Veranstaltungen am Mathematischen Institut

Abstract: Many equivariant phenomena arise uniformly in families of groups, giving rise to so-called global objects. Such objects can be efficiently described using the language of parametrized homotopy theory, which allows one to characterize constructions in (global) equivariant homotopy theory via universal properties. I will discuss parametrized categories of G-(global) spaces and G-(global) spectra for a compact Lie group G, construct a Thom spectrum functor in this setting, and explain that many classical Thom spectra admit multiple equivariant enhancements. This is joint work with Tobias Lenz.

Abstract: The cohomology ring of moduli spaces of manifolds is an important object in geometric topology, as it classifies characteristic classes of manifold bundles. For even-dimensional manifolds, Galatius and Randal-Williams established a complete homotopy theoretic formula for this object after a certain stabilization procedure. In this talk, I will explain an odd-dimensional analog of Galatius and Randal-Williams' work in the context of odd-dimensional manifold triads. After stating this result, I will explain the strategy of proof and discuss some applications and examples.

Pureness is a regularity property that plays a prominent role in the structure of C*-algebras. It can be thought of the $\mathcal{Z}$-stability at the level of the Cuntz semigroup of the algebra and at the same time is proved to be equivalent to certain weak comparison and divisibility properties. This characterization implies that every simple, non-elementary C*-algebra with a unique quasitrace and mild comparison has strict comparison. The characterization is also used to show that the class of pure C*-algebras behaves well under extensions. Further, for a separable C*-algebra $A$, one has that $\mathcal{Z}$-stability is equivalent to demanding that the uncorrected central sequence algebra of $A$ is pure, which in turn is equivalent to saying that said algebra is almost divisible. This is joint work with Ramon Antoine, Hannes Thiel, and Eduard Vilalta; and with Hannes Thiel and Eduard Vilalta.