Termine

Solving equations over groups while satisfying first-order conditions is difficult, yet necessary when studying elementary classes of expansions of groups. For certain groups coming from pure group theory, o-minimality or commutative algebra, we may rely on valuations that behave well with respect to commutativity on such groups. I will explain how to use valuations in order to solve equations over groups.

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.

Further information

Abstract: It is a curious fact of life in geometric topology, that the classification of closed manifolds by surgery theory becomes easier as one passes from smooth to piecewise linear and finally to topological manifolds. It was long conjectured that an even cleaner statement should be expected in the somewhat arcane world of homology manifolds of the title, which ought to fill the role of some "missing manifolds". This was finally proven by Bryant, Ferry, Mio and Weinberger in the 90's in the form a surgery sequence for homology manifolds, building on an earlier theorem of Ferry and Pedersen that any homology manifold admits a euclidean normal bundle. In the talk I will try to explain this surgery sequence, and further that its existence is incompatible with the result of Ferry and Pedersen. The latter is therefore incorrect and/or the proof of the former incomplete.

Let G be a group and R a set of elements of G with normal closure N. In general, it is not easy to understand the quotient G/N. Small Cancellation Theory is, broadly speaking, a family of conditions of the following form: Let G be a negatively curved group, R a family of ?independent enough? elements. Then G/N is itself negatively curved, and we understand many properties of this quotient. In this talk, I will review the classical Small Cancellation Theory, as well as some of its variants (Graphical Small Cancellation and Geometric Small Cancellation), and exhibit some of the groups with exotic properties that can be constructed using these methods

We review characterization of lower Ricci curvature bounds. Of particular interest for us are characterization which generalize to nonsmooth spaces. We further investigate in Ricci bounds, i.e. we combine lower with upper curvature bounds.

We will discuss a topological version of Kesten's theorem and its connections with the asymptotic properties of group actions on the orbits of an amenable countable Borel equivalence relation. The talk is based on a joint work with Kate Juschenko and Friedrich Martin Schneider.