Masterclass: Continuous K-theory, dualizable and rigid categories

There has been a joint effort from participants of the masterclass to tex the lectures and we wish to thank all the people involved in the project. The following notes are kindly offered to us by Claudius Heyer.

Clausen Lecture

Efimov Lecture

Nikolaus Lecture

Dustin Clausen:

Title: Three perspectives on Deligne cohomology

Abstract: Deligne cohomology is a refined cohomology theory for complex manifolds, known in number theory for its role both in Arakelov geometry and in Beilinson's conjectures on special values of L-functions. In these lectures I will describe three interrelated perspectives on Deligne cohomology. The first represents an attempt to determine for which kinds of "analytic spaces" the standard definition of Deligne cohomology makes sense and has good properties. The second and third, which describe theories only conjecturally equivalent to Deligne cohomology, represent attempts to articulate that Deligne cohomology is the complex-analytic analog of motivic cohomology. In the second, which is joint with with Peter Scholze, we define (using Efimov's continuous K-theory) a version of algebraic K-theory for analytic spaces and conjecture that it identifies with the (easily defined) K-theory analog of Deligne cohomology. In the third we more directly articulate a conjectural universal property for Deligne cohomology inspired by the recent "non-A^1-invariant motivic" formalism of Annala-Iwasa in the algebraic setting. The first and second perspectives make use of the theory of analytic geometry developed in joint work with Peter Scholze; we will review the necessary aspects of that theory.

Alexander Efimov:

Title: Dualizable categories and localizing motives

Abstract: I will give an introduction to dualizable categories and their localizing invariants. I will start with the general theory of dualizable categories, and explain some non-trivial results, such as equivalence between dualizability and flatness for a presentable stable category. Then we will compute the localizing invariants of various dualizable categories which come from topology and non-Archimedean analysis. These include sheaves on locally compact Hausdorff spaces and categories of nuclear modules on formal schemes. I will also explain a deep connection between the algebra of dualizable categories and the category of localizing motives -- the target of the universal finitary localizing invariant (of categories over some base). We will see that the category of localizing motives (considered as a symmetric monoidal category) is in fact rigid in the sense of Gaitsgory and Rozenblyum, and sketch some applications.

Thomas Nikolaus:

Title: Continuous K-Theory and Geometric Topology

Abstract: we will begin by providing equivalent characterizations of dualizable categories and describing various aspects of their general theory, complementing the courses of Clausen and Efimov. We will review Efimov's definition of K-Theory for such categories. Additionally, we will discuss the theory of (locally) rigid categories as developed by Gaitsgory and Rozenblyum, incorporating insights from Efimov, Clausen, and Ramzi. Finally, we will explain the application of this theory to geometric topology, particularly in the characterization of simple homotopy types. A key aspect is to describe assembly maps, inspired by similar descriptions given for operator algebras. This approach leads to new proofs of results by West and Chapman.

We kindly ask the participants to arrange their own accommodation.

We recommend Hotel 9 Små Hjem, which is pleasant and inexpensive and offers rooms with a kitchen. Other inexpensive alternatives are CabInn, which has several locations in Copenhagen: the Hotel City (close to Tivoli), Hotel Scandinavia (Frederiksberg, close to the lakes), and Hotel Express (Frederiksberg) are the most convenient locations; the latter two are 2.5-3 km from the math department. Somewhat more expensive – and still recommended – options are Hotel Nora and  Ibsen's Hotel.

An additional option is to combine a stay at the CabInn Metro Hotel with a pass for Copenhagen public transportation (efficient and reliable). See information about tickets & prices.