GAUS-Workshop "Condensed Mathematics and K-Theory"

Workshop organised by Christian Dahlhausen and Georg Tamme and financially supported by CRC326 - GAUS: "Geometry and Arithmetic of Uniformizing Structures".

Background

In the last decade, the systematic study of continuous and analytic K-theory of non-archimedean rings and spaces led to several groundbreaking results in algebraic K-theory. On the other hand, the existing definitions and constructions of the K-theory of non-archimedean rings are unsatisfying from a conceptual point of view. A promising new approach is the use of the language of condensed mathematics for these constructions. The aim of this event is to bring together researchers working in K-theory, condensed mathematics, or related fields within an informal atmosphere with the goal of exchanging ideas and identifying open questions for future research.

Time & Place

The workshop takes place from Monday, 27th February until Wednesday, 1st March, 2023 at Universität Mainz. There will be a social Welcome Event on Sunday, 26th February; meeting point is 19:30 in the lobby of Intercity Hotel (Binger Straße 21, 55131 Mainz). All talks are in room 05-426 at the Institute of Mathematics (Staudingerweg 9, 55128 Mainz). On Monday and Tuesday, the coffee breaks take place at Hilbertraum next to the seminar room. On Wednesday, the coffee break takes place at Siegelscher Halbraum on the 4th floor.

Schedule

 Time Monday Tuesday Wednesday 08:00--09:00 Registration 09:00--10:00 Welcome Coffee Discussions Discussions 10:00--11:00 Andreychev Tamme Wolf 11:00--11:30 Coffee Break Coffee Break Coffee Break 11:30--12:30 Mair Dahlhausen Bräunling 12:30--14:00 Lunch Break Lunch Break Lunch Break 14:00--15:30 Discussions Discussions Discussions 15:30--16:30 Brink Aoki 16:30--17:00 Coffee Break Coffee Break 17:00--18:00 Efimov Cordova Fedeli 19:00-- Dinner Dinner

Titles & Abstracts

Grigory Andreychev: K-theory of adic spaces

Ko Aoki: Analytic K-theory: an attempt in the archimedean case
This is a progress report on my thesis work in progress, which is about analytic K-theory in the archimedean case. I start by describing the expected picture for analytic K-theory of liquid quasicoherent sheaves on complex analytic spaces or differentiable manifolds. Next, I outline my first approach, which defines K-theory of an arbitrary analytic ring as the universal localizing invariants for enriched categories. I first discuss some structural results, including how it can be written using the usual algebraic K-theory. Then I describe a specific result in the liquid case, which connects the notion of truncatedness and Bott periodicity. If time permits, I describe a different approach of the K-theory of an analytic ring based on very nuclear modules. I explain why this new approach may be more promising than the enriched approach.

Oliver Bräunling: The K(1)-local dual of the K-theory of S-integers
Before Selmer K-theory and Condensed Maths hit the planet, Clausen gave us his PhD thesis and his nice 2017 preprint on K-theoretic Artin maps. I´ll talk about this from my personal angle and update some parts of the story to 2023 (eg the Blumberg-Mandell paper). I propose to think about it as a K(1)-local duality between the K-theory of finitely generated modules versus locally compact (as in "LCA") topologized ones. This cross-connects in various ways to aspects of class field theory which previous generations of mathematicians have addressed by using topologies. The result is a proper mess. I´ll address some points which might deserve a condensed overhaul.

Emma Brink: Condensed Group Cohomology
We compare condensed group cohomology to continuous group cohomology and show that for a large class of groups, continuous group cohomology with solid coefficients can be described as a cohomological $\delta$-functor in the condensed setting. This is based on results of Johannes Anschütz and Arthur-César Le Bras who discussed the case of locally profinite groups.

Adriano Cordova Fedeli: Localising invariants for nuclear categories

Christian Dahlhausen: Representability of continuous K-theory
Algebraic K-theory of smooth schemes (over a regular noetherian base scheme) is representable within Morel-Voevodsky's unstable motivic homotopy category, wherein the affine line is contractible. For rigid analytic spaces, Ayoub developped an analoguous theory wherein the closed unit ball B^1 is contractible. Within Ayoub's category, Morrow's continuous K-theory and Kerz-Strunk-Tamme's analytic K-theory are not representable for two reasons: First, they are not B^1-invariant and, secondly, the mapping objects are not pro-spaces. In this talk, I will sketch the construction of an unstable condensed motivic homotopy category wherein the rigid affine line is contractible and whose mapping objects are condensed spaces. I will explain how this yields that -- after passing from pro-spaces to condensed spaces -- continuous K-theory and analytic K-theory shall be representable. In future work, I intend to employ this representablity in order to study Adams operations, similarly to Riou's work on Adams operations on higher algebraic K-theory. In the long run, this might be useful for studying the problem of lifitng cycles as studied by Bloch-Esnault-Kerz.

Alexander Efimov: On the category of localizing motives
I will explain some new results about the category of localizing motives. This category is the target of the universal localizing invariant of stable infinity-categories (over some base ring k), commuting with filtered colimits. It turns out that this category is dualizable and rigid (w.r.t. the natural symmetric monoidal structure). Further, the morphisms between the motives of categories can be effectively computed in many cases. In particular, it turns out that for a connective E_1-ring R the spectrum TR(R) is the spectrum of morphisms from the reduced motive of affine line to the motive of R (this can be considered as an analogue of a classical theorem of Almkwist about rational Witt vectors). I will also explain the “corepresentability” of topological cyclic homology (again, when restricted to connective E_1-rings).

Catrin Mair: Condensed Shape Theory
In this talk, I want to address the question of how to lift shape theory of infty-topoi to the setting of condensed mathematics. I will justify my interest in this question from the perspective of homotopy theory for topological spaces, and I will demonstrate its relevance by presenting a result about a condensed shape refining the \'etale homotopy type of a scheme. Overall, my talk aims to present ideas and open questions for discussion rather than ready-made answers.

Georg Tamme: Analytic and continuous K-theory
In this talk, analytic K-theory refers to a version of K-theory for non-archimedean analytic spaces. I will try to motivate and explain its construction and indicate its basic properties, in particular how it relates to continuous K-theory of formal models. Joint work with Moritz Kerz and Shuji Saito.

Sebastian Wolf: Internal higher categories and dualizable sheaves
In this talk, I will give an introduction to the theory of "Internal higher categories", that me and Louis Martini have been developing for the past two years. The motivitating problem I will adress using these methods is, to give a good definition (at least for certain purposes) of a condensed category of perfect complexes over a condensed ring. In fact, the methods I'm going to introduce, will yield a more general characterization of dualizable objects in categories of sheaves.