Alternatively have a look at the program.

## Conley index theory and condensed sets

The Conley index is a spatial refinement of the Morse index. Informally speaking, it is a ‘space’ that describes the local dynamics around an isolated invariant subset of a topological dynamical system. In this talk, I will explain a new formulation of Conley index theory, which I think is simpler and more flexible than the traditional formulation. One important point is that the Conley index should be defined as a based equivariant condensed set/anima, not as a mere homotopy type of topological spaces.

## Quasicoherent Sheaves on Perfectoid Spaces

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.

## Condensed Shape of a Scheme

The $\infty$-category Cond(Ani) of condensed anima combines homotopy theory 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. In my talk, I will focus on explaining how to define a refinement of the étale homotopy type of a scheme as an object in Cond(Ani) following constructions from Shape Theory.

## Embracing condensed mathematics for the study of cohomological finiteness conditions of profinite groups

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.

## Continuous K-theory and polynomial functors

Waldhausen K-theory can be defined as an invariant of small stable $\infty$-categories, and is characterized (following work of Blumberg-Gepner-Tabuada and Barwick) by the property that it converts Verdier quotient sequences of stable $\infty$-categories to fiber sequences. Recently, Efimov defined the continuous K-theory of the larger class of dualizable presentable stable $\infty$-categories, which is important for applications to the K-theory of analytic spaces.

## The de Rham stack for rigid analytic varieties

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.

## Condensed math $\&$ exodromy

This talk will continue discussing the condensed homotopy type of a scheme introduced in Catrin’s talk. We’ll explain how to access the étale homotopy type and condensed homotopy types of schemes from the perspective of exodromy. We’ll also talk about work in progress with a number of different people that makes use of this condensed homotopy type.

## Algebraic bivariant K-theory and nuclear modules

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.

## Localizing Invarints in Adic Geometry

## Algebraic $K$-theory and the telescope conjecture

© MPI f. Mathematik, Bonn | Impressum & Datenschutz |