples of how these apply to Gaussian covariance models and Gaussian mixture models.
Janin
Heuer
Technische Universität Braunschweig
What is a nonnegativity certificate?
Abstract.
Mathematicians have been studying nonnegativity of real polynomials since as early as the 19th century. Nonnegativity certificates are an important tool in these investigations, giving easier to check, sufficient conditions for nonnegativity. In this talk we will motivate the study of nonnegativity by relating it to polynomial optimization. Furthermore, we will define the nonnegativity certificates sums of squares (SOS) and sums of nonnegative circuit polynomials (SONC).
Josué
Tonelli-Cueto
TU Berlin
What is the probabilistic analysis of a condition number?
Abstract.
For a given problem, a condition number is a quantity depending on the data that measure the numerical sensitivity of the data to perturbations. This parameter plays a fundamental role in the complexity analysis of numerical algorithms, both from a run-time and precision control perspective. However, because of this, numerical algorithms tend to have complexity estimates that do not depend solely on the input size. The main philosophy to solve this is to perform a probabilistic analysis of the condition number assuming some reasonable probability distribution of the input. In this talk, we introduce the different ways in which such a probabilistic analysis can be done and the differences between the different approaches.
July 2019
Benjamin
Gardiner
TU Berlin
What is a ranking algorithm?
Abstract.
A search engine is a software, similar to Google, that presents a list of relevant content from a large data set, where criteria for relevance is usually based off of user input. Further, a search engine will usually rank the data according to relevance to maximize utility for the user. A ranking algorithm assigns a numerical rank to each element of its data set, and then returns the sorted list. In this talk we will explore a couple of well known ranking algorithms including Page Rank and HITS (Hyperlink-Induced Topic Search).
Alp
Müyesser
Carnegie Mellon
What is a positional game?
Abstract.
We introduce a systematic study of Tic-Tac-Toe-like games, starting with concrete examples like Hex, and then moving on to a more abstract treatment via Maker-Breaker games played on arbitrary hyper-graphs. As time permits, we will explore connections with other fields, including algebraic topology, the probabilistic method, Ramsey theory, and computational complexity theory.
Anna-Lisa
Sokol
TU Berlin
What is a random walk in random environment?
Abstract.
The random walk on a graph is a widely known process in natural science. But what happens if every edge of the graph is equipped with a random variable deciding if it is open or closed? Meaning the graph itself becomes random. How does this process behave on large timescales? In this talk I will give an overview of the matter and state some interesting results.
June 2019
Josué
Tonelli-Cueto
TU Berlin
What is semialgebraic sets in context?
Abstract.
Semialgebraic sets are a quite general class of sets that can be described by reals polynomials and inequalities. In this talk, we will show how semialgebraic sets appear naturally in many different contexts: a) real algebraic geometry, b) discrete geometry, c) robotic arms and d) rigid models of molecules.
Gari Yamel
Peralta Alvarez
HU Berlin
What is Klein's $j$-invariant?
Abstract.
The $j$-invariant is a special example of a modular form: a complex-valued function on the upper half-plane which behaves “nicely” respect to a group of symmetries. Modular forms have surprising connections with several areas of mathematics, in particular number theory. In this seminar we cover basic definitions of the theory of modular forms making emphasis on its connection with complex elliptic curves, from which the $j$-invariant arise naturally. Finally, we will try to illustrate how the $j$-invariant encodes meaningful information for several mathematical objects.
Tom
Klose
TU Berlin
What is a regularity structure?
Abstract.
Commonly, the regularity of a function describes how well it is approximated by polynomials. This understanding breaks down when we consider solutions to PDEs perturbed by a highly irregular stochastic object called white noise: Polynomials are simply too crude to be the right approximating quantity in this case, but what is? To answer this question, I will introduce the notion of a regularity structure pioneered by Martin Hairer. We will see that it is a far-reaching generalisation of Taylor series that is robust enough to set up a solution theory for afore-mentioned (stochastic) PDEs.
Peter
Krautzberger
What is an ultrafilter?
Abstract.
A historic re-enactment of the very first “What is ...?” Seminar talk, 0% fewer mistakes guaranteed. We'll introduce the basic notions for ultrafilters, discuss fundamental examples and properties and, if time permits, build a simple model of hyperreals.
Marta
Panizzut
TU Berlin
What is Betti numbers?
Abstract.
Betti numbers are nonnegative integers that helps classifying topological spaces. Loosely speaking, the $n$th Betti number counts the number of $n$-dimensional holes. They are defined as ranks of homology groups. In this talk we will introduce them by looking at simplicial homology and focusing on many examples.
May 2019
Riccardo
Morandin
TU Berlin
What is a differential-algebraic equation?
Abstract.
The dynamical behavior of physical processes is usually modeled via differential equations. But if the states of the physical system are in some ways constrained, like for example by conservation laws or position constraints, then the mathematical model also contains algebraic equations to describe these constraints. Such systems, consisting of both differential and algebraic equations, are called differential-algebraic systems. In this talk, we introduce linear differential-algebraic equations, both with constant and variable coefficients. In particular, we will present their canonical forms, and we will discuss what do they imply about the existence, uniqueness and smoothness of solutions.
Fei
Xue
TU Berlin
What is Minkowski
geometry?
Abstract.
A Minkowski space is a finite-dimensional real Banach space whose unit ball is a centrally symmetric convex body. In this talk, we will introduce some basic concepts of Minkowski spaces, and we will show some main concepts like the length and the area in the two-dimensional Minkowski space in comparison with the Euclidean space.
Levent
Doğan
TU Berlin
What is Kronecker coefficients?
Abstract.
The Kronecker coefficients are some natural numbers that naturally arise from the decomposition of the tensor product of two irreducible representations of the symmetric group. Even though they are fundamental in algebra, representation theory and somehow quantum information theory, there are many open problems surrounding the theory of Kronecker coefficients. One such problem is the existence of a combinatorial description of these numbers. In this talk we will briefly describe what Kronecker coefficients are and then we will discuss the complexity of their computation.
Carlos
Maestro Pérez
HU Berlin
What is a g_d^r?
Abstract.
The notion of linear series, or $g^r_d$'s (collections of cuts of a given variety by hyperplanes), is critical in algebraic geometry, where it yields a rich theory in which both classical and modern techniques beautifully come together. In this seminar we discuss some of the basic tools of this theory, and how they provide us with a better understanding of the geometry of algebraic curves. If time permits, a geometric interpretation of the Riemann-Roch theorem involving linear series on curves will be discussed.
April 2019
Sophia
Elia
FU Berlin
What is Steinitz's theorem?
Abstract.
Steinitz's theorem states that a graph is the graph of a $3$-dimensional polytope if and only if it is simple, planar, and 3-connected. In this What Is seminar we cover the notions in the theorem and the proof of the easier implication. We discuss polytopes and their graphs, and we look at Schlegel diagrams and $d$-connected graphs. We sketch a proof of Balinski's theorem stating that the graph of a $d$-dimensional polytope is $d$-connected.
Katerina
Papagiannouli
HU Berlin
What is support vector machines?
Abstract.
In this talk, we will introduce Support vector machines (SVMs). SVMs is a supervised learning method for classification. We will discuss about a particularly useful family of hypothesis spaces called Reproducing Kernel Hilbert spaces (RKHS) for non-linear classification. Finally, we will illustrate SVMs algorithm for pattern recognition using IRIS dataset.
February 2019
Eugenio
Buzzoni
TU Berlin
What is the Wright-Fisher diffusion?
Abstract.
The Wright-Fisher diffusion — mainly expressed as the unique solution of a stochastic differential equation — is a key object in probabilistic population genetics. We will see what it looks like, defining Brownian motion and stochastic integrals along the way.
Dimitris
Bogiokas
FU Berlin
What is it like to lower the mixed volume?
Abstract.
The study of mixed volumes plays a central role in the Brunn-Minkowski theory, since it combines two fundamental notions of the area: Minkowski sums and volume. In this presentation we introduce the notion of mixed volume and examine some first properties.
Tommaso Cornelis
Rosati
HU Berlin
What is a hydrodynamic limit?
Abstract.
We will consider one, a few or an infinite number of particles which move, interact or reproduce randomly. We explain in what sense a law of large numbers and a central limit theorem can hold for such systems. We exploit these structures to (re-)discover deep interactions between probability theory and the analysis of PDEs. We will discuss several prototypical examples and take a look at some rather ugly simulations.
January 2019
Florian
Nie
TU Berlin
What is a competing species model?
Abstract.
We will introduce a certain model arising from mathematical population genetics – the competing species model using the theory of Stochastic Differential Equations. Using this we will discuss how the different terms in the model may affect the evolution of the modelled population. In particular, we are interested in how real life biological effects can be represented through these terms and what kind of mathematical (and biological) questions one can answer in the model.
Simona
Boyadzhiyska
FU Berlin
What is interval graphs, interval orders, and their friends?
Abstract.
Interval graphs and interval orders are two classes of discrete structures that arise naturally in many real-world problems. They find applications in scheduling, archaeology, genetics, psychology, and circuit design, among others. In this talk, we will give a short introduction to the theory of interval graphs and orders. In particular, we will discuss the connection between these two types of structures, how they can be characterized, and why they are important from both a theoretical and a practical point of view. We will conclude by mentioning some special cases and generalizations.
December 2018
Alex
Nietner
FU Berlin
What is tensor networks, topological order and entanglement renormalization?
Abstract.
In this introductory talk we will look at many body physics by means of the partition function as its central object. Starting with an introduction to tensor networks as a general formalism we will use the language of tensor networks in order to get an intuition for the partition function and hence many body physics. The notion of equivalence classes of tensor networks will lead us directly to a notion of equivalence classes of many body systems usually referred to as phases. As a special case we will dig deeper into the equivalence class with respect to topological moves (Pachner moves) leading directly to gapped topological phases. Combining the properties of tensor networks and topological moves we will introduce the entanglement renormalization scheme as a method to detect topological phases. We will conclude the talk with some general remarks on renormalization, regularization and discretization.
November 2018
Max
Krause
TU Berlin
What is the Alexander polynomial of a knot?
Abstract.
In the 1920s, J. W. Alexander discovered the first polynomial invariant for knots. It would take several more decades for topologists to realize the full potential of this construction, which has inspired several similar invariants and now is a cornerstone of modern knot theory. In this talk we will see three different ways of calculating the Alexander polynomial, discuss the connections between them.
October 2018
Barbara
Jung
HU Berlin
What is the secret of zeta function?
Abstract.
The Riemann zeta function, in its simplicity, is the key to a bunch of well-known theorems in algebra and number theory. We will introduce the zeta function in its various shapes and forms, and discover these connections. Finally, we will have a glance at the Riemann hypothesis, find evidence for its correctness and shed light on some of its consequences.
June 2018
Marco
D'Addezio
FU Berlin
What is a scheme?
Abstract.
In this talk I will introduce schemes and I will explain why and how they were discovered in the 50's.
Alejandro
López
FU Berlin
What is control theory?
Abstract.
I will give a short survey on control theory, some of the many ways in which it can be approached and its applications in the real world. After this the talk will focus on Pyragas control, a special technique used to modify the stability of particular elements within a dynamical system.
April 2018
Jorge Alberto
Olarte
FU Berlin
What is a matroid?
Abstract.
Matroids are rich combinatorial structures that can be seen as a general notion of independence. However, matroids are particular in that they have dozens of different cryptomorphic definitions. Therefore they appear underlying in many mathematical objects and applications can been found in several different fields including algebra, geometry, graph theory, model theory and optimization. In this talk we will briefly describe the main concepts of matroid theory as well as explaining how to abstract matroids from matrices and graphs.
February 2018
Mohsen
Sadeghi
FU Berlin
What is molecular dynamics?
Abstract.
Molecular dynamics has been around since the 60's, and has gained ever increasing popularity as the method of choice for doing computer experiments on molecular systems. Despite the immensity of its forms and applications, the method at its core is built upon rather simple concepts. In this talk, we will look behind the curtain of a generic molecular dynamics simulation, and investigate its main components. We will talk about dynamics of particle systems, forcefields, and integrators, and look at the challenges we face in making the connection between molecular dynamics simulations and the physical world.
Héctor
Andrade Loarca
TU Berlin
What is an inverse problem?
Abstract.
What is an inverse problem? Is it just another kind of problem? In this talk I will present an intuitive approach to inverse problems as well as a rigorous mathematical way to define them and some techniques to solve them, known as regularization theory. I will also focus on inverse problems applied in imaging science, showing some examples and implementations of typical algorithms. At the end if the time is enough I would like to give an small introduction to deep learning techniques in inverse problem regularization.
Alexander
Müller
FU Berlin
What is a homotopy invariant?
Abstract.
We will introduce and explain the fundamental notions of homotopy theory underlying all of algebraic topology. Path components, homotopy groups of spheres, Betti numbers: homotopy invariants are diverse and powerful tools to understand spaces of all flavours.
January 2018
Marek
Gluza
FU Berlin
What is Maxwell's equations?
