Kolloquium Wilhelm Killing

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.

Ricci curvature is ubiquitous in mathematics: it appears in Hamilton's Ricci flow (a key tool in Perelman's resolution of the Poincaré conjecture), as well as in Einstein's equations of general relativity. Understanding its interplay with the global shape of Riemannian manifolds has been one of the key broad themes in geometric analysis since its early developments. While this interplay is well understood for manifolds with dimensions less than or equal to 3, several questions remain in dimension 4. After a gentle introduction to Ricci curvature, I will discuss joint work with Elia Bruè and Alessandro Pigati, in which we prove that any Riemannian 4-manifold with nonnegative Ricci curvature and Euclidean volume growth looks like a cone over a spherical space form at infinity. I will provide all the background needed for the precise statement, explain in which sense it is optimal, and explain why one might expect it to be true.

tba

We consider suspensions of many small rigid particles in a fluid. The theoretical study of the effective rheology, i.e. fluid properties, of such complex flows dates back to Einstein's doctoral research, where he predicted an effective increase of the fluid viscosity due to the presence of the particle. Despite its relevance for many applications and a vast physics and engineering literature on the topic, a mathematical rigorous derivation of effective rheology has been elusive until recent years. In this talk, I give an overview of results that not only justify Einstein's formal calculations but go far beyond. In particular, fascinating non-Newtonian effects will be presented.

tba

tba

tba

tba