Oberseminare und sonstige Vorträge

I will discuss an approach to the Dwyer-Weiss-Williams index theorems for topological and for smooth manifold bundles, which is based on a formalism for bivariant theories and known results about cobordism categories. In the smooth case, the theorem implies that a canonical transformation from stable homotopy to the algebraic K-theory of spaces is natural with respect to transfer maps. I will then present some results on the analogous question for Waldhausen's splitting map from the algebraic K-theory of spaces to stable homotopy. This splitting map gives a splitting of the algebraic K-theory of spaces into stable homotopy and the Diff-Whitehead space. If time permits, I will present some results on the homotopy type of the h-cobordism category and discuss its connection with the Diff-Whitehead space. (This is joint work with W. Steimle.)

We introduce the notion of stationary actions in the context of C*-algebras, and prove a new characterization of C*-simplicity in terms of unique stationarity of the canonical trace. This ergodic theoretical characterization provides an intrinsic understanding for the relation between C*-simplicity and the unique trace property, and provides a framework in which C*-simplicity and random walks interact.
Joint work with Mehrdad Kalantar.

Branching random walks are systems of particles which branch and diffuse randomly. They arise in many different contexts such as population models, disordered systems in statistical physics or reaction-diffusion equations. We speak of a heterogeneous environment if the reproduction or the motion of the particles depends on space and/or time on a macroscopic scale. This extra freedom is useful from a modelling perspective, moreover, these models exhibit a much wider range of behaviour than the classical branching random walk. In this talk, I will review their basic properties and present two recent results. The first concerns the efficiency of algorithms for finding low-energy states in Derrida?s continuous random energy model, a Gaussian branching random walk with time-dependent variance. The second concerns the speed of propagation of branching random walks with local selection in heterogenous environment. Based on joint work with (first) Louigi Addario-Berry and (second) Gaël Raoul and Julie Tourniaire.

Any group G of permutations can be endowed with the so called standard topology, the group topology in which a system of neighbourhoods of the identity consists of the collection of all fix-point stabilizers of finite sets and in case G is the automorphism group of a countable structure M it is a Polish group topology. For certain classes of highly homogeneous M such as the random graph or the dense linear order this yields interesting examples of Polish groups with remarkable dynamical properties. Here we take a look at the question of minimality, i.e. of whether there are no Hausdorff group topologies on G strictly coarser than the standard topology. We present a couple of new minimality results in case M is the Fraïssé limit of a class with free amalgamation, as well as for the isometry group of the Urysohn space with the point-wise convergence topology. (Joint work with Zaniar Ghadernezhad)