Abstract.
The talk is an invitation to Maxwell's equations, a set of four partial differential equations that describe the classical electromagnetic phenomena. In physics, they were a milestone with huge impact and influence on subsequent physics such as the theory of special relativity and the quantum gauge field theories, like the Standard Model of particle physics. In engineering, the numerical solution to Maxwell's equations allows important technical advances, such as the improvement of the building blocks of electronic devices and our communication channels (which allows us to watch cats over Internet in always better quality). In this talk, I will introduce electromagnetic fields in the Euclidean space and the Maxwell's equations governing them, explain their physical meaning and show how one can derive that the light is an electromagnetic wave from Maxwell's equations. So wonder no more about the fundamental laws making it all possible, let's see together what are the Maxwell's equations!
András
Tóbiás
TU Berlin
What is large deviations?
Abstract.
The field of large deviations deals with rare events in probability theory. It considers events whose probability tends to zero exponentially fast. In many cases, the exponential rate of decay can be identified precisely, and the rate function helps understanding the rare event itself. It shows the most likely way how the event is realized, and it can often be used for proving a law of large numbers. Thanks to these properties, large deviations are used, e.g., for Markov chains, statistical mechanics, SDEs, random walks in a random environment or extremal combinatorics. In this talk, I will introduce large deviations starting with the example of coin tosses, I will present Cramér's theorem and sketch some applications of large deviation theory.
December 2017
Lara
Skuppin
TU Berlin
What is the Teichmüller space of a surface?
Abstract.
The aim of this talk is to give an introduction to Teichmüller space. Roughly speaking, Teichmüller space is a space of geometric surfaces sharing the underlying topological surface. An important property is its relation to the so-called moduli space. This presentation has a connection to Anna Wienhard's colloquium talk. It will contain a review of concepts related to the topology of an orientable compact surface, such as its genus and fundamental group, and concepts related to its geometry, such as hyperbolic structures, conformal structures and the uniformization theorem. Teichmüller space parametrizes these structures. Further, it is related to representations of the fundamental group as a discrete group of isometries of the hyperbolic plane. As final outlook, we will discuss some results relating to Teichmüller spaces.
Georg
Loho
TU Berlin
What is monomial tropical conesfor multicriteria optimization?
Abstract.
We introduce a special class of tropical cones which we call `monomial tropical cones'. They arise as a helpful tool in the description of discrete multicriteria optimization problems. After an introduction to tropical convexity with an emphasis on these particular tropical cones, we explain the algorithmic implications. We finish with connections to commutative algebra.
November 2017
Mario
Kummer
MPI Leipzig
What is a spectrahedron?
Abstract.
Spectrahedra are a central object in convex algebraic geometry. We will present some basic properties.
Monika
Eisenmann
TU Berlin
What is a finite element method?
Abstract.
Partial differential equations appear in many applications, therefore it is of great importance to have efficient approximation concepts. For a wide range of problems the finite element method can be used to obtain numerical approximations. In this talk, I will explain the idea of this method on a simple one dimensional example problem.
Carlos
Améndola
TU Berlin
What is algebraic statistics?
Abstract.
Algebraic statistics is a relatively new field that has developed rapidly in recent years, using algebraic-geometric methods to provide new tools to analyze and solve statistical problems. In this talk I will present examples that illustrate the exciting interplay between algebra and statistics. No background in either of these areas will be assumed!
October 2017
Jannes
Münchmeyer
HU Berlin
What is a random graph?
Abstract.
Graphs are a crucial model in many parts of science nowadays. As often the structure of the underlying graphs is not fully known, the field of random graphs has hugely gained significance. This talk will give an introduction on random graphs. It will also present the main stochastic models for random graphs and their properties.
July 2017
Irem
Portakal
FU Berlin
What is Gröbner basis doing in Sudoku?
Abstract.
For most mathematicians, it is a difficult task to answer the question "What are you studying?", or even harder "What is it good for?". Algebraic geometry is one of the most abstract research areas in mathematics, and these kind of questions leave us speechless. But luckily, almost everyone knows what Sudoku is. It turns out that one can use algebraic geometry to solve these games via the so-called Gröbner bases. We will introduce this technique and apply it to some examples.
Robert
Lasarzik
TU Berlin
What is Onsager's conjecture?
Abstract.
In this talk, the equations of motion for an idealized fluid are motivated from basic laws of physics. I will comment on some properties of solutions to this equations, like existence, uniqueness and conservation of energy, and link them to Onsager's conjecture.
Paul
Breiding
TU Berlin
What is a condition number?
Abstract.
In this talk I will motivate and introduce a fundamental notion in numerical analysis — the condition number. Condition numbers serve as a measure of how sensitive a function is with respect to perturbations in the input. I will give an introductory example demonstrating that small changes in the input data may lead to large deviations in the output, even for seemingly simple problems such as solving linear equations. Using this example, I will give the definition of condition number and then go over to define condition numbers à la Shub and Smale within a more general framework.
June 2017
Josué
Tonelli-Cueto
TU Berlin
What is BPP, RP and the other probabilistic complexity classes?
Abstract.
When one faces problems to solve, randomness can be used in order obtain faster answers at the cost of some uncertainty. Probabilistic complexity classes capture the different ways in which this can be done. In this talk, we introduce the basic probabilistic complexity classes, their interrelations and we illustrate by outlining the solution to concrete problems.
Marcin
Lara
FU Berlin
What is Iwasawa theory?
Abstract.
Iwasawa theory is the study of arithmetic objects over infinite towers of number fields. The main example is the extension of $\mathbb{Q}$ by the $p$-power roots of unity. The theory origins in the following insight of Iwasawa: instead of working with a fixed finite Galois extension and modules under its Galois group, it is often easier to describe every Galois module in an infinite tower of fields at once. This talk will be an introduction to Iwasawa theory.
May 2017
Patrick
Jähnichen
HU Berlin
What is variational inference?
Abstract.
In my talk I will give a (very) brief introduction into Bayesian probabilistic models. In general, we are interested in the posterior distribution over the latent variables in such a model. I will explain how we can turn this estimation problem into an optimization one and how to make use of properties of the exponential family of distributions to derive an elegant solution to it.
April 2017
Giulia
Codenotti
FU Berlin
What is triangulated spheres and Pachner moves?
Abstract.
In the world of discrete geometry, somewhere between combinatorics and geometry, Pachner moves (or bistellar flips) are an important tool to "build up" complex triangulations from more simple ones, while preserving topological properties. I will show with examples how Pachner moves pop up in old and new questions, spanning topics from combinatorics to topology.
Alexander
Fauck
HU Berlin
What is Hamiltonian dynamics?
Abstract.
Classical mechanical systems are systems where the time evolution depends only on position and momentum. One can describe them with a vector field on the so called phase space — the Hamiltonian vector field. The studies of the dynamics of such a vector field (for example the motion of our planetary system or the 3-body problem) motivated the notion of symplectic and contact structures. In my talk I will explain some of these concepts and describe how they arise from the classical setup.
February 2017
Georg
Loho
TU Berlin
What is linear programming?
Abstract.
Linear programming is a special optimization problem which is widely applicable for solving real-world problems. It has a rich discrete geometric structure. Furthermore, there are still several open complexity questions concerning the algorithms to solve linear programs. In this talk we will give a geometric intuition for the problem. We present the simplex method which is the major tool to solve linear programs.
Lasse
Hinrichsen
FU Berlin
What is a (discontinuous) Galerkin finite element method?
Abstract.
Many physical phenomena are modeled by partial differential equations (PDEs) that cannot be solved analytically anymore. Thus our aim is to compute a — hopefully — reliable approximation of the solution to such a PDE. One of the basic steps is to transfer the infinite-dimensional problem to something that computers — with finite memory — can handle. In this "What is..." talk, we will get an idea how the Galerkin Finite Element Method (FEM) serves this purpose. In the end, we shall also see a special variant, namely the Discontinuous Galerkin (DG) method, and discuss its uses.
January 2017
Josué
Tonelli-Cueto
TU Berlin
What is the length of a potato?
Abstract.
It's clear what someone means when E refers to the volume or the area of a potato, but if this same person starts to talk about its length, a sensible doubt about the meaningfulness of this concept arises. In this talk we will see how the length of a potato is a meaningful concept by introducing the so-called intrinsic volumes and reviewing some of their main properties. Note: Title stolen from one article of Schanuel.
Shagnik
Das
FU Berlin
What is randomness good for?
Abstract.
We all know that randomness has several useful applications, ranging from quantum physics to cryptography to experimental design. What, though, does it offer the poor pure mathematician, who cannot afford a quantum computer, has nothing to hide, and has even less interest in real-world phenomena? In this talk we will give a partial answer to this question by demonstrating the use of the probabilistic method in extremal combinatorics. No prior knowledge of combinatorics is necessary, although some familiarity with colours — in particular, red and blue — could be useful.
December 2016
Martin
Skrodzki
FU Berlin
What is formal mathematics?
Abstract.
To call something Formal Mathematics seems to be redundant. However, the field has its own merits, some of which shall be exemplified in the talk. We will take some initial steps to clarify what we understand by the notion of "proof". Having done this, we go on to formally prove a part of de Morgan's law. Furthermore, the "Donkey Sentence" will raise our awareness for difficulties in formal proofs. Finally, we will briefly get into the topic of Automated Theorem Provers and consider the famous proof of the irrationality of sqrt(2), formalized in Isabelle.
November 2016
Antareep
Mandal
HU Berlin
What is a modular form?
Abstract.
We define one of the most important invariants for a lattice in the Euclidean space called its theta series and show that it belongs to an interesting class of functions called modular forms, whose symmetry and nice properties makes it possible to prove strong statements about them.
October 2016
Stanley
Schade
FU Berlin
What is mixed integer linear programming?
Abstract.
Defining mixed integer linear programming (MILP) is easy, but not very instructive. What we want to understand is how we can approach MILP problems in practice and why they are so relevant. To do so, we will make a quick tour through the field of combinatorial optimization. We will explore some of its problems, mathematical concepts and tools and also gain a few geometric insights on MILP.
July 2016
Albert
Haase
FU Berlin
What is sphere packing?
Abstract.
We will discuss what a (lattice) sphere packing is and how its density can be defined. Then, we will look at examples in dimensions 2, 3, and perhaps 4. We will also introduce the related kissing number problem. Finally, we will survey known results and look at a rather puzzling plot of the densities — by dimension — of the densest sphere packings known. If we have time, we can discuss the two lattices (E_8 and Leech lattice) that lead to the packings in dimensions 8 and 24 that were recently shown to have highest possible density.
June 2016
Kathlén
Kohn
TU Berlin
What is the eigenvector of a tensor?
Abstract.
This talk gives an introduction to tensors and their eigenvectors. It is especially addressed to people that are not algebraists. We will see that tensors and their eigenvectors appear naturally when using Lagrange-multipliers in optimization and when studying higher order Markov chains in stochastics.
May 2016
Fei
Ren
FU Berlin
What is a Chow Ring?
Abstract.
In algebraic geometry, the Chow ring (named after W. L. Chow by Chevalley (1958)) of a smooth algebraic variety over a field is an algebro-geometric analogue of the cohomology ring of a complex variety considered as a topological space. The elements of the Chow ring are formed out of actual subvarieties (so-called algebraic cycles), and the multiplicative structure is derived from the intersection of subvarieties. In this talk, we will define what is a Chow ring, introduce basic properties and see a few examples if time permits.
April 2016
Mara
Ungureanu
HU Berlin
What is symplectic geometry?
Abstract.
Symplectic geometry is an even dimensional geometry that lives on even dimensional spaces and measures two dimensional quantities rather than the one dimensional quantities like lengths and angles familiar from Riemannian geometry. Moreover, symplectic geometry displays an intriguing interplay between rigidity and flabbiness, which makes the question of constructing invariants that distinguish between between different symplectic structures especially interesting. In this talk we shall explore some of these aspects and motivate the definition of a certain symplectic invariant, namely the symplectic capacity.
February 2016
Irem
Portakal
FU Berlin
What is a rational function?
Abstract.
In this talk, we will investigate functions on affine space over a field. In the projective case, we naturally come across the notion of rational functions. More concretely, when we change the field to the complex numbers, for the projective space, one gets an identification of the field of rational functions with the field of meromorphic functions, which is a consequence of Chow's Theorem. We will prove this for the projective line.
Miriam
Schröder
TU Berlin
What is the disjoint
paths problem?
Abstract.
Imagine you are an electrical engineer and responsible for the power supply of a town. By building a large network of transmission towers you connected the town to the nearest power plant. Now your boss asks you to prove to him that the power supply of the town is still ensured if k transmission towers fail. How can you convince your boss that the network you built is failsafe? Of course, you can simply switch out every possible k-subset of transmission towers and check whether the city is still on power supply after that. Unfortunately, you built a large network, so this approach would take a long time. A good solution in this situaion would be to use Menger's theorem which provides us with a fast to check certificate that ensures that the network is failsafe. In this talk I will introduce Menger's Theorem and the related disjoint paths problem in different variants.
