Veranstaltungen am Mathematischen Institut

Equipments, a special kind of double categories, have shown to be a powerful environment to express formal category theory. We build a model structure on the category of double categories and double functors whose fibrant objects are the equipments, and combine this together with Makkai?s early approach to equivalence invariant statements in higher category theory via FOLDS (First Order Logic with Dependent Sorts) and Henry?s recent connection between model structures and formal languages, to show a result on the equivalence invariance of formal category theory.

Recent years have seen substantial progress in understanding algebraic K-theory of singular varieties. Still, computations of (higher) K-theory of singular varieties are quite rare, especially in dimension >1. In this talk, I will explain some techniques that allow one to perform computations in higher dimensions, with a particular focus on cubic surfaces and threefolds, where complete computations can be obtained in many cases.

It is well-established that Patterson-Sullivan measures on the boundary of a hyperbolic space, along with the associated Bowen-Margulis-Sullivan measure, provide valuable insights into the action of a group of isometries on the space's boundary through analysis of the geodesic flow. Given that paths of a random walk on a hyperbolic groups lie close to the group's quasi-geodesics, it is natural to ask whether similar behaviour can also be seen in the flow along bi-infinite random walk paths. In this talk, I will give an insight into classical Patterson-Sullivan theory, and show how studying bi-infinite random walks on a discrete hyperbolic group $G$ leads to an analogue of the Patterson-Sullivan measure on $\partial^2G$. This talk is based on joint work with Mahan Mj and Chiranjib Mukherjee.