Oberseminare und sonstige Vorträge

In the first part of the talk I will motivate the Breuil--Mezard conjecture, which describes the geometry of moduli spaces of mod p Galois representations in terms of the representation theory of groups like GLn(Fq). In the second part I will explain how new cases of this conjecture can be proven by describing explicit subschemes inside the affine grassmannian whose geometry locally models the geometry of moduli spaces of Galois representations.

This semester we have had several talks on positive scalar curvature already and in this talk we will explore yet another aspect: the question of uniqueness of psc metrics. Phrased properly this is the study of the homotopy type of the space of psc metrics. One of the main tools used in its study is the index difference by Hitchin, which provides a map to K-theory. This map is known to be surjective on homotopy groups if the dimension of the manifold is at least 6. In my talk I will explain the methods used to prove this non-triviality result and present enhancements, which allow for a simultaneous extension to dimension 5 and incorporation of the fundamental group (yielding an index difference mapping to the K-theory of group C*-algebras). These enhancements use the original Madsen?Weiss theory and its extension by Perlmutter.

This talk is about groupoids and C*-algebras generated by left regular representations of certain left cancellative small categories called Garside categories. I will explain these objects and constructions with the help of many concrete example classes. I will also give an overview of structural results for C*-algebras, groupoids and topological full groups arising from Garside categories.