Josué
Tonelli Cueto
HU Berlin
What is an origami
constructible number?
Abstract.
When the people of the Greek city of Delos asked the Delphic oracle how to stop a plague that Apollo sent to them, the oracle answered that they should duplicate the cubic altar of this god. Although the efforts of the great mathematicians of the day, none of them was able to double the cube through straightedge and compass (as the fashion of the day requested in geometry). As we know nowadays, this latter task is impossible; however, if we allow ourselves to use origami, the duplication of the cube and other classical geometric problems not solvable by straightedge and compass can be solved. In the talk, we will explore the constructions that the use of origami permits in classical geometry.
January 2016
Stefan
Rüdrich
FU Berlin
What is a stochastic
process?
Abstract.
Stochastic processes are families of random variables whose trajectories may differ with each realization, unlike deterministic processes, yet allow for analysis and simulation of dynamical systems, conserving some non-probabilistic quantities. This talk will be a rough introduction to some key concepts of stochastic processes, e.g. random walks, Markovianity and metastable sets. The latter typically correspond to almost independent substructures of the state space, such as functional subunits of a cellular network or stable conformations of molecules.
Claudius
Heyer
HU Berlin
What is a $p-$adic
number?
Abstract.
$p-$adic numbers are an integral part of algebraic number theory. In this talk we will define the field $Q_p$ of $p-$adic numbers and observe some differences and similarities between $Q_p$ and the field of real numbers. Furthermore, we will state Hensel's Lemma and deduce some immediate, yet interesting, properties of $Q_p$.
December 2015
Christoph
Spiegel
FU Berlin
What is discrete Fourier analysis?
Abstract.
Discrete Fourier analysis can be a powerful tool when studying the additive structure of sets. Sets whose characteristic functions have very small Fourier coefficients act like pseudo-random sets. On the other hand well structured sets (such as arithmetic progressions) have characteristic functions with a large Fourier coefficient. This dichotomy plays an integral role in many proofs in additive combinatorics from Roth?s Theorem and Gower?s proof of Szemer�di?s Theorem up to the celebrated Green-Tao Theorem. We will introduce the discrete Fourier transform of (balanced) characteristic functions of sets as well some basic properties, inequalities and exercises. No prior knowledge of combinatorics or number theory is necessary.
November 2015
Jan
Hofmann
FU Berlin
What is reciprocity?
Abstract.
Pick's theorem allows us to compute the area of any lattice polygon just by counting integer points. We will sketch a proof of Pick's theorem. It is now natural to ask if there are analogues in higher dimensions. This leads to the introduction of the Ehrhart polynomial and Ehrhart reciprocity.
October 2015
Filippo
Nuccio
Université St. Étienne
What is a period?
Abstract.
Not available.
Patrick
Da Silva
HU Berlin
What is representation theory?
Abstract.
This What Is seminar is meant to be a rough introduction to representation theory to be able to follow Geordie Williamson's talk. We will sketch the basic concepts : definition of a representation, Maschke's theorem, Schur's Lemma, character theory and if we have some time perhaps the link between representations and \(k[G]-\)modules. Since Geordie will mention modular representation theory I will give an example in characteristic p, where Maschke's theorem fails.
July 2015
Manh Tuan
Tran
FU Berlin
What is Ramsey theory?
Abstract.
Ramsey theory refers to a branch of mathematics whose underlying philosophy is captured by the statement that "Every large system contains a large well-organized subsystem." There are examples of such statements in many areas, including geometry, number theory and analysis. In this talk, we shall discuss some examples and focus on Roth's theorem, which guarantees that every set of integers with positive density contains a 3-term arithmetic progression. No prior knowledge of combinatorics and number theory is assumed.
June 2015
Asilya
Suleymanova
HU Berlin
What is the heat kernel?
Abstract.
First we will discuss the notion of a kernel of an operator and the heat operator itself. On the Euclidean space the scalar heat kernel is given by the exact formula . For an arbitrary Riemannian manifold it is usually impossible to find an exact expression for the heat kernel. However for many problems approximate solution suffices. For example, I will show how the local Atyiah-Singer index theorem can be proven using heat kernel approach.
Peter
Patzt
FU Berlin
What is group homology?
Abstract.
In this talk we recall a few notions and definitions from homological algebra. Then we give a quick-and-dirty definition of group homology. The talk finishes with Shapiro's Lemma, which is an easy implication from the definitions.
Adam
Nielsen
ZIB
What is a transfer operator?
Abstract.
The transfer operator is a very important tool in applied mathematics, originating from measure theory. Its beauty and simplicity is challenged by a horrible wikipedia article which will be rectified during the talk with a clean and plain introduction of transfer operators.
May 2015
Tuan Anh
Nguyen
TU Berlin
What is a supercritical percolation cluster?
Abstract.
With a lot of pictures my talk will explain you what a supercritical percolation cluster is and give you an example of two random walks among random conductances on this cluster: i) the constant speed and ii) the variable speed random walk, which are typical examples of discrete and continuous time Markov chains. (They may not be exactly the objects in Nina Gantert's talk, but they will give you some feeling about this field.) Very easy examples will show one of the main difficulties when studying the macroscopic properties of the random walks. That is, the trap problem, which can be roughly understood as follows. While the constant speed random walk stays a very long time in very high conductances, the variable speed one cannot easily get out of very low conductances. To get rid of this difficulty, one needs some assumptions on the law of the random conductances, which is introduced at the end of the talk.
April 2015
Pierre
Lairez
TU Berlin
What is the complexity of matrix multiplication?
Abstract.
Studies in computer algebra led to numerous efficient algorithms to solve many kinds of difficult problems (e.g., large integer multiplication). Complexity theory focuses on problems for which an efficient solution is not known and tries to grasp the intrinsic difficulty of problems. This involves both searching for lower bounds and upper bounds. Matrix multiplication is one of these hard problems. The obvious algorithm needs $n^3$ scalar multiplications to multiply two n by n matrices, and it is far from optimal. Simple methods and sophisticated tools allow to lower this bound, but it seems likely that we are still not close to optimal.
February 2015
Thomas
von Larcher
FU Berlin
What is a scale in motion?
Jesko
Hüttenhain
TU Berlin
What is (algebraic) complexity theory?
Abstract.
Complexity theory is generally the study of algorithms, and the notion of an algorithm is mathematically not among the most accessible. In many cases however, we want to solve problems with an inherent mathematical structure, like multiplication of matrices. In algebraic complexity theory, we only look at very special classes of algorithms, those which have algebraic descriptions and interpretations. This way, stronger mathematical tools can be employed to answer computational questions. We give a short introduction to some of these algebraic models.
January 2015
Shagnik
Das
FU Berlin
What is the probabilistic method?
Abstract.
Has your mathematics become mundane? Are you suffering through a mid-Masters crisis? If so (or even if not), come along and witness the wonder of the mysterious probabilistic method at work. Guaranteed* to reignite your passion for mathematics. In this talk, we shall see some random number of applications of the probabilistic method in various fields. No prior knowledge** of combinatorics is assumed, and only very basic probability will be used. * Or your money back. **Indeed, for maximum dramatic effect, it is hoped no such knowledge exists.
Isaac
Mabillard
Institute of Science and Technology Austria
What is a van Kampen obstruction cocycle?
Abstract.
The Kuratowski theorem provides a nice criterion for graph planarity, ie, to decide whether a simplicial $1$-complex can be embedded into $\mathbb{R}^2$. A natural generalization of the problem is to find a criterion to decide whether a simplicial $n$-complex $K$ can be embedded into $\mathbb{R}^{2n}$. This is what the van Kampen obstruction cocycle gives us. By using standard tricks in PL topology, one can show that $K$ is embeddable if and only if (the class of) its cocycle is zero. This is (maybe?) surprising because embeddability is a geometric question, whereas a cocycle is an algebraic object, but it still carries enough information to solve the geometric problem.
Wang-Q
Lim
TU Berlin
What is sparse and redundantrepresentation modelingfor image processing?
Abstract.
Recently, sparse and redundant representation modeling has received extensive attention and shown to be quite effective in signal and image processing. This framework allows for the successful reconstruction of the true image from severely corrupted or undersampled data. In this talk, we will introduce an image reconstruction scheme based on this framework and show the effectiveness of such a scheme in various applications.
December 2014
Jonathan
Youett
FU Berlin
What is a Hamiltonian system?
Abstract.
Hamiltonian systems provide an elegant way to describe the evolution equations of certain mechanical systems. The corresponding flow mappings can be shown to be symplectic and usually possess conserved quantities like momentum or energy. In this talk we will introduce the basic concepts, first for Euclidean space and then extend them to finite dimensional manifolds.
Florian
Frick
TU Berlin
What is the ham sandwich theorem?
Abstract.
Some problems in discrete geometry can be solved using topological methods by exploiting inherent symmetries. We will discuss some examples of this and focus on the Ham sandwich theorem, which guarantees that $d$ finite measures with continuous density in Euclidean $d$-space can be simultaneously cut in half by an affine hyperplane.
November 2014
Arnau
Padrol Sureda
FU Berlin
What is a (complex) hyperplane arrangement?
Abstract.
Every arrangement of real linear hyperplanes induces a decomposition of the sphere into cells. We will study the combinatorics of such a decomposition (explained by an oriented matroid) and define an analogous notion for complex hyperplane arrangements.
Sebastian
Meinert
FU Berlin
What is a $\operatorname{CAT}(0)$ space?
Abstract.
A $\operatorname{CAT}(0)$ space is a metric space in which geodesic triangles are at most as thick as in Euclidean space. We will make this notion precise and explain some basic features of $\operatorname{CAT}(0)$ spaces, like contractibility.
October 2014
Daniele
Agostini
HU Berlin
What is a syzygy?
Abstract.
In astronomy, a "syzygy" is an alignment of three celestial bodies. The term was imported into mathematics by Sylvester to denote a linear relation between the generators of a module. Nowadays, syzygies and, more generally, free resolutions are powerful tools to relate the algebra and geometry of a projective variety. In this talk I will introduce these concepts and present some concrete examples.
July 2014
Ulrich
Reitebuch
FU Berlin
What is a gyroid?
Abstract.
The gyroid is a minimal surface which belongs to the family of the Schwarz P and Schwarz D surfaces. We will first have a short introduction to minimal surfaces and their symmetries and then see some minimal surfaces in visualizations and 3D models.
Jonas
Krone
FU Berlin
and
Zoi
Tokoutsi
FU Berlin
What is time-of-death modeling?
Abstract.
In forensics, there is a natural interest in determining the exact time of death of murder victims. Thus far this was done by measuring the victim's body temperature and extrapolating the time of death based on a rather crude physical model. Recently, there have been attempts to come up with more realistic models that take into account all the different body properties (height, weight, clothing, etc.) as well as environmental factors (outside temperature, wind, heat sources, etc.). Our goal is to develop a physically accurate, three-dimensional heat flow simulation for the body that can be adjusted to a multitude of scenarios, and estimate the time of death by a curve-fitting scheme for the heat curve.
Lara
Skuppin
TU Berlin
What is a Riemannian manifold?
Abstract.
A Riemannian manifold is a smooth manifold together with a Riemannian metric giving rise to notions of geometric quantities such as length, angle, distance, volume and curvature. After discussing surfaces in $\R^3$, I will give the definition of a Riemannian manifold and some examples, state the Nash embedding theorem, and explain a few properties.
June 2014
Lucas
Braune
IMPA Rio de Janeiro
What is the Atiyah-Singer Index Theorem?
Abstract.
The Atiyah-Singer index theorem is a general result that gives an integral formula for the index of an elliptic operator on a compact manifold. It has as immediate corollaries, fundamental theorems in different areas of geometry — theorems whose statements have seemingly nothing to do with an index. The main examples are the Chern-Gauss-Bonnet theorem, the Hirzebruch signature formula, and the Riemann-Roch-Hirzebruch theorem. My purpose for this talk is to show each of these theorems as a solution to an index problem and, with this as the motivation, to explain the statement of a version of the Atiyah-Singer index theorem.
Giulia
Battiston
FU Berlin
What is the moduli spaces of curves?
Abstract.
Moduli spaces are constructed in order to parametrize specific classes of objects. We will explicitly construct moduli spaces for simple parametrization problems, and then introduce the moduli space of curves of genus g and discuss some of their basic geometric properties.
Michael
Joos
TU Berlin
What is an isohedral tiling?
Abstract.
A tiling is a tessellation of the plane and an isohedral tiling is a tiling in which all tiles are related by a symmetry of the tiling. One can categorize these tilings into 93 types. The definition leads to a compact symbolic encoding. We will focus on some computational aspects of isohedral tilings and also have a look at an implementation.
May 2014
Miguel
Grados
HU Berlin
What is a partial zeta function?
Abstract.
Zeta functions are ubiquitous in number theory and usually its residue at s=1 contains relevant information of the problem in question. This talk is two-fold. First, we introduce two variants of the Riemann zeta function: one that appears in practice, and the other that is good for theoretical purposes. Second, we show an identity between these two variants and its consequences.
