Warwick Algebraic Topology Seminar 26/27

A list of the seminar talks of the previous years can be found here.

Term 1

The talks will take place on Tuesdays at 5pm in B3.02.

Date	Speaker	Affiliation	Title
6 Oct	Kelly Wang	Cambridge	Periodic families in the homology of GL_n(F₂)
13 Oct	Yorick Fuhrmann	Warwick	Smooth Artin motives and equivariant homotopy theory
20 Oct	Lukas Waas	Oxford	Presenting stratified homotopy theory
27 Oct	Ulrich Pennig	Cardiff	Units of K-theory and Their Applications in Operator Algebras
3 Nov	Behrooz Mirzaii	Sao Paulo	The integral homology of SL_2(Z[1/n])
10 Nov	Isambard Goodbody	Warwick
17 Nov	Matthew Antrobus	Manchester
24 Nov
1 Dec	Henning Krause	Bielefeld
8 Dec	Jelena Grbic	Southampton

Abstracts

Kelly Wang, 6 October

Title: Periodic families in the homology of GL_n(F₂)

Abstract: In his 1972 paper Quillen completely determined the cohomology ring of GL_n(F_q) away from the characteristic of F_q. At the characteristic he showed that the stable value is zero, but the unstable part remains largely mysterious. From a slightly different perspective Galatius-Kupers-Randal-Williams showed that when q = 2, H_*(GL_n(F₂);F₂) exhibits stability of slope 2/3. In this talk I will explain how to construct periodic families of non-zero classes in H_*(GL_n(F₂),GL_n-1(F₂);F₂) that lie on line of slope 2/3. In particular they contribute to the unstable homologies and establish sharpness of the 2/3 stability. This also has consequences for GL_n(Z) and Aut(F_n).

Yorick Fuhrmann, 13 October

Title: Smooth Artin motives and equivariant homotopy theory

Abstract: For a scheme S, an interesting part of the category of Voevodsky motives over S can be expressed in the language of equivariant homotopy theory for the étale fundamental group of S. This is the category of smooth Artin motives. It admits a natural passage to étale motives, induced by étale sheafification. We will explain what this means on the equivariant side of the story and why it requires new notions of Borel completeness for profinite groups.

Lukas Waas, 20 October

Title: Presenting stratified homotopy theory

Abstract: Historically speaking, one of the original aims of stratified homotopy theory was to develop a homotopy theory that retains the finer information encoded by the strata and their interactions in classical geometric examples of stratified spaces.

More recently, the emphasis has shifted towards the exodromical perspective: stratified homotopy theