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Abstracts for Workshop on "Dualisable Categories $\&$ Continuous $K$-theory"

Alternatively have a look at the program.

Talk 1: Refined $TC^-$ over perfectoid rings

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Speaker: 
Peter Scholze
Affiliation: 
MPIM
Date: 
Mon, 09/09/2024 - 16:30 - 17:30
Location: 
MPIM Lecture Hall
Parent event: 
Workshop on "Dualisable Categories $\&$ Continuous $K$-theory"

For p-adic formal schemes, there is a tight link between prismatic cohomology and $TC^-$. A little more than a year ago, I was trying to understand how prismatic cohomology should work for rigid-analytic varieties, and realized that Efimov's result that localizing motives form a dualizable category allows one to decomplete the functor $TC^-$. This refined $TC^-$ induces a nontrivial functor for localizing motives over the generic fibre; yielding in particular such a functor for rigid-analytic varieties over $\mathbb{C}_p$ (by applying it to the dualizable category of nuclear modules).

Contributed talk: Dualizability in the context of invariants for $C^*$-algebras

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Speaker: 
Benjamin Dünzinger
Affiliation: 
University of Regensburg
Date: 
Tue, 10/09/2024 - 15:00 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Workshop on "Dualisable Categories $\&$ Continuous $K$-theory"

The goal of this talk is to introduce $E$-theory for $C^*$-algebras, which is a slight variant of $C^*$-algebraic $KK$-theory, and explain why $E$-theory is dualizable.

Contributed talk: Norm, Assembly and Coassembly

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Speaker: 
Georg Lehner
Affiliation: 
University of Regensburg
Date: 
Tue, 10/09/2024 - 15:15 - 15:30
Location: 
MPIM Lecture Hall
Parent event: 
Workshop on "Dualisable Categories $\&$ Continuous $K$-theory"

A major open problem in topology are (rational) injectivity results about assembly in $K$- and $L$-theory, e.g. the Borel and Novikov conjectures.

Contributed talk: Smooth and proper categories in analytic geometry

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Speaker: 
Matteo Montagnani
Affiliation: 
SISSA
Date: 
Tue, 10/09/2024 - 15:30 - 15:45
Location: 
MPIM Lecture Hall
Parent event: 
Workshop on "Dualisable Categories $\&$ Continuous $K$-theory"

In this talk, I will initially introduce the definition of smooth and proper category, and then explore how it can be used in algebraic and analytic geometry. This definition fits very well within the algebraic setting, as it allows for the characterization of smooth and proper algebraic varieties by examining their category of quasi-coherent sheaves. In the analytic context, such a characterization is more challenging to achieve; during the talk, I will aim to explain the difficulties and some results that we can obtain in this setting.

Contributed talk: Refined $TC^-$ over $ku$ and derived q-Hodge complexes

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Speaker: 
Ferdinand Wagner
Affiliation: 
MPIM Bonn/University of Bonn
Date: 
Tue, 10/09/2024 - 15:45 - 16:00
Location: 
MPIM Lecture Hall
Parent event: 
Workshop on "Dualisable Categories $\&$ Continuous $K$-theory"

As a consequence of his proof of rigidity of the category of localising motives, Efimov has constructed refinements of localising invariants. Such refined invariants often contain a lot more information than the original ones. For example, the refined $TC^-$ of the rational numbers is not a rational spectrum; it contains very subtle p-complete information as well. In this talk, we'll explain how to compute it after base change to $ku$.

Talk 2: Universality of algebraic $K$-theory

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Speaker: 
Maria Yakerson
Affiliation: 
Oxford University
Date: 
Tue, 10/09/2024 - 16:30 - 17:30
Location: 
MPIM Lecture Hall
Parent event: 
Workshop on "Dualisable Categories $\&$ Continuous $K$-theory"

Among various features of algebraic $K$-theory, there is known to be covariance with respect to finite flat morphisms of schemes. In this talk we will see, in which sense $K$-theory is universal as a cohomology theory with such covariance. Time permitting, we will discuss an analogous universality property for hermitian $K$-theory. This is joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, and Burt Totaro.

 

Talk 3: Motives as a localization of categories

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Speaker: 
Maxime Ramzi
Affiliation: 
University of Copenhagen
Date: 
Thu, 12/09/2024 - 16:30 - 17:30
Location: 
MPIM Lecture Hall
Parent event: 
Workshop on "Dualisable Categories $\&$ Continuous $K$-theory"

The category of localizing motives is the recipient of the universal localizing invariant, and will be one of the main characters of this conference. I will report on joint work with Vova Sosnilo and Christoph Winges, where we give a new perspective on this category, closer in spirit to the operator theoretic $KK$-category. Namely, we prove that this category is a localization of the category of stable categories. I will discuss a proof of this result as well as applications thereof.

 

 

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