Adam
Streck
FU Berlin
What is a random graph?
Abstract.
Unsurprisingly, the random graph is a graph that has been constructed in a random manner. However, the exact nature of this construction is of interest as it has been shown that many real-world networks share the structural properties of particular sorts of randomly constructed networks. We will focus on the two most common approaches, Erdős-Rényi and Barabási-Albert models, and show some well known emergent patterns of these approaches. We will also refresh some of the graph-related notions that will be discussed in the following talk.
April 2014
Ariane
Papke
FU Berlin
What is the Navier-Stokes equation?
Abstract.
The motion of a fluid can be described by the Navier-Stokes equations. We discuss the forces acting on a fluid element and explore some applications.
February 2014
Ananyo
Dan
HU Berlin
What is the prime number theorem?
January 2014
Yizhi
Sun
TU Berlin
What is a Carleson operator?
Abstract.
We recall some fundamental facts about Fourier series and different types of convergence. Then we will see how the Carleson operator and its boundedness work in the proof of Lusin's conjecture for $L^2$ functions.
Hannes
Pollehn
TU Berlin
What is an Ehrhard Polynomial?
Abstract.
A lattice polygon is a shape in the plane bounded by a sequence of line segments $\overline{v_1 v_2},\ldots,\overline{v_{k-1} v_k}$, where the vertices $v_i$ have integral coordinates. Georg Alexander Pick described a relation between the area of the polygon and the number of lattice points contained in the polygon by a formula which is now known as Pick's Theorem. Ehrhart later generalized this result to higher dimensions, which is now an important tool in the field of Geometry of Numbers.
Faniry
Razafindrazaka
FU Berlin
What is a 4D object?
Abstract.
As human beings, we are only able to see 2D images. The extra dimension comes from our ability to understand shadows as depth information. While 3D objects have 2D shadows, 4D objects have 3D shadows. In this talk, we are going to explore, first, the 4D variant of the platonic solids together with their visualization and second, we will construct some surfaces embedded in 4-dimensional space.
Peter
Patzt
FU Berlin
What is the representation theory of finite groups?
Abstract.
A representation of a group is a linear group action. The representations of finite groups are well understood and their study led to new results formulated without representations. Also linear group actions occur naturally and representations theory yields more inside to the underlying vector spaces. We give an introduction to the representation theory of finite groups and some examples.
December 2013
Juan Camilo
Orduz
HU Berlin
What is the Burger's equation and heat equations?
Abstract.
In this short talk we will describe two concrete examples of partial differential equations (PDEs): Burger's and the heat equation. We will mention some useful techniques to solve them: method of characteristics and Fourier theory.
Eva
Martínez
FU Berlin
What is a Grassmannian?
Abstract.
Although the definition of a Grassmannian can be understood by anyone who has taken a course on linear algebra, it actually has a very rich and subtle structure and has become a very important tool in many fields. In this talk we will introduce the definition of a Grassmannian and some of its basic properties in a very elementary way with the aim to convince the audience about its significance.
November 2013
Matthias
Liero
WIAS Berlin
What is a Wasserstein distance?
Abstract.
Gradient flows are an important subclass of evolution equations. A gradient-flow structure can be exploited to obtain additional information about the evolution, such as existence and the stability of solutions. Moreover, gradient flows can provide additional physical and analytical insight, such as the maximum dissipation of entropy and energy, or the geometric structure induced by the dissipation distance. A special subclass is formed by gradient flows with respect to the Wasserstein distance. This class was first identified in the seminal work by Jordan, Kinderlehrer, and Otto in the late nineties. In my talk I will introduce the Wasserstein distance and discuss its relation to diffusion equations.
Christoph
von Tycowicz
FU Berlin
What is the spacetime constraints paradigm?
Abstract.
Creating motions of objects or characters that are physically plausible and follow an animator's intent is a key task in computer animation. The spacetime constraints paradigm is a valuable approach to this problem. It computes optimal physical trajectories that are solutions of a variational spacetime problem. Such techniques calculate acting forces that minimize an objective functional while guaranteeing that the resulting motion satisfies prescribed spacetime constraints, e.g., interpolates a set of keyframes. Resulting forces are optimally distributed over the whole animation and show effects like squash-and-stretch, timing, or anticipation that are desired in animation. In this talk, we give an introduction to spacetime constraints and — if time permits — sketch a recent approach to interactive spacetime optimization for deformable objects.
Keita
Kunikawa
Tohoku University
What is mean curvature flow and its self-shrinkers?
Abstract.
This talk is a short introduction to mean curvature flow (MCF). MCF is one of the geometric flows (PDE on manifolds). Start with a generic initial surface having singularities. After rescaling near a singularity, we can see a self-shrinker of the flow. They are special solutions of MCF and they do not change their shape under the flow (up to scaling). Classifying self-shrinkers is an important topic of research. In general, however, it is impossible to do so without some additional conditions. I will explain a method using the Gauss map to approach this problem.
Wayne
Lam
TU Berlin
What is the conformal geometry of surfaces?
Abstract.
A brief introduction to conformal geometry and its relation to complex analysis is shown. Some basic results, like the Riemann mapping theorem, are emphasized for their relations to the discrete theory. The goal is to give the audience a taste of discrete differential geometry.
Deniz
Genlik
TU Berlin
What is the relation between Laplace transform and Z-transform?
Abstract.
The Laplace transform plays an important role in the analysis of continuous linear time invariant systems and the Z-transform does the same for Discrete Linear Shift Invariant Systems. In this talk, these systems will be briefly characterized and these transformations will be related to the characterizations. The relation between the Laplace transform and the Z-Transform will then be introduced, which is important for the analysis of mixed-type CLTI and DLSI Systems.
Codrut
Grosu
FU Berlin
What is a graph limit?
Abstract.
The goal of this talk is to define the notion of graph limit. I will present the relevant definitions and state the Lovász-Szegedy theorem characterizing graph limits. I will also compute a graph limit in a special case as an example, and as time permits, consider some generalizations of graph limits to digraphs, and different metric distances on graphs and graphons.
October 2013
Uğur
Doğan
HU Berlin
What is a hyperreal number?
Abstract.
The set of hyperreal numbers is a field which contains $\mathbb R$ (real numbers) and some "infinitesimal" and "infinite" numbers. It will be constructed using model theory, so some basic facts and theorems in model theory will also be mentioned, such as Łoś's theorem and compactness of first order logic (and some applications if time permits).
Louis
Theran
FU Berlin
What is combinatorial rigidity?
Abstract.
A bar-joint framework is a structure made of fixed-length bars connected by joints with full rotational degrees of freedom. The allowed continuous motions preserve the lengths and connectivity of the bars. If all the allowed motions come from Euclidean isometries, the the framework is rigid and otherwise it is flexible. Combinatorial rigidity theory is concerned with how much geometric information about a framework (e.g., if it is rigid or what its motions are like) from just the graph that has as its edges the bars. In this talk, I'll introduce frameworks in more detail and discuss some basic results and techniques in the area.
July 2013
Carsten
Gräser
FU Berlin
What is a variational inequality?
Abstract.
Solutions of optimization problems can be characterized nicely by variational equations if their associated functional is smooth. For nonsmooth functionals, variational inequalities provide a useful generalization while only requiring that part of the functional is differentiable. After discussing the relation of minimization problems to variational inequalities, I will show how we can use the latter to extract some important features of such problems.
Giovanni
Conforti
HU Berlin
What is a self-avoiding random walk?
Abstract.
In this talk we will give an informal introduction to random walks on graphs and see how typical quantities of interest can be computed with the help of the Markov property. We will also see how, in the case of the self avoiding random walk, very basic questions remain unanswered and the Markov property, in the usual sense, is lost. Motivating examples from polymer science and particular graphs and lattices where the self avoiding random walks have been intensively studied will be presented.
June 2013
Franz
Király
TU Berlin
What is matrix completion?
Abstract.
Matrix completion is the task of reconstructing missing entries in a matrix of known rank which has gained wide attention through the \$1,000,000 NetFlix prize and the subsequent class action lawsuit. In the talk I will briefly explain the problem and its many interrelated connections to different fields of mathematics and computer science such as statistics, machine learning, convex optimization, functional analysis, algebraic geometry, commutative algebra, graph theory and combinatorics — highlighting some interesting results and viewpoints which can serve as different but related starting points for approaching the problem of matrix completion.
Christian
Kreusler
TU Berlin
What is abstract functions and weak convergence?
Abstract.
Most of the modern theory of Partial Differential Equations is formulated via the help of abstract functions, i.e., functions taking values in Banach spaces of infinite dimension. We will discuss the spaces of those functions and problems that arise when working in infinite dimensions such as lack of compactness. The concept of weak convergence is one of the major tools of interest.
Elias
Pipping
FU Berlin
What is a subgradient?
Abstract.
The subgradient of a convex function can be seen as a generalisation of the gradient in that it does not require differentiability. After a brief introduction, I will say a word or two about its connection to nonsmooth constitutive laws and the Legendre transform.
May 2013
Ariane
Beier
University of Potsdam
What is the Nash embedding theorem?
Abstract.
The notion of a Riemannian manifold evolved from more concrete objects like surfaces in three-dimensional Euclidean space. But how much more general is this concept? Nash's embedding theorem gives one answer to the question whether or not a Riemannian manifold can be isometrically embedded into Euclidean space. It provides surprising and, at first glance, inconsistent results which we want to illustrate by considering the example of a flat torus.
April 2013
Miguel
Grados
HU Berlin
What is an $L$-function?
Abstract.
$L$-functions are analytical objects containing meaningful information for the underlying context in which they are defined. For example, take the Riemann zeta function; since it can be written in terms of primes, it will encode arithmetical information of $\mathbb Z$. Also, the proof of the prime number theorem was possible thanks to the nice properties of $L$-functions. In this talk we will take a quick tour through different $L$-functions appearing in mathematics. We will point out some features they have in common, and finally we will arrive to an approach of what an $L$-function should be (in a broad context).
Max
Pfeffer
TU Berlin
What is a higher order tensor decomposition?
Abstract.
Low-rank matrix structures can be exploited in many ways. We seek to generalize this concept to higher order tensors by generalizing the Singular Value Decomposition (HOSVD). Several ideas have been put forward, each proving to have certain advantages and disadvantages. Different tensor decompositions will be briefly discussed and the more recent approach of TT tensors will be introduced. The alternating least squares (ALS) algorithm will be presented as one of the most basic yet reliable tools in tensor optimization.
February 2013
Dennis
Clemens
FU Berlin
What is a positional game?
Abstract.
Using some specific examples, we will introduce positional games, in particular, the class of maker-breaker games. Moreover, we will take a deeper look at the Erdős-Selfridge theorem from 1973 which often is seen as the starting point in the history of maker-breaker games.
Christian
Schröder
TU Berlin
What is a palindromic eigenvalue problem?
Abstract.
"Mom", "Dad", "I prefer pi", and " A man, a plan, a canal — Panama" are palindromes — they can be read form left to right and vice versa. Analogously, a polynomial is palindromic if its sequence of coefficients is the same in both directions (i.e., $2x^4 + 7x^3 + 5x^2 + 7x + 2$). In just a few more small steps, we get to palindromic matrix polynomials and their corresponding eigenvalue problems. The talk will be on palindromic eigenvalue problems: their introduction, where they arise, their properties and suitable algorithms.
Silvia
de Toffoli
HU Berlin
What is a rational tangle?
Abstract.
"During the period from the end of the '60s through the beginning of the '70s, Conway pursued the objective of forming a complete table of knots. […] Therefore, he pulled another jewel from his bag of cornucopia and introduced the concept of tangle." - Murasugi A $n$-tangle is an embedding of a collection of $n$ arcs in a $3$-ball such that the endpoints are on specific 2$n$ points on the boundary sphere. The focus will be on $2$-tangles. A rational tangle is a $2$-tangle that is homeomorphic to the trivial tangle, which is formed by two unlinked arcs, vertical or horizontal. By taking the numerator closure of a tangle we obtain a knot and, in particular, the numerator of a rational tangle is a rational knot. Rational tangles are associated, as the name suggests, to rational numbers (union infinity) and there is a deep connection between them and the theory of continued fractions. Apart from their mathematical importance in knot theory, rational tangles are crucial to the study of DNA topology and, in general, to biological applications of knot theory.
Ahmad
Afuni
FU Berlin
What is a monotonicity formula?
Abstract.
Monotonicity formulae are an indispensable tool for extracting information about solutions of ODEs and PDEs directly from the structure of the equations, especially when explicit solutions are not readily available. In this talk, we introduce the notion of a monotonicity formula and outline how such formulae may be used to draw certain conclusions about solutions of certain differential equations.
January 2013
Isabella
Thiesen
TU Berlin
What is a quasiconformal mapping?
Abstract.
