Vorträge des SFB 1442

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.

This talk is based on a joint work in progress with Gurbir Dhillon. A remarkable theorem of Feigin and E. Frenkel from the early 90's describes the center of the universal enveloping algebra of an (untwisted) affine Kac-Moody Lie algebra at the so called critical level proving a conjecture of Drinfeld: the center in question is the algebra of polynomial functions on an infinite dimensional affine space known as the space of opers. In our work we study a part of the center in positive characteristic p at an arbitrary non-critical level. Namely, we prove that the loop group invariants in the completed universal enveloping algebra is still the algebra of polynomials on an infinite dimensional affine space that is ``p times smaller than the Feigin-Frenkel center''. In my talk I will introduce all necessary notions, state the result, explain examples and, time permitting, motivations.

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: 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.

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.