Veranstaltungen am Mathematischen Institut

Abstract: Chern characters are natural transformations from K-theory like coarse homology theories to coarse homology theories derived from chain-complex constructions. Transgressions are natural transformations from coarse homology theories to functors applied to the Higson corona. I will explain instances of such constructions and how they are possibly related. One goal of this talk is to highlight the richness of the world of coarse homology theories and its connection with interesting developments, classical and modern, in other fields.

We propose a generalization of K-theory to operator systems. Motivated by spectral truncations of noncommutative spaces described by C*-algebras and inspired by the realization of the K-theory of a C*-algebra as the Witt group of hermitian forms, we introduce new operator system invariants indexed by the corresponding matrix size. A direct system is constructed whose direct limit possesses a semigroup structure, and we define the K0-group as the corresponding Grothendieck group. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. For C*-algebras it reduces to the usual definition. We illustrate our invariant by means of the spectral localizer.

Abstract: The signature of a closed manifold is an important invariant in geometric topology. Recently, Higson, Xie, and Schick proved an invariance theorem for the K-theoretic signature in codimension 2. However, the L-theoretic counterpart of this result remains an open problem. In this talk, I will describe how to achieve the L-theoretic counterpart of their result and provide some examples. I will also explain the basic ideas of the proof, highlighting the key steps and challenges involved.

Abstract: The Higman-Thompson groups consist of certain piecewise linear automorphisms of intervals. Szymik and Wahl prove homological stability for this family of groups, and compute the stable homology to be that of the infinite loop space of the Moore spectrum. We give a new proof of this result using scanning methods on a topological model for the disjoint union of these groups, using Thumann's framework of operad groups.