Have you ever tried to find a conformal mapping between a square and a non-square rectangle that maps corners to corners? This won't work, so the German mathematician Grötzsch asked for the most conformal way instead. That was in 1928, and he laid the foundation for what became later known as quasiconformal mappings, a natural generalization of conformal mappings. Some people love to use quasiconformal mappings as a tool for proving theorems, and (more often) other people love to compute them for applications in computer graphics. Both groups use it for the same reason; in some sense they are almost as good as conformal maps but much more flexible. I will talk about the different approaches to quasiconformal mappings and try to get you interested in this topic.
Emre
Sertöz
HU Berlin
What is a $j$-invariant?
Abstract.
The elliptic curves (or complex tori) can be parametrized in 2 different ways. The first method parametrizes lattices in the complex plane in a rather obvious way. The second parametrization gives to each elliptic curve a more geometric value in the sense that this value corresponds more closely to how the curve is embedded in the plane. Then there is a function mapping the first parametrization to the other. This function is called the $j$-invariant and it is a modular form of weight zero, where number theory comes to join geometry and algebra. We will discuss briefly what modular forms are and what a fundamental domain is--all the absolute basics you need to know.
December 2012
Florian
Frick
TU Berlin
What is $SU(2)$ and why does it double-cover $SO(3)$?
Abstract.
We will investigate the relationship between rotations in vector spaces of dimension at most four and multiplication of complex numbers or quaternions. We will see that complex unitary 2-by-2 matrices with determinant one, which is a subgroup of isometries of a complex two-dimensional vector space, and rotations of three-dimensional Euclidean space are closely related.
November 2012
Philippo
Lappicy
FU Berlin
What is a spacetime
singularity
Abstract.
Trying to answer questions about space, time and gravity leads us to the notion of singularities — where the theory doesn't hold. In this talk, we'll introduce the basics of General Relativity and how it is related to dynamics.
Faniry
Razafindrazaka
FU Berlin
What is a regular map?
Abstract.
The notion of a regular map is everywhere. It sits at the intersection of arithmetic geometry, hyperbolic geometry, topology, and computational group theory. Recently, it is also being used in mathematical visualization. Even people in string theory use the notion in some applications. In this talk, we use the geometrical definition related to its symmetry group. We will investigate problems that have been solved and those that are still open in the process of visualizing them.
Ananyo
Dan
HU Berlin
What is a Hodge locus?
Abstract.
Given a family of smooth projective varieties $B$, the Hodge locus corresponding to a Hodge class $\gamma$ parametrizes all $b \in B$ where $\gamma$ remains a Hodge class. In this talk we discuss the definition of a Hodge class, variation of Hodge structures and the statement of the Hodge conjecture. We also look at the Lefschetz (1,1)-theorem which states that the Hodge conjecture is true in the case of divisors.
Inder
Kaur
HU Berlin
What is a Bruhat-Tits tree?
Abstract.
In this talk I will define a Bruhat-Tits tree and describe the construction of the tree associated to $PGL(2,F)$ for $F$ a local field. If time permits, I will also discuss some applications of the tree in representation theory.
Christian
Wald
HU Berlin
What is a quaternion algebra?
Abstract.
Quaternion algebras over a field $K$ are special noncommutative $K$-algebras of dimension four. We will consider the classical example of the Hamilton quaternions followed by quaternion algebras over arbitrary fields and different characterizations. We will discuss basic properties and, if time permits, explain the notion of automorphic forms on quaternion algebras.
October 2012
Mijail
Guillemard
TU Berlin
What is support vector machines?
Abstract.
Persistent homology is an important modern subject in computational topology, and it offers useful tools for signal and data analysis. This short talk is intended for a general audience, and we cover elementary aspects of persistent homology, including some basics on simplicial homology and the concept of persistent diagrams.
Felix
Knöppel
TU Berlin
What is a smoke ring?
Abstract.
This talk will give a short introduction to smoke rings and the rich theory to which they are connected. Smoke rings are closed curves that evolve by the vortex filament equation. This equation is connected to the non-linear cubic Schrödinger equation, which is well-known in the theory of solitons.
July 2012
Hannah
Enders
HU Berlin
What is
Ext and its applications?
Abstract.
From the module of homomorphisms $Hom(M,N)$ between two modules $M$ and $N$, one can derive other modules $Ext^i (M,N)$, for all $i$. These modules contain certain information about $M$ and $N$. We will show how to construct $Ext$ explicitly and give some further results on how to use it.
Adrián
Gonzalez Casanova
TU Berlin
What is Brownian motion?
Abstract.
Rather than giving a precise mathematical statement to describe Brownian motion, here is a great quote from "Stochastic Calculus," by Richard Durrett. "If you run Brownian motion in two dimensions for a positive amount of time, it will write your name. Of course, on top of your name it will write everybody else's name, as well as all the works of Shakespeare, several pornographic novels, and a lot of nonsense." In this seminar, we will better understand one of the most fascinating and complex mathematical objects: the Brownian motion.
June 2012
Giovanni
de Gaetano
HU Berlin
What is a group representation?
Abstract.
Representation theory is the main bridge between abstract and linear algebra. Moreover, representations can be found in virtually any field in mathematics and can generalize many commonly used objects. In this talk, we will approach the fundamentals of the theory, with specific attention to the representations of groups. If time permits, we will clarify some notions useful for the following Friday talk by Prof. Bridson, such as finitely presented groups and ${\rm SL}(n,\mathbb{Z})$ representations.
Felix
Günther
TU Berlin
What is the Fabricius-Bjerre theorem?
Abstract.
The Fabricius-Bjerre theorem states that for a generic curve in the plane, the number of crossings plus half the number of inflections plus the number of opposite-side double tangencies is equal to the number of same-side double tangencies. We will define each of the quantities referred to in the theorem and look at some examples before we give the original (and very beautiful) proof of Fabricius-Bjerre himself. In the case of polygons with vertices in general position, similar definitions are given for crossings, opposite-side and same-side double tagencies, and inflection edges. We will sketch the proof of Tom Banchoff for the polygonal version of the Fabricius-Bjerre theorem, which is based on a deformation argument. In the end, we will look at generic curves in the sphere, and give the Spherical Fabricius-Bjerre formula.
Juan
Orduz
HU Berlin
What is
a Dirac operator?
Abstract.
The main idea of this talk is to introduce the notion of a Dirac Operator from the physical problem of finding the quantum equation for a relativistic electron. After that we will explore how these ideas can be generalized in order to find a general geometric setting for these operators and how they can be used to construct a bridge between analysis and topology via index theorems.
Mikola
Lysenko
UW Madison
What is a group action?
Abstract.
This short expository talk is an attempt to answer the "What?" and "Why?" of group theory. Because no familiarity with abstract algebra is assumed, we will start from first principles in order to keep this presentation self contained. The main goal here is to impart some conceptual understanding, not to study any single problem in much depth. We will focus on motivation, and survey a few interesting applications.
Julio
Backhoff
HU Berlin
What is
a problem in mathematical finance?
Abstract.
In this talk, we introduce stochastic financial models for both discrete time and continuous time and discuss some of their defining properties. Then we present some of the most relevant problems encountered in financial mathematics.
May 2012
Dror
Atariah
FU Berlin
What is
a configuration space?
Abstract.
In a nutshell, a configuration space is one where each point represents a unique pose/configuration of some physical system. In this talk we will introduce a special instance of a configuration space. For the physical system, we will consider the case of a planar polygonal robot which is free to rotate and translate amid planar polygonal obstacles. By means of explicit parameterization of the configuration space, we will obtain a clear picture of the geometrical properties of the space and, in particular, of the configuration space obstacles. The talk will be accompanied with a short video visualizing the discussed case.
Shahrad
Jamshidi
FU Berlin
What is
a differential inclusion?
Abstract.
Differential equations with discontinuous right hand sides crop up when modelling switches or relays in control systems. In order to deal with these discontinuities we need to generalise the concept of solution so that the differential equations are satisfied almost everywhere. Differential inclusions incorporate the discontinuities of the differential equation in order to obtain a well defined solution. In this talk we focus on the mathematical complications that arise with the discontinuous right hand side and how they are overcome using differential inclusions.
Kristian
Rother
What is
software development?
Abstract.
Scientists who depend on programming for their work face uncertainty; they need to adapt their programs as their research projects evolve. This typically makes it impossible to design a complete software up front. How, then, can you write good software under this condition of uncertainty? Software development is the engineering solution to that question. This talk will present three general methodologies to writing software: Waterfall (old), Agile (new), and Lean Development (very new). In addition, best practices for developing software faster, making it more reliable, and communicating with peers are briefly presented. The goal of this talk is to enable you to apply development techniques in your programming practice and evaluate their usefulness.
April 2012
Jennifer
Rasch
HU Berlin
What is
a path-following method?
Abstract.
Mathematical modeling of real-world phenomena often leads to formulations of problems in infinite dimensions containing variational inequalities. In this talk, we will focus on so-called elasto-plastic problems which require the pointwise bounding of the gradient of the displacement, i.e., the stress on a body at each point in the presence of a given force. We will then derive the first order optimality conditions. To use the semismooth Newton method to solve the problem numerically, we require a regularized penalization of our problem. Numerical path-following strategies will be developed and analytical results can be numerically verified.
Romain
Nguyen
FU Berlin
What is a dissipative weak solution?
Abstract.
Hydrodynamical models are descriptions of fluids based on transport equations for macroscopic quantities such as flow field, temperature, density, etc. Two distinct terms are found in these equations: advection terms, which describe transport by the fluid flow itself and are reversible and independent on the nature of the fluid, and diffusion terms, which describe the irreversible transport due to disordered molecular motion. In many interesting cases, after non-dimensionalizing the equations, one finds that diffusion coefficients are extremely small. For example, the momentum diffusion coefficient which applies when pouring a glass of water is about 10^{-5}! When dealing with such cases, although it is very tempting to omit diffusion terms, we know that the water quickly comes to rest, its momentum being irreversibly dissipated. How could this process be described without dissipative terms in the equations?? We will show that asking this simple question leads to the mathematical concept of a dissipative weak solution, and to some of the most difficult open problems in mathematical fluid dynamics.
Maciek
Korzec
TU Berlin
What is
a spectral method?
Abstract.
Besides the usual techniques, such as finite difference methods and finite element methods, spectral methods are another important tool for numerically solving partial differential equations. If the solutions are sufficiently smooth these methods yield spectral accuracy and hence one can achieve a faster convergence rate than with any local polynomial based interpolation method. In this talk the general framework of spectral methods, i.e., collocation methods based on trigonometric interpolation, is introduced. Example nonlinear partial differential equations are solved numerically.
February 2012
Fabian
Lenhardt
FU Berlin
What is the fundamental group and covering space?
Abstract.
We define the fundamental group of a topological space, which is one of the most fundamental topological invariants. One of the main tools for computing fundamental groups is the theory of covering spaces; we give a fast introduction and try to compute some fundamental groups in the end.
Christian
Bogner
HU Berlin
What is a Feynman integral?
Abstract.
Computations in perturbative Quantum Field Theory typically involve a certain kind of integral, the so-called Feynman integrals. Without assuming prior knowledge of particle physics, I will point out some of the main ideas on why we need these integrals and how they arise from certain graphs. In the end I will briefly point out some recently developed interrelations with modern mathematics.
Samuel
Drapeau
HU Berlin
What is risk?
Abstract.
Risk as a notion, even if very intuitive, is still ambiguous. In this seminar, we will briefly discuss the characteristics of risk and how it can then be modeled and studied mathematically. Finally, we discuss the insights we gain from various interpretations of this approach.
January 2012
Özge
Ekin
FU Berlin
What is the connection between Kantian intuition of mathematical
objects and diagrams?
Abstract.
In this talk, I will reveal the connection between particular representations (symbols, diagrams) and the Kantian intuition of mathematical objects. I will construct an interpretation of Kant's philosophy of mathematics that explains the requirement of pure intuitions, space and time and reveals the a priori nature of mathematics in Kant's doctrine. Kantian characterization of mathematics exposes a different reasoning model from the current method of mathematics, namely the formal sentential reasoning. I argue that by understanding this approach and by recognizing the roles of diagrams in mathematics, it is possible to realize the advantages of heterogeneous reasoning in mathematics.
Lothar
Narins
HU Berlin
What is an independent
transversal?
Abstract.
In this talk I introduce independent transversals and give a combinatorial proof of their existence for certain classes of graphs.
Wang-Q
Lim
HU Berlin
What is a shearlet?
Abstract.
Shearlets were introduced as means to sparsely encode anisotropic singularities of multivariate data while providing a unified treatment of the continuous and digital realm. In this talk, recent results on the construction of compactly supported shearlet systems will be presented, in particular, showing that these shearlet frames provide optimally sparse approximations of piecewise smooth 2D as well as 3D functions. Finally, we will discuss various applications of shearlets such as image restoration, data separation and numerical PDEs.
December 2011
Antoine
Laurain
HU Berlin
What is propagating interfaces and level set methods?
Abstract.
