Interactions between C*-algebraic KK-theory and homotopy theory

Aim

Throughout the mid to late 20th century, K-theory has been an important tool in its various algebraic, topological, and analytic manifestations. Homotopy theory has long been a central language for studying algebraic and topological K-theory, in particular. Recently, there has been more work connecting the methods of homotopy theory and infinity categories to C*-algebraic K-theory as well, and the potential applications of this new perspective are still to be seen.

The aim of this 2-week online workshop will be to bring together researchers and students with backgrounds in either C*-algebraic KK-theory or homotopy theory to learn about the growing interactions between the two fields. The first week will consist of separate introductions to classical C*-algebraic KK-theory and infinity categories, which will be followed by a modern infinity categorical approach to C*-algebraic KK-theory. The second week will consist of research talks related to the interactions of these two fields.

There will be time during the second week for shorter contributed talks by early career mathematicians (PhD students and postdocs). We encourage those who are interested in speaking to submit a title and abstract in the registration form below by December 11, 2024.

Time

Dates: 7 - 10 January, and 13 - 17 January, 2025
Time: 16:00 - 18:30 CET / 09:00 - 11:30 CST
Zoom access data: Will be sent out to registered participants via E-Mail.
The specific time is chosen so that the workshop is hopefully accessible to people in various different time zones.

Speakers

Pierre Albin
Ulrich Bunke
Bastiaan Cnossen
Marius Dadarlat
Eugenia Ellis
Alexander Engel
Markus Land
Ralf Meyer
Devarshi Mukherjee
George Nadareishvili
Shintaro Nishikawa
Ulrich Pennig
Christian Voigt

Schedule

Week 1:

	Tuesday 07 Jan	Wednesday 08 Jan	Thursday 09 Jan	Friday 10 Jan
16:00 - 17:00 (CET)	Pierre Albin	Shintaro Nishikawa	Ulrich Bunke	Ulrich Pennig	17:30 - 18:30	Bastiaan Cnossen	Bastiaan Cnossen	Christian Voigt	Contributed Talks 1

Abstracts

Week 1:

Pierre Albin: An introduction to KK theory

I will give a brief introduction to KK theory from the point of view of the Atiyah-Singer index theorem.

Ulrich Bunke: KK-theory from the point of view of homotopy theory

The classical group-valued KK-theory can be organized into a triangulated category. In this talk I will explain that classical KK-theory is the homotopy category of a stable infinity category. I will discuss various aspects of constructions of such a stable infinity category.

Shintaro Nishikawa: The Baum-Connes Conjecture: An Introduction and Survey

This talk introduces the Baum-Connes Conjecture (BC) and explores a few subtopics. After a standard/traditional introduction to the BC, we discuss the distinctions and parallels between the BC and the Farrell-Jones Conjecture, examine the Higson-Kasparov theorem, and highlight the contributions of Julg and Lafforgue to the BC in the context of groups with Kazhdan's Property (T).

Week 2:

Ulrich Bunke: Homotopy theory for K-theory of crossed products

If a group G acts on a C*-algebra A, then one can form the crossed product A ⋊ G. In this talk I discuss the problem of calculating the K-theory K(A ⋊ G) of the crossed product in terms of the spectrum K(A) with the induced G-action. This talk should motivate to consider the K-theory of a C*-algebra as a spectrum rather than a graded abelian group.

Eugenia Ellis: Homotopy structures realizing algebraic kk-theory

Algebraic kk-theory, introduced by Cortiñas - Thom is a bivariant K-theory defined on the category of algebras over a commutative unital ring l. It consists of a triangulated category kk endowed with a functor j : Alg → kk that is the universal excisive, homotopy invariant and matrix-stable homology theory. Moreover, we can recover Weibel's homotopy K-theory from kk since we have kk(l,A) = KH(A) for any algebra A. In this work we see that the category of algebras with fibrations the split surjections and weak equivalences the kk-equivalences is a stable category of fibrant objects, whose homotopy category is kk. Using this we prove that kk_∞ the Dwyer-Kan localization of the ∞-category of algebras at kk-equivalences is a stable infinity category whose homotopy category is kk. (joint with Emanuel Rodriguez-Cirone)

Ralf Meyer: Classifying C*-algebras up to KK-equivalence

Several classification results for C*-algebras say that an equivalence between two C*-algebras in a suitable bivariant KK-theory that encodes extra structure such as ideals or a group action, lifts to an isomorphism of those C*-algebras, suitably compatible with the extra structure. This leads to the question how to classify objects in a bivariant KK-theory up to isomorphism using some simpler K-theoretic invariant. One method to produce prove that an invariant works for this is to show that any object has a projective resolution of length 1 in a certain sense. There are several examples where this exists. A rather sophisticated theorem about this due to my doctoral student Köhler is for C*-algebras with an action of a group of prime order. More generally, if an object has a projective resolution of length 2, then one may use a more general classification result proven by Bentmann and me. Besides the invariant used to set up the projective resolutions, this invariant contains an extra piece of information, which we call the obstruction class. This classification method applies, for instance, to C*-algebras with an action of the circle group or to graph C*-algebras with finite ideal lattice.

Registration

You can register for the workshop and submit an abstract for a contributed talk via this Google form.

Organizers

Benjamin Dünzinger (Regensburg)
Yigal Kamel (Illinois)
Fredrick Mooers (Illinois)

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