Testbrett

Given some large positive integer N, what is the largest possible size of a subset of {1,...,N} which does not contain a three-term arithmetic progression (i.e. without three distinct elements x,y,z satisfying x+z=2y)? Similarly, given a prime p and a large positive integer n, what is the largest possible size of a subset of the vector space F_p^n which does not contain a three-term arithmetic progression (i..e without three distinct vectors x,y,z satisfying x+z=2y)? These are long-standing problems in additive combinatorics. This talk will explain the known bounds for these problems, give an overview of some of the proof techniques, and discuss additional applications of these techniques to other additive combinatorics problems.

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Optimal transport provides an intuitive and robust way to compare probability measures with applications in many areas of mathematics. This holds in particular for the Wasserstein-2 distance with its formal Riemannian structure. While entropic regularization of optimal transport has several favourable effects, such as improved statistical sample complexity, it destroys this metric structure. The de-biased Sinkhorn divergence is a partial remedy, as it is positive, definite, and its sublevel sets induce the weak* topology. However, it does not satisfy the triangle inequality. We resolve this issue by considering the Hessian of the Sinkhorn divergence as a Riemannian tensor and study the induced distance. In this talk we outline the key steps of this construction, the corresponding induced notion of tangent space, some early results on the distance, and open directions for future work.

Problems involving multiple scales are ubiquitous in applications. The typical situation, for which analytical as well as numerical methods are quite well developed, is a partial differential equation (PDE) posed on a (macroscopic) domain with coefficients that oscillate on a much smaller (microscopic) scale. In this talk, we consider two areas where additional scales of oscillations occur and the inter-relation of the different scales crucially influences the overall behavior. The first example is wave propagation in spatial multiscale media with a high contrast, where resonances can cause unusual wave phenomena such as all-angle band gaps. The second example are spatial multiscale media with additional temporal variations, in the heat or wave equation. For both examples, we discuss the design and analysis of numerical methods to capture these effects.

The topology and total curvature of a Riemann surface is determined by a single integer, the Euler characteristic (Leonhard Euler, 1707-1783). While this is not the case in dimension four, the Euler characteristic still gives an interesting invariant for finitely presented groups. For example, what is the minimum possible value for the Euler characteristic of a closed 4-manifold with a given fundamental group ? The talk will survey some recent joint work with Alejandro Adem on this theme.

Following an introduction to general relativity, the Einstein field equations, and an overview of recent results on the stability of black holes, I will discuss the construction of singular perturbations of spacetimes via the insertion of small black holes. This provides the first rigorous examples of spacetimes describing the merger of two black holes with extreme mass ratios.

Stochastic geometry is the branch of probability theory dealing with spatial random structures such as random tessellations, random sets, random polytopes or spatial random graphs. Such objects are often constructed from underlying point samples. In many cases and also throughout this talk, it is assumed that these points are given by a Poisson process. Thus, quantities of interest are random variables depending only on a Poisson process, so-called Poisson functionals. Since random geometric structures and associated random variables usually exhibit an extremely complex behaviour, which does not admit explicit finite size descriptions, one studies the asymptotic behaviour as the number of underlying points tend to infinity. In order to establish central limit theorems for this situation, one is interested in approximating distributions of Poisson functionals by normal distributions. A powerful tool to establish such results is the Malliavin-Stein method, which will be discussed in this talk. It combines Stein's method, a collection of techniques to derive quantitative limit theorems, with Malliavin calculus, a variational calculus for random variables. To illustrate the use of the Malliavin-Stein method, some problems from stochastic geometry will be considered.

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