Level Set Methods are numerical techniques which allow to model the evolution of interfaces between geometrical domains. Unlike parametrization methods, complex geometrical changes such as sharp corners, appearance of holes, and merging together may be modeled using level set methods. The techniques have a wide range of applications, including problems in fluid mechanics, combustion, structural optimization, computer animation, image processing, and the shape of soap bubbles.
Barbara
Jung
HU Berlin
What is a lattice in a
Lie group good for?
Abstract.
The talk of Igor Rivin will have discrete subgroups of matrix groups as a topic. But why? In the seminar I will give two examples of such lattices. I will connect them with elliptic curves and with each other, so at the end we see how to understand elliptic curves without algebraic geometry.
November 2011
Jonad
Pulaj
ZIB
What is the Lagrangian relaxation of an integer program?
Abstract.
The Lagrangian relaxation is a well known technique which provides bounds for integer programs through penalty adjustments. We give an overview of the method and illustrate its application to multi-period network design.
Andre
Mialebama
FU Berlin
What is a Gröbner basis in a polynomial ring?
Abstract.
Our main goal in this talk is to define a Gröbner basis in a polynomial ring over a field and show how it is important to solve the ideal membership problem and other problems.
Andreas
Hochenegger
FU Berlin
What is a derived category?
Abstract.
The derived category is an important object in algebraic geometry. The derived category associated to a variety is an important invariant that is even stronger than cohomology. I will try to give a gentle introduction by showing how the definition of this category evolved.
David
Ouwehand
HU Berlin
What is a zeta function?
Abstract.
Zeta functions are objects that arise in many areas of mathematics. This talk will be about the Dedekind zeta function of a number field and the Hasse-Weil zeta function of a smooth curve over a finite field; the goal is to explain how these zeta functions contain (respectively) arithmetic and geometric information. If time permits, I will also talk about the zeta functions of schemes that are of finite type over the integers. These generalize the previous types of zeta functions.
October 2011
Cesar
Ceballos
FU Berlin
What is tropical
geometry?
Abstract.
Tropical geometry is a relatively recent area of mathematics with strong applications to algebraic geometry, mirror symmetry, combinatorics, mathematical biology and enumerative geometry among others. It manipulates, in a purely combinatorial way, geometric objects that take over the role of classical algebraic varieties. In this talk, we present an elementary introduction to the subject.
Maciek
Korzek
TU Berlin
What is diffusion?
Abstract.
One of the most fundamental differential operators appearing in partial differential equations (PDEs) is the Laplace operator. Its understanding is essential to be able to treat models describing real-world problems. One of the most basic PDEs relates the rate of change of some quantity to the Laplace operator applied to the same function: the diffusion equation $u_t = k \nabla^2 u$, also known as heat equation. In this talk several examples for diffusion will be explained. A connection between random walks and continuous diffusion will be established, a Gaussian filter will be linked to diffusion in image processing and the anisotropic diffusion equation $u_t = \nabla \cdot k(x) \nabla u$ will be used to improve an image by advocating diffusion in small slope regions only. In this way the edges in an image remain intact while noise or similar image failures are diffused out. Finally the heat equation will be derived in a bulk material, and also on a regular surface, resulting in a surface diffusion equation, $u_t = \nabla_{\!s} \cdot k(x) \nabla_{\!s} u$.
July 2011
Markus
Hihn
HU Berlin
What is a matroid? And how is it related to quantum field theory?
Abstract.
In quantum field theory, the study of periods associated to Feynman graphs is of growing interest. However, some graphs define the same period (if it exists). The question which graph defines the same period is answered by their corresponding cycle matroid. This is an easy construction, but they are also appearing in other topics in quantum field theory: the falsification of Kontsevich conjecture and Feynman integrals with tensor structure.
Marcel
Ortgiese
TU Berlin
What is a Green's function?
Abstract.
The concept of a Green's function arises when trying to solve certain partial differential equations. We will highlight the interplay between analysis and probability theory and see how to solve these equations in a probabilistic way and in particular we will give an interpretation of the corresponding Green's functions.
June 2011
Ambros
Gleixner
ZIB
What is linear programming?
Abstract.
Linear programming is arguably the single most essential technique in theory and practice of mathematical optimisation. We will introduce the underlying duality theory including complementary slackness and the central path. As an example for the powerful algorithms available to solve linear programs, we will sketch a primal-dual interior point method. This algorithm solves a linear program by following the central path to an optimal solution.
Nicolai
Beck
FU Berlin
What is a sheaf?
Abstract.
A basic object in mathematics is the set of certain functions on a space, for example continuous functions on a topological space or differentiable functions on a manifold. A presheaf is a first generalization of this notion, which is easily defined but can have unexpected properties. Sheaves are presheaves which behave nicely. This definition leads to some technical difficulties. However, it is just this difficulty which allows us to define sheaf cohomology. Finally one can show that for many spaces sheaf cohomology agrees with singular cohomology.
Fabian
Müller
HU Berlin
What is a moduli space?
Abstract.
In modern algebraic geometry, moduli spaces provide a way of describing the set of isomorphism classes of various kinds of objects, such as curves, maps or vector bundles. While the points of the moduli space correspond just to these isomorphism classes, these spaces can be endowed with a much richer algebraic structure reflecting the way in which the objects under consideration behave in families. The talk will give a low-level introduction to the concepts of fine and coarse moduli spaces, classifying maps and universal families. With an eye towards the subsequent talk by Valery Alexeev, we will also take a short look at the issues one encounters when one tries to compactify such spaces.
May 2011
Beatrice
Bugert
WIAS
What is the direct method in the calculus of variations?
Abstract.
One of the fundamental problems in the calculus of variations consists of finding a function $u$ minimizing the integral functional \[ I(u) = \int_\Omega f(x, u(x), Du(x)) \ dx \] over all the functions $u$ satisfying $u = u_0$ on the boundary $\partial \Omega$ of $\Omega$, where $u_0$ is a given function. Euler--often referred to as the founder of the calculus of variations--treated this problem by deducing the so-called Euler-Lagrange equation from the integral functional. He proved that in the case of convex functionals solutions of this equation are already minimizers of $I(u)$. As this method is hard to implement for higher dimensional integrals (i.e., not one-dimensional ones), there was a great need to find an alternative method avoiding the Euler-Lagrange equations. It was Riemann who finally succeeded at this task and introduced the so-called direct method in the calculus of variations, which provides the existence of minimizing functions $u$ directly from the properties of the functional $I$. This talk will give an overview of Riemann's method for convex functionals and show how it has further developed over almost two centuries under the influence of the Italian mathematicians Tonelli and De Giorgi.
Daniel
Heldt
TU Berlin
What is a graph?
Abstract.
A short introduction to graphs, which contains some pictures (of graphs) as well as (some of) the definitions in graph theory, like cliques, bipartite graphs, spanning trees, colorings, adjacency matrices, and so on.
Thomas
El Khatib
TU Berlin
What is
Who?
Thomas El Khatib (TU Berlin)
When?
2011/05/06, 16:00
Where?
TU Berlin, at the BMS Lounge, MA 212
About what?
In the 17th century, Leibniz and Newton invented calculus using infinitesimally small quantities. 200 years later, Bolzano, Cauchy and Weierstraß made calculus rigorous by introducing the modern epsilon-delta-formulation of limits, bereaving mathematics of intuitively appealing objects. Still, another 150 years later, students of physics and engineering are still taught to think in terms of infinitesimals, with or without the warning never to mention them in the presence of a mathematician.
In this talk, I will discuss some ways of rigorously introducing infinitesimals in modern day mathematics, and — if time admits — I will talk about a model theoretic approach by Robinson/Zakon in more detail.
Abstract.
In the 17th century, Leibniz and Newton invented calculus using infinitesimally small quantities. 200 years later, Bolzano, Cauchy and Weierstraß made calculus rigorous by introducing the modern epsilon-delta-formulation of limits, bereaving mathematics of intuitively appealing objects. Still, another 150 years later, students of physics and engineering are still taught to think in terms of infinitesimals, with or without the warning never to mention them in the presence of a mathematician. In this talk, I will discuss some ways of rigorously introducing infinitesimals in modern day mathematics, and — if time admits — I will talk about a model theoretic approach by Robinson/Zakon in more detail.
April 2011
Thomas
Lehmann
ZIB
What is the Kuhn-Tucker theorem?
Abstract.
The Kuhn-Tucker theorem is the foundation of nonlinear programming. Requiring a constraint qualification, it states necessary conditions for local solutions of nonlinear programs. Even more, these (Karush-)Kuhn-Tucker conditions are the basis of many efficient nonlinear programming algorithms.
Anilatmaja
Aryasomayajula
HU Berlin
What is a modular form?
Abstract.
Modular forms are complex analytic functions on the upper half plane satisfying a certain kind of functional equation and growth condition. In this talk we will see how the theory of modular forms answers a classical problem in number theory, namely: "Which natural numbers can be represented as the sum of four squares, and in how many ways can that be done?".
February 2011
Matthias
Lenz
TU Berlin
What is a matroid?
Stefan W.
von Deylen
FU Berlin
What is a weak derivative?
Abstract.
...or: Plugging objects into an equation that it was absolutely not thought for. Of course, all of you know the partial integration rule: $\int f' g = -\int f g'$ plus some boundary term we will impudently ignore. In your Analysis I course, both functions needed to be differentiable. But what if $f$ were piecewise differentiable? Perhaps even with jumps in the function values? Or unbounded, yet still integrable? In this talk, we will think about other ways to define $f'$ while keeping this equation valid. More precisely, we require nothing from real calculus, only this single equation. Surprisingly, this requirement, all alone in the world, is not as lonely, lost and feeble as it seems, but already leads to a huge and beautiful theory (which could answer many questions from partial differential equations or calculus of variations, but that would be going too far). I will first introduce the most general notion for weak derivatives, the language of distributions. But you may recall your analysis courses on continuity or connectedness: What seems the most natural general definition is, in the domain of analysis, often a very nasty-behaving object. We will take a short look at the pitfalls we foolishly did not exclude in that first attempt and quickly move on to Sobolev spaces. As much as we can fit in half an hour, this will be the ultimate answer to life, universe and the Dirichlet problem.
Jannik
Matuschke
TU Berlin
What is P vs. NP?
Abstract.
Complexity theory is a branch of theoretical computer science that deals with the limits of efficient computability. Complexity classes allow us to compare the computational hardness of problems from a theoretical point of view. We will give formal definitions of the classes P (“problems that can be solved efficiently by a deterministic computer as we all know it”) and NP (“problems that can be solved efficiently by a non-deterministic computer that can guess the answer and only needs to verify its correctness”) and motivate the importance of the great open question whether or not P = NP.
Han
Lie
HU Berlin
What is the exit problem?
Abstract.
Consider the behaviour of a deterministic dynamical system in a domain containing a stable attracting equilibrium — for example, a particle trapped in a potential well, or a marble dropped into a large bowl. Given suitable initial conditions, the system will move towards the equilibrium and stay there once it has arrived at the equilibrium. However, if the dynamical system is perturbed by white noise, is it possible that the system might exit the domain of attraction? If it is possible, what conditions do we need? Where and when is the system most likely to exit the domain? These are the questions involved in the “exit problem”, which is often encountered in large deviations theory. In this talk we will take a different perspective from large deviations theory, and instead apply ideas from control theory to answer these questions in the context of linear systems.
January 2011
Felix
Breuer
FU Berlin
What is math blogging?
Abstract.
Social media are omnipresent today! But what is the role of social media in mathematics? This question is still wide open and in this talk I want to give a glimpse of the developments in this area. The focus will lie on mathematical blogs, but mathoverflow, the polymath project and the use of blogs in other sciences will also be addressed. Moreover, I will use the opportunity to present Mathblogging.org, an aggregator for mathematical blogs that can serve both as an index as well as a starting point for exploring the mathematical blogosphere.
Natasa
Djurdjevac
FU Berlin
What is a random walk on a network?
Abstract.
Networks are widely used to characterize and model a broad range of complex systems in various fields from biology to the social sciences. Important information about the topology and dynamics of the network can be obtained by analyzing random walks on the network. This talk will give the theoretical background of the random-walker-based approach for complex networks and address some of the current challenges, such as partitioning of networks and identifying important nodes.
Stefan
Keil
HU Berlin
What is a Diophantine equation?
Abstract.
As a warmup to Professor Funke's lecture, we introduce Diophantine equations and present a famous example of how elliptic curves can be employed in their solution: namely, Fermat's Last Theorem.
December 2010
Roland
Friedrich
HU Berlin
What is cognitive neuroscience?
Abstract.
In this short talk we shall focus on probably the single most important tool in doing research in cognitive neuroscience, namely, functional magnetic resonance imaging, (fMRI). We shall explain how with this method questions concerning higher cognitive processes in the brain are investigated and what kind of inferences can be made.
Anita
Liebenau
FU Berlin
What is the probabilistic method?
Abstract.
