Oberseminare und sonstige Vorträge

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.

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.

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.

Abstract: Let G be a measurable semigroup. Consider a stationary ergodic Markov chain ? = (?n)n ?? taking values in G. We are interested in conditions on ? ensuring that for every measure preserving and ergodic action {Tg}g G? of G on a probability space (X,?) and every f ? L1(X) the random averages (f + f T? ?0 + ... + f T? ?0...?N-2)/N converge a.s. to the integral ? f d?. Considering the shift map S on the path space (?,P) of the Markov chain we may associate to every such action {Tg}g G? the skew product T of S and {Tg}g G? . This allows us to reduce the above problem to the question whether for every ergodic action {Tg}g G? of G the respective skew product T is also ergodic. For iid processes ? this question is answered in the affirmative by a classical result of Kakutani. As a consequence on obtains Kakutani?s well known random ergodic theorem, which has found wide applications in the study of random walk dynamics. For finite state Markov chains Bufetov introduced the condition of strict irreducibility and showed that Markov chains satisfying this condition also satisfy the above property. Such Markov chains arise for instance naturally from Markov codings of certain word hyperbolic groups such as free groups and Fuchsian groups. In this talk I will explain how the notion of strict irreducibility can be extended to Markov chains with arbitrary state spaces. The main result I will present shows that the strictly irreducible Markov chains are precisely the Markov chains satisfying the above property. This provides us with random ergodic theorems for Markovian averages of general semigroup actions generalizing Kakutani?s as well as Bufetovs results. The talk is based on joint work with Pablo Lummerzheim and Felix Pogorzelski.

This talk is based on joint work with Fredrik Viklund. In the classical Hele-Shaw flow a domain in the complex plane grows according to the gradient of its Green's function, thus modelling the propagation of a viscous fluid trapped in a thin layer. We introduce a competitive version of the flow where several domains in the complex plane (or more generally in a Riemann surface of finite type) similarly strive to expand but at the same time hinder each other. Interestingly, stationary flows correspond to a special class of quadratic differentials whose associated half-translation surfaces have a simple description. We also introduce a discrete model, closely related to Propp's competitive erosion model, which conjecturally allows us to simulate the flow.