Abstracts for Arbeitstagung 2023 on Condensed Mathematics

We present v-descent results for quasicoherent sheaves on perfectoid spaces and show some applications. For sheaves over $\mathcal O^{+a}/\pi$ we obtain a 6-functor formalism for p-torsion étale cohomology on diamonds and v-stacks. If one is willing to slightly modify the definition of solid modules over adic rings, then similar descent results can be established for $\mathcal O^{+a}$-modules and $\mathcal O$-modules, thus providing 6-functor formalisms for completed cohomology and pro-étale $\mathbb{Z}_p$- and $\mathbb{Q}_p$-cohomology. This is ongoing joint work with J.

A toe in the condensate: We study cohomological finiteness conditions for profinite groups by using Clausen-Scholze condensed mathematics. This approach allows one to emulate known strategies for discrete groups. We need only the most elementary constructions in condensed maths up to the notion of solidification. We outline how this approach can be used to complete the proof of a conjecture that is stated in the Ribes-Zalesski book on cohomology of profinite groups.

The de Rham stack of a smooth scheme of finite type over a field of characteristic zero is a geometric object that encodes the theory of $D$-modules in its theory of quasi-coherent sheaves. In this talk, using analytic geometry of Clausen and Scholze, I will explain how to construct similar de Rham stacks for rigid spaces (and some relatives) that encode the theory of solid $\hat{D}$-modules and locally analytic representations of $p$-adic Lie groups.

We will introduce the purely algebraic version of Kasparov's KK-theory via the category of localizing motives. Namely, this category is the target of the universal localizing invariant of small stable infinity-categories (over some base ring spectrum), commuting with filtered colimits. Now, the KK-theory spectrum for a pair of categories $A$ and $B$ is just the spectrum of morphisms from the motive of $A$ to the motive of $B$. We will explain how to compute this KK-theory, in particular the K-homology.