Oberseminare und sonstige Vorträge

Automorphism groups of right-angled Artin groups (RAAGs) lie at the intersection of geometric group theory and algebraic topology and have been studied a lot. In this talk, we focus on computing the virtual cohomological dimension (vcd) of outer automorphism groups of RAAGs. I will present an algorithm recently developed by Day and Wade that enables such computations. The algorithm breaks the outer automorphism group down using short exact sequences and can be visualized as a tree-like structure, reflecting the recursive decomposition process. It is quite intricate, and I will illustrate the method using concrete examples.

Every (hyperfinite) subfactor comes with an invariant that measures relative noncommutativity for the ambient algebra over the subalgebra. I will explain this notion and show explicit examples that feature different degrees of noncommutativity. This structure reveals the existence of potentially interesting quantum channels.

Abstract: We briefly review the tautological ring of the moduli space of surfaces, before discussing how to define the analogous ring for the moduli space of 3-dimensional handlebodies (and more generally, odd-dimensional manifolds with boundary). We then discuss some sheaf-theoretic methods of probing the structure of this ring. The talk is based on ongoing work that will (hopefully) appear in my PhD thesis this summer.

Abstract: The ?-category Cond(Ani) of condensed anima combines the homotopy-theoretic direction of anima with the topological space direction of condensed sets. For example, we can recover the shape of a sufficiently nice topological space from the corresponding condensed anima. My talk will focus on a joint refinement of the etale homotopy type and the pro-etale fundamental group of a scheme, realised as an object in Cond(Ani). This is closely related to work of Barwick, Glasman and Haine on exodromy. I will give an overview of results from my dissertation and an ongoing project jointly with Haine, Holzschuh, Lara and Wolf.

Abstract: Given a connected compact Lie group G and a degree 4 cohomology class of its classifying space, one obtains a central extension of the loop group LG by a procedure known as ?transgression?. K. Waldorf proved that this procedure is reversible if one remembers the ?fusion structure? on the central extension. In this talk, I will explain an analogue of Waldorf?s theorem in the context of algebraic geometry.

Guentner, Willet and Yu defined a notion of dynamic asymptotic dimension for an etale groupoid that can be used to bound the nuclear dimension of its groupoid C*-algebra. To have finite dynamic asymptotic dimension, the isotropy subgroups of the groupoid must be locally finite. I will discuss 1) how to use similar ideas to bound the nuclear dimension of the C*-algebra of a groupoid with `large' isotropy subgroups and 2) the limitations of that approach. In an application to the C*-algebra of a directed graph, if the C*-algebra is stably finite, then its nuclear dimension is at most 1. This is joint work with Dana Williams.