To give an idea of the power of probabilistic arguments in graph theory, I will show two results of different flavour. The first comprises the statement that for an arbitrary graph G there a is partition of the vertices such that the resulting bipartite graph has at least half the number of edges of G. For the second, I will introduce the random graph model G(n,p) and the notion of a threshold function. We will then take a short look at the property of having isolated vertices. Hopefully, this will pave the way to a better understanding of the talk by Joel Spencer for people from outside the area.
November 2010
Barbara
Jung
HU Berlin
What is the origin of elliptic functions?
Abstract.
Elliptic functions are double-periodic meromorphic functions on C, that means basically $$f(a+ib)=f((a+n)+i(b+m)) \qquad \forall a, b \in \mathbb{R},\; n, m \in \mathbb{Z}.$$ So they define a function on a torus. But what has this to do with an ellipse? To find the answer, we go on a journey to the origins of Riemann surfaces in the times of Euler and Lagrange and see the theory beautifully arising from the observation of integrals over some simple curves we already know from school.
Klebert
Kentia
HU Berlin
What is a multifunction?
Abstract.
Multifunctions, or multivalued mappings or correspondences, are “functions” that assign to a fixed point one or several values. They can be viewed as set-valued mappings and turn out to be very interesting objects in many areas of mathematics (e.g. optimization, probability, functional analysis). This motivates the need for a nice and thorough analysis of these objects. During this analysis, some questions arise when one indeed considers correspondences as mappings taking values in the power set of a given set. Of particular importance is how one defines a useful (mainly in application) concept of measurability (i.e. conservation of information by inverse image) for such mappings. If this is at all possible, then can one associate to a measurable correspondence a suitable notion of integral? An even more interesting question is that of the existence of a measurable selection of a correspondence (i.e. a measurable function that takes values in the values of the correspondence). The purpose of this talk will be to attempt to address some of these questions.
Heinrich
Mellmann
HU Berlin
What is a selflocator?
Abstract.
Whatever we do, wherever we go, most of the time we know “where” we are—in Berlin, at home, in front of the fridge, etc. For a successful interaction with the environment (e.g., switching on the TV set) the knowledge of one's own position relative to the object is essential. Thus, an artificial being, like a robot, which should perform similar to a human also has to know its environment and especially its position in it. In this talk we introduce some mathematical groundings for Bayesian modeling and discuss, in particular, the Monte-Carlo particle filter which can be used to model the position of a robot in a dynamic world. We will also illustrate the gray theory with interesting videos and examples from robot soccer.
Ágnes
Cseh
TU Berlin
What is a stable marriage?
Abstract.
This definitely important question can be answered with the help of graph theory. The stable marriage theorem of Gale and Shapley states that for some men and women there always exists a stable marriage scheme, that is, a set of pairs such that no man and woman mutually prefer each other to their partners in the matching. The stable marriage problem can be extended in several directions, one of the most recent topics deals with network flows. Besides sketching some theorems and unanswered questions we will give some useful hints to find a stable partner in real life.
October 2010
Jean-Philippe
Labbé
Technische University Berlin
What is a Catalan number?
Abstract.
Catalan numbers form a sequence of natural numbers that counts a plethora of objects in mathematics. Thus, they appear constantly in enumerative combinatorics and many related fields like algebra, discrete geometry, representation theory and more. I will present their origin, discuss about some recursive and closed formulas and finally show some objects that are counted by Catalan numbers.
July 2010
Kaie
Kubjas
FU Berlin
What is phylogenetic algebraic geometry?
Abstract.
Phylogenetic algebraic geometry studies algebraic varieties arising from evolutionary trees. In this talk I will explain based on examples how to construct these algebraic varieties.
Hanna
Döring
TU Berlin
What is large deviations?
Abstract.
A classical result in probability theory is the law of large numbers. For a long sequence of coin tosses we expect half the coins to show head. Large deviations describe the probability of so-called rare events, that differ from its expected behaviour. Starting with the example of coin tossing we learn about Cramér's theorem and point out some applications.
June 2010
Shahrad
Jamshidi
FU Berlin
What is a hybrid automaton?
Abstract.
A thermostat maintains a constant temperature via a switching mechanism. This discrete switch and the evolution of the temperature can be modelled using a hybrid automaton. In this talk we will discuss the advantages and disadvantages of this modelling method.
Emerson
Leon
TU Berlin
What is Pólya theory?
Abstract.
Pólya enumeration methods help to count objects up to some symmetry. I will briefly explain how it works and then show couple of applications, were other combinatorial objects will be involved.
Stephan
Müller
HU Berlin
What is the classifying space of a category?
Abstract.
I will describe an easy way to associate a (nice) topological space to a (small) category — its classifying space. Using homotopy theory we can discover Quillen's higher $K$-theories.
May 2010
Aaron
Greicius
HU Berlin
What is the big deal with math and music?
Abstract.
The connection between math and music has fascinated scientists for millenia. The ancient Greek investigated relations between numerical ratios and musical scales, nowadays new composition techniques have led to applications of set theory, abstract algebra and number theory in musical theory. This talk will shed some light on the implicit and explicit math in music and vice versa.
Jerry
Gagelman
TU Berlin
What is a Fourier Series?
Abstract.
Fourier's method of analyzing differential equations has inspired so many ideas in other areas of mathematics that the name has become seemingly ubiquitous — in harmonic analysis, representation theory, number theory, and so on. In this talk I'll motivate one perspective on Fourier series through basic linear algebra. From only the most minimal theory (e.g., basic calculus), I'll explain a couple things that Fourier series reveal.
April 2010
Mimi
Tsuruga
FU Berlin
What is a Killing vector field?
Abstract.
A Killing vector field is a vector field on a Riemannian manifold that preserves the metric. We will go through some basic concepts in differential geometry including Riemannian metric, isometries, and Lie derivatives, and note some nice properties of Killing fields.
Annie
Raymond
TU Berlin
What is a Theta Body?
Abstract.
The cold war is raging outside and our young hero, László Lovász, stumbles upon a body that has been stabbed! Or, in math terms: in 1979, Lovász found an upper bound for the Shannon capacity of a graph and introduced the theta body, a powerful relaxation of the stable set polytope $\text{STAB}(G)$.
Eyal
Ron
FU Berlin
What is Hilbert's 16th problem?
Abstract.
Ever think that the Millennium problems were the first of their kind to be posed? Not at all! In 1900, David Hilbert, the godfather of math at that time, posed a list of no less than 23 unsolved problems in various math disciplines. These problems received remarkably large attention from the math community and a solution of one bestowed the solver with huge appreciation — and, much more importantly — a modest field medal. As of today, ten of the problems have been completely solved. Another seven were "solved", where the quotation marks denote that the solution is either not fully accepted, or, worse, nobody is really sure what Hilbert meant when posing the problem. The remaining six problems still lay in the dark, waiting for a brave mathematician to one day come and save (=solve) them. An example? the 16th problem: the problem of the topology of algebraic curves and surfaces. There are two equivalent phrasings of the 16th problem. The incomprehensible one — at least for me — and the one that concerns phase plane analysis and dynamical systems tools. What are these? what's the problem? and how can you solve it and earn an easy ticket to a good post-doc position and a comfortable tenure? All of these, and a myriad of other questions, will be answered during this seminar talk! Bonus: It seems that bounty-problems generate much more media interest than regular ones. Therefore, I will personally offer a 100-Euro prize to anybody that solves this problem during the seminar talk!* *In return the solver would be obliged to include my name as a co-author in the resulting paper. Fair is fair, right?
February 2010
Carsten
Lange
FU Berlin
What is a graph
associaedron?
Abstract.
Graph associahedra are a special breed of polytopes but appear in many different branches of mathematics. We start with basic definitions of a convex polytope, see many examples and explore some fundamental properties.
January 2010
Oliver
Sander
FU Berlin
What is a multigrid?
Abstract.
All known solvers for elliptic finite element problems get slower with increasing problem size. All solvers? No! Linear multigrid can solve even large problems in optimal time. This makes it the method of choice for many challenging application problems.
Hartwig
Mayer
HU Berlin
What is a language game?
Abstract.
Around the end of 19th century a new paradigm arose in philosophy known as the "linguistic turn". Philosophers, like Gottlob Frege, were focused on analyzing language and formalized its logical structure in order to distinguish between absurd, meaningless, and meaningful sentences hoping to make misunderstandings disappear. Wittgenstein's work was strongly influencing this kind of philosophy. In his early stage he was a supporter of an "ideal language". After finishing his "Tractatus Logico-Philosophicus" and keeping silent for some while (he believed he had solved all essential questions) Wittgenstein again became interested in philosophy. But now his interest was devoted to "normal language". Instead of searching for some ideal structure behind language he took it then as it is in the first place, namely as a performative act - a language game. I will talk about some basic ideas of Wittgenstein's later philosophy which will give some background for his understanding of mathematics.
Plamen
Turkedjiev
HU Berlin
What is a
semimartingale?
Abstract.
The theory of integration is well known to most mathematics students. We are interested here in the theory of 'good integrators'. The Riemann-Stieltjes construction of the integral allows only integrators that have finite variation to be integrators of continuous integrands. This is a problem for even the most fundamental stochastic process-the Brownian Motion-and it seems impossible to build a theory of stochastic integration with this approach. In this talk, I first show how the Riemann-Stieltjes approach fails, and then introduce the semimartingale as the general 'good integrator' for stochastic integration, and finally recover the integral for the Brownian Motion. For this talk, it will be useful for the audience to know basic measure theory and probability theory (in particular, the meaning of the term 'convergence in probability'). Due to the nature of the topic, some ideas are very technical. I will sketch these ideas very lightly. The intention of the talk is to give audience a grasp of the particularities of the theory of stochastic integrals.
Tobias
Pfeiffer
FU Berlin
and
Dror
Atariah
FU Berlin
What is a geodesic on a Riemannian manifold?
Abstract.
What does it mean to "go straight" on a sphere? What is the shortest distance between two points in a space other than the ordinary Euclidean $\mathbb{R}^n$? These two questions, and many more, are of geometrical nature, and are treated within the framework of differential geometry. The key object that is used is the manifold, which we will define in this talk. We will start from the broadest definition of a topological manifold, and end at the Riemannian one. Then we will give a basic idea and definitions of what a geodesic on a Riemannian manifold is, together with some examples.
December 2009
Joscha
Diehl
HU Berlin
What is a rough path?
Abstract.
Consider a $\mathbb{R}^d$-valued continuous function on $[0,1]$ that has finite length (i.e. finite variation). One can define integration with respect to such a function via the classical Riemann-Stieltjes integral. Rough path theory enables us to define integration with respect to functions of infinite length. It turns out that these paths must first be endowed (non-canonically) with more information, which leads to paths not taking values in $\mathbb{R}^d$, but some bigger space (a certain Lie group). One important area of application is stochastic analysis, where most processes are of infinite variation. Nonetheless this talk will focus on deterministic aspects and aims to be understandable with knowledge of only undergraduate mathematics.
Silvia
de Toffoli
TU Berlin
What is a Seifert
surface?
Abstract.
Giving the basic definitions and explaining the some important ideas, we will introduce the field of knot theory. We will explain the relation between a knot and its diagram and how to find invariants which allow us to distinguish different knots (or links). Thanks to an easy invariant we will prove the existence of non-trivial knots! We will introduce the concept of Seifert Surface of a knot: An orientable, compact connected surface whose boundary is the knot. We will show an algorithm to create a Seifert Surface starting form an arbitrary projection of a knot. We will see that this algorithm will be useful to calculate the genus, a knot invariant, of a certain class of knots. If time will permit we will talk of the signature of a knot, its relation with the unknotting number (Gordian distance), and the general context in which Seifert was working when he introduced his surface.
November 2009
Inna
Lukyanenko
TU Berlin
What is a quantum
group?
Abstract.
The term "Quantum Group" is due to V. Drinfeld and it refers to special Hopf algebras, which are the non-trivial deformations ("quantizations") of the enveloping Hopf algebras of semisimple Lie algebras or of the algebras of regular functions on the corresponding algebraic groups. These objects first appeared in physics, namely in the theory of quantum integrable systems, in the 1980's, and were later formalized independently by Vladimir Drinfeld and Michio Jimbo. In this talk I will explain the main idea of deformation and introduce the simplest and historically the first example of a Quantum Group: $U_q(\mathfrak{sl}(2))$.
Nicola
Tarasca
HU Berlin
What is a moduli
space?
Abstract.
Moduli Spaces are of vital importance in Algebraic Geometry. Given algebraic varieties with some fixed properties, one tries to understand the isomorphism classes and to put an algebraic structure on them. In this talk I will introduce Elliptic Curves, and I will show the classic example of Moduli Space of Elliptic Curves.
Carsten
Schultz
What is Morse theory?
Abstract.
A Morse function on a differentiable manifold is a real valued function with only nice critical points. The idea behind Morse Theory is to use a function like this to obtain global information on the manifold. We will have a look at some instances of this approach.
Noemi
Kurt
What is a self-avoiding random walk?
Abstract.
