In this talk, I am going to explain the main results of my recent preprint (arXiv:2105.12591). The primary goal will be to prove that for every affinoid analytic adic space $X$, pseudocoherent complexes, perfect complexes, and finite projective modules over $\mathcal{O}_X(X)$ form a stack with respect to the analytic topology on $X$. The proof relies on the new approach to analytic geometry developed by Clausen and Scholze by means of condensed mathematics; therefore, I will also explain how to apply their formalism of condensed analytic rings to the study of adic geometry.

We give an outline of a (conjectural) construction of cohomology groups for smooth and proper varieties over local fields with values in the derived category of locally compact groups satisfying a Pontryagin duality. For certain weights, we give an ad hoc construction which satisfies such a duality unconditionally. We then explain how this leads to … Continue reading Thomas Geisser (Rikkyo University Tokyo): Duality for motivic cohomology over local fields and applications to class field theory →

In classical algebra, the prime fields are Q and for every prime number p the finite field F_p. In higher algebra, one has for every prime number p an additional sequence of prime fields K(p,n), n a natural number, which in some sense interpolates between Q and F_p. Associated with these prime fields one has … Continue reading Georg Tamme (Universität Mainz): Purity in chromatically localized algebraic K-theory →

Yujie Xu (Harvard University)
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The specialization morphism for the étale fundamental groups of Grothendieck cannot be generalized word-for-word to the more general pro-\'etale fundamental group of Bhatt and Scholze. It turns out, that one can deal with this problem by applying a rigid-geometric point of view: for a formal scheme X of finite type over a complete rank one valuation ring, … Continue reading Dr. Marcin Lara: Specialization for the pro-étale fundamental group and fundamental groups in rigid geometry →

Thomas Nikolaus (Münster
Segal's Burnside ring conjecture and a generalization for norms