Vorträge des SFB 1442

Abstract: I will discuss an equivariant version of Quillen's theorem that the complex bordism ring carries the universal formal group law, both over a fixed abelian group and in a global equivariant setting

Abstract: A standard question in group theory is to ask if we can categorise the subgroups of a group in terms of their growth. In this talk we will be asking this question for uniform product set growth, a property that is stronger than the more widely studied notion of uniform exponential growth. We will see how quasi-trees could help us answer this question for acylindrically hyperbolic groups, and give a particular application to right-angled Artin groups

A minimal surface is one whose area is stationary under small perturbations. If you dip a loop of wire in soapy water and pull it out, the soap film will naturally form a minimal surface. The Plateau problem is to find a minimal surface which bounds a given closed loop. From the point of view of analysis, this involves solving a non-linear PDE. I will explain how counting the number of solutions to a version of the Plateau problem can lead to knot invariants. Given a knot K in the three-sphere, one can count minimal surfaces in 4-dimensional hyperbolic space which are asymptotic at infinity to K and the answer does not change when you deform the knot. I will explain a conjecture that this count of minimal surfaces actually recovers the "HOMFLYPT" polynomial of the knot. This polynomial is purely combinatorial and simple to calculate from a planar diagram of the knot. If the conjecture is true, it would mean that easy combinatorial calculations can give existence of minimal surfaces! I will assume no knowledge of either minimal surfaces or knots. There will be lots of pictures

ZOOM-Link: https://wwu.zoom.us/j/65940821521?pwd=cFB6cVUwSDhtSFloaStiUUJpQzJ6Zz09#success Meeting-ID: 65940821521 Kenncode: 516480

Abstract: One-relator groups G=F/<> pose a challenge to geometric group theorists. On the one hand, they satisfy strong algebraic constraints. On the other hand, they are not susceptible to geometric techniques, since some of them ? such as the famous Baumslag?Solitar groups ? exhibit extremely pathological behaviour. I will relate the subgroup structure of one-relator groups to a measure of complexity for the relator w introduced by Puder ? the *primitivity rank* \pi(w). A sample application is that every subgroup of G of rank < \pi(w) is free. These results in turn provoke geometric conjectures that suggest a beginning of a geometric theory of one-relator groups. This is joint work with Larsen Louder.

We introduce the macroscopic point of view for questions around scalar curvature. Then we discuss the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, 3) a conjectural bound of ?²-Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. Group actions on Cantor spaces surprisingly play an important role in the proof. The talk is based on joint work with Sabine Braun.