Imagine you are a tourist, visiting New York for the first time, going for a stroll in Manhattan. Since you don't know the city, at each crossroad you decide at random where to go next — left, right or straight — with one restriction: you never want to go back to a place you've visited before. After $n$ crossroads, how far will you be from your starting point? What kind of path will you have walked along? In this talk, we will present the probabilistic model for self-avoiding walk, and tell you what probabilists know about the answers to these questions.
October 2009
Bruno
Benedetti
What is a facet
massacre?
Abstract.
We explain what it means to collapse away the faces of a given complex. This simple idea has applications in combinatorics, topology, commutative algebra and even physics. If time permits, we'll have pizza earlier than scheduled.
Hernan
Leovey
HU Berlin
What is algorithmic
differentiation?
Abstract.
Practically all calculus based numerical methods for nonlinear computations are based on truncated Taylor expansions of problem-specific functions. The basic assumption of AD is that the function to be differentiated is at least conceptually evaluated by a sequence of "elemental" statements. We will see how the basic forward mode of Automatic Differentiation works and also some advanced applications to the field of high dimensional integration.
René
Birkner
FU Berlin
What is a Gröbner basis?
Abstract.
For ideals $I$ of some algebra $k[x_1,...,x_n]$ a set of generators is in general not unique. When considering the technique of term orders and initial ideals one can compute a subset of the ideal, the so-called Gröbner Basis, with respect to this term order. A reduced form of this Gröbner Basis is in fact unique for a given term order and a generates the ideal. These Gröbner Bases have proven to be useful for many applications from solving polynomial equations to moduli spaces.
July 2009
Peter
Krautzberger
FU Berlin
What is topological dynamics?
Abstract.
Topological dynamics is the abstract version of dynamical systems consisting of nothing else but a compact, Hausdorff space $X$ and a continuous function $f$ on $X$. Since this setting is very basic, I hope to offer an easy introduction to the basic phenomena, e.g. recurrence, proximality, maybe even chaos, while allowing for the discussion of some deeper results, e.g. the Ellis-Auslander Theorem.
June 2009
Robin
Scholz
U Potsdam
What is the square root of a graph?
Abstract.
For a certain class of finite graphs we consider the concept of square root of a graph. This concept, as well as the examined class of graphs, arises from a special decision problem. We will step by step develop criteria which characterize the graphs that have a square root. This is essential for the solution of the original problem.
Laura
Hinsch
FU Berlin
What is the field with one element?
Abstract.
The field with one element is a recurring legend. In this talk I will try to explain where the idea came from, what it could be, what it might be and what it cannot be.
Axel
Werner
ZIB Berlin
What is a flag vector?
Abstract.
We'll define $f$- and flag vectors (mainly of polytopes), consider a few examples and see why flag vectors of polytopes span a space of affine dimension given by the Fibonacci numbers. If time permits, we'll also consider linear inequalities for the flag vectors which can be used to 'map' polytpes and see some maps of 3-, 4- and 5-dimensional polytopes.
May 2009
Nathan
Ilten
FU Berlin
What is a Goppa code?
Abstract.
Reed-Solomon codes are widely used in coding theory due to their good parameters and ease of decoding. Goppa codes can in a way be viewed as a generalization of Reed-Solomon codes which, for the most part, maintains these nice properties. In this talk, I hope to briefly recall some basics of coding theory and the definition of Reed-Solomon codes. In what appears to be tagential, I will talk a little about divisors in algebraic curves. However, the definition of Goppa codes should bring everything back together. Time permitting, I will talk about what makes these codes so great.
Yvon
Vignaud
TU Berlin
What is entropy?
Abstract.
Entropy is a state function measuring microscopic disorder of a system. Its importance is related to its nature of "arrow of time": considering an isolated system, its entropy is increasing over time, unlike its energy which is a conserved quantity. It is a rather subtle and yet fundamental notion, which is especially useful in information theory or statistical mechanics. We aim to follow an intuitive approach so as to help grasping the concept of entropy.
Anna
Posingies
HU Berlin
What is the Dedekind zeta function?
Abstract.
Euler's zeta function in which the sum is taken over all natural numbers is well-known. There is a corresponding function called Dedekind zeta function for number fields where the sum is taken over all ideals of the ring of integers. We will introduce the Dedekind zeta function with a detailed definition of all number theoretical objects that occur and illustrate these objects with examples.
April 2009
Joscha
Gedicke
HU Berlin
What is the adaptive finite element method?
Abstract.
The adaptive finite element method (AFEM) is a powerful tool for numerical simulations. It is used to solve various problems of different kind numerically with high accuracy but less computational effort. This introductory talk explains key terms such as linear finite elements, a posteriori error estimators and adaptive mesh refinement. For a simple Poisson model problem the basic analytical setting such as the variational formulation is explained. The linear finite elements lead to a numerical discretization of the variational equation. In order to save computation time it is important to involve adaptivity into the algorithms. It will be explained what adaptivity means and how adaptive mesh refinement and a posteriori error estimates lead to the AFEM.
February 2009
Plamen
Turkedjiev
HU Berlin
What is Brownian motion?
Abstract.
Brownian Motion is a canonical process in stochastic analysis. Properties and existence will be discussed. Slides are available here.
Mareike
Massow
TU Berlin
What is Helly's theorem?
Abstract.
Helly's Theorem is one of the most famous results of a combinatorial nature about convex sets. It states that if we have $n$ convex sets in $\mathbb R^d$, where $n>d$, and the intersection of every $d+1$ of these sets is nonempty, then the intersection of all sets is nonempty. In preparation of Gil Kalai's BMS talk, we will see a basic proof of this theorem using (a basic proof of) Radon's Lemma. Hopefully we will also have a look at some application(s).
January 2009
Artem
Chernikov
HU Berlin
What is a Fraisse construction?
Abstract.
Take all finite subraphs of your infinite graph, now forget it and ask yourself whether you can recover the graph you started with just from this bunch of finite graphs or not? There is a precise and very simple answer, which is actually delivered by a general method of taking limits in certain categories called Fraisse construction. For example, the limit of all finite graphs is exactly the random graph, or say limit of all finite linear orders is the dense linear order (like in rationals). But you can apply the same procedure to groups, partial orders, metric spaces, fields and whatever else getting lots of fancy objects, sometimes well-known and sometimes totally new. I will show a couple of exotic species hopefully.
Fritz
Hörmann
HU Berlin
What is the Birch and Swinnerton-Dyer conjecture?
Abstract.
Elliptic curves — which can be given by equations of degree 3 (e.g. $X^3+Y^3=1$) — are the most interesting among all algebraic curves. It is an old question in number theory, called a Diophantine problem, to determine the set of rational points on such a curve. The elliptic case, again, is the most interesting and mysterious of all Diophantine problems of dimension 1. For example, there may or may not be infinitely many rational solutions. At present no known algorithm can determine this. However, already in the 60's, Birch and Swinnerton-Dyer experimentally found a deep and mysterious relation of this question to analytic properties of the zeta-function of the curve, which encodes the easily determined solutions of the congruences (e.g. $X^3+Y^3 \equiv 1 \pmod N$). It later became one of the most famous conjectures of mathematics, and is one of the millenium prize problems, for whose solution the Clay Mathematical Institute offers a reward of \$1,000,000.
Carsten
Hartmann
FU Berlin
What is an averaging principle?
Abstract.
The talk gives a high-level overview of problems that involve dynamical systems with slow and fast time scales. Prominent examples are the sun-earth-moon system in celestial mechanics or climate models in which the weather appears as a fast perturbation to the slowly varying climate. I will explain how the fast dynamics can be systematically eliminated from the equations of motion (averaging) thereby yielding closed-form equations that govern the motion the slow variables.
Matthias
Lenz
TU Berlin
What is the $\mathsf P$ versus $\mathsf{NP}$ problem?
Abstract.
Which boxes in the picture above should be chosen to maximize the amount of money while still keeping the overall weight under or equal to 15 kg? Problems of this type are called knapsack problems. They belong to the class of $\mathsf{NP}$ problems, which are in general very hard to solve. Let $\mathsf P$ be the class of problems that can be solved in polynomial time. $\mathsf P$ is a subset of $\mathsf{NP}$. One of the Millenium problems is to decide whether $\mathsf P$ equals $\mathsf{NP}$. Put differently, we want to know if there are problems whose answer can be quickly checked, but which require an impossibly long time to solve even on a supercomputer. We will define the classes $\mathsf P$ and $\mathsf{NP}$ using Turing machines, give plenty of examples and explain why the $\mathsf P$ vs. $\mathsf{NP}$ problem is important.
December 2008
Daria
Schymura
FU Berlin
What is similarity of shapes?
Abstract.
This talk is about shape matching, an interesting problem from Theoretical Computer Science. I will introduce the general matching problem, several classes of shapes and distance measures. I will also present results on how to compute the similarity of shapes.
Anna
Pippich
HU Berlin
What is spectral expansion?
Abstract.
Starting from the Christmas guitar we will study a very basic example to answer the above question and we will indicate some applications to number theory. Check also the attached abstract.
Peter
Krautzberger
FU Berlin
What is Hindman's theorem?
Abstract.
In the talk I will introduce Hindman's theorem, a result in (infinite) Ramsey theory important for both its content and its historical role. If time allows it, I will try to sketch the proof due to Galvin and Glazer using ultrafilters.
November 2008
Barbara
Jablonska
TU Berlin
What is geometric knot theory?
Abstract.
Using visualisation softare based on JReality I will introduce some basic notions of the classical (topologic) and geometric knot theory. I will also discuss some results from the latter field and if time allows I will present my recent result. The whole talk is ment to be very interactive and will teach you geometric knot theory mainly through graphics. No previous knowledge is required. Please note the attached PDF version.
Dror
Atariah
FU Berlin
What is loops and the fundamental group?
Abstract.
In this talk I will introduce an important tool for studying topological spaces. We will see how loops can characterize spaces, and define the fundamental group of a topological space.
Nathan
Ilten
FU Berlin
What is deformation?
Abstract.
I plan to give a concise introduction to deformations of singularities. After showing some very pretty pictures, I will define what a deformation is. Additionally, I hope to make remarks concerning concepts such as flatness, induced deformations, and versality. Time permitting, I will present Pinkham's famous example of the cone over the rational normal curve of degree 4.
October 2008
Anton
Dochtermann
TU Berlin
What is homotopical algebra?
Abstract.
Homotopy theory of topological spaces represents a rich and interesting interplay between relatively easy-to-define notions such as homotopy of maps, homotopy groups of spaces, fibrations, etc. In the sixties Quillen realized that topological structures like these could be encoded in a set of axioms which, if satisfied, allow one to talk about 'homotopy theory' in a more abstract setting. Any category that satisfies these axioms is called a '(closed) model category'. Although many instances arise in a geometric context, a perhaps surprising application of model categories is in a more algebraic setting: one of the early success of model categories was the proof that the combinatorial notion of 'simplicial sets' sufficiently 'models' the homotopy category of topological spaces. It turns out that chain complexes of modules also satisfy the axioms (this lead Quillen to the notion of 'homotopical algebra') and hence we can talk about such things as the 'suspension of a chain complex', etc. More recently model categories have been introduced in algebraic geometry in the context of '$A^1$ homotopy' of schemes. In this talk we will introduce the axioms for a model category and discuss a couple of examples and applications (among those mentioned above).
Stefanie
Frick
FU Berlin
What is the random graph?
Abstract.
The random graph can be defined on the set of natural numbers: For each pair $(i,j)$ a coin flip decides whether or not the two numbers are connected with an edge. The resulting graph is universal in the sense that every countable graph is contained as an induced subgraph. We will define the graph and use it to show the so called 0-1-laws. If the probability that a graph on $n$ nodes has a given property converges to 1 (for increasing $n$), we say that the property applies 'almost for all'. If the probability converges to 0, we say the property applies 'almost never'. The 0-1-law states the rather surprising fact that for all first order order statements always one of the two cases applies. The elegant and for a general audience accessible proof will be presented. It combines logical arguments (Gödelian completeness, compactness) with probabilistic considerations. The random graph has been studied in different setups, under different angles and in different disciplines. If there is time left, I will present some results in combinatorics. That is, we will consider colorings of edges and vertices of the random graph.
July 2008
Michael
Payne
FU Berlin
What is the chromatic number of the plane?
Gido
Scharfenberger-Fabian
FU Berlin
Was ist eigentlich das Kontinuum und was ist sein Problem?
June 2008
Peter
Krautzberger
FU Berlin
Was ist eigentlich eine sinnvolle Erweiterung der Halbgruppe der natürlichen Zahlen?
Felix
Breuer
FU Berlin
Was ist eigentlich eine sinnvolle Verallgemeinerung des Zwischenwertsatz?
Nathan
Ilten
FU Berlin
Was ist eigentlich eine Garbe?
Frederik
von Heymann
FU Berlin
Was ist eigentlich die Schälung eines Polytops?
May 2008
Tobias
Marxen
FU Berlin
Was ist eigentlich der Ricci-Fluss?
Peter
Krautzberger
FU Berlin
Was ist eigentlich ein Ultrafilter