Oberseminare und sonstige Vorträge

The use of Baire category methods to study generic properties of group actions traces back to the beginnings of ergodic theory. In this talk, I will discuss recent results on generic properties of free group actions on the Cantor set. While motivated by questions in the theory of C*-algebras, the talk will not assume any background on this subject. Based on joint work with Shirly Geffen and David Kerr.

Abstrakt: Scalar curvature is a very weak and flexible local invariant in Riemannian geometry. So it is surprising that coupling lower scalar curvature bounds with seemingly harmless topological conditions can recover very powerful rigidity statements, such as Llarull?s theorem or the positive mass theorem. In this talk I will begin by introducing some aspects related to the comparison geometry of lower scalar curvature bounds, focussing in particular on Llarull?s theorem. After this I will discuss a recent rigidity theorem in this setting. This is joint work with Thomas Tony.

Abstract: In the higher-categorical setting, Galois theory extends beyond finite groups: one can study Galois extensions for an arbitrary E_1-group. We develop such a theory for higher semiadditive categories, where finite groups are naturally replaced by pi-finite groups. Under suitable height assumptions, we construct an n-truncated pro-pi-finite "absolute Galois group" classifying all Galois extensions. We also relate abelian extensions to the Picard spectrum through a higher version of Kummer theory. Time permitting, we will compute the absolute Galois group in some chromatic and categorified settings.

Abstract: A longstanding problem in differential geometry, known as the Bonnet problem, is whether a compact surface is uniquely determined by its metric and mean curvature function. It is known that this is the case for generic surfaces, and also for topological spheres. We explicitly construct a pair of immersed tori in three-dimensional Euclidean space that are related by a mean curvature preserving isometry. These tori are the first examples of compact Bonnet pairs. Moreover, we prove these isometric tori are real analytic. This resolves also a second longstanding open problem on whether real analyticity of the metric already determines a unique compact immersion. Discrete differential geometry is used to find crucial geometric properties of surfaces. This is a joint work with Tim Hoffmann and Andrew Sageman-Furnas.