Vorträge des SFB 1442

In joint work, very much in progress, with Johannes Anschütz, Juan Esteban Rodriguez Camargo and Peter Scholze, we define an analogue of prismatic cohomology for rigid analytic varieties. Our construction is formulated using analytic stacks and takes inspiration both from Scholze's theory of diamonds and the work of Bhatt-Lurie and Drinfeld on the prismatization of p-adic formal schemes. It also furnishes a potential formulation of the geometrization of the p-adic local Langlands correspondence. In this talk, I would like to survey various aspects of this work.

I will talk about my recent result regarding the separability of products of subgroups in virtually special cubulated groups. My talk will also contain lots of background on cube complexes, cubulated groups and (virtual) specialness, which has been an important topic in geometric group theory over the last 20 years, particularly with the connection to 3-manifold theory.

Abstract: In this talk we will discuss the structure of $\mathbb E_\infty$-monoids on which a prime $p$ acts invertibly, which we call $p$-perfect. In particular, we prove that in many examples, they almost embed in their group-completion. We further study the $p$-perfection functor, and describe it in terms of Quillen's $+$-construction, similarly to group completion. This is joint work with Maxime Ramzi.

If we complete the rational numbers $\mathbb{Q}$ with respect to the standard absolute value, we obtain the reals $\mathbb{R}$, wich is a connected topological space. The completion $\mathbb{Q}_p$ of $\mathbb{Q}$ with respect to the $p$-adic valuation for a prime $p$, however, is totally disconnected. So it seems that the concept of a path connecting two points in the $p$-adic numbers $\mathbb{Q}_p$ (or $\mathbb{Q}_p^n$) does not make sense. In fact, the space $\mathbb{Q}_p$ itself is not quite suitable for doing geometry. Instead one can consider the affine line $\mathbb{A}_{\mathbb{Q}_p}^1$ as an adic space. It contains $\mathbb{Q}_p$ as so called \emph{classical points} but has many more points. In this talk we will convince ourselves that $\mathbb{A}_{\mathbb{Q}_p}^1$ is indeed path connected if we use the right notion of a path.

Abstract: Llarull proved in the late '90s that the round n-sphere is area-extremal, meaning that one cannot simultaneously increase both its scalar curvature and its metric. Goette and Semmelmann generalized Llarull's rigidity statement to certain area-non-increasing spin maps $f\colon M\to N$ of non-zero $\hat{A}$-degree. In this talk, I will begin with a brief introduction to scalar curvature comparison geometry and review the Dirac operator method. I will then explain how higher index theory can be used to generalize classical extremality and rigidity statements. More specifically, I will present a recent generalization of Goette and Semmelmann?s theorem, in which the topological condition on the $\hat{A}$-degree is replaced by a weaker condition involving the so-called higher mapping degree. A key challenge in the proof is that, in general, a non-vanishing higher index does not necessarily give rise to a non-trivial kernel of the corresponding Dirac operator. I will present a new method that extracts geometrically useful information even in this more general setting.

Abstract: The Chang-Skjelbred method is a popular technique to compute the the (equivariant) cohomology of a space with a certain type of torus action. While traditionally employed for coefficients in fields of characteristic 0, the strategy is apt for integral calculations as well under the right conditions. We will give an overview over the method and in particular discuss integral developments and applications.

A group is called W*-superrigid if its group von Neumann algebra completely remembers the original group. In this talk I will present a joint work with Stefaan Vaes in which we generalize W*-superrigidity for groups in two directions. Firstly, we find a class of groups for which W*-superrigidity holds in the presence of a twist by an arbitrary 2-cocyle: the twisted group von Neumann algebra completely remembers both the original group and the 2-cocycle. Secondly, for the same class of groups, the superrigidity also holds up to virtual isomorphism. If time permits, I will moreover explain how this result and the technical machinery behind it led to other new W*-superrigidity theorems in two different contexts.