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.

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## Programm discussion

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## 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.

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## Program discussion

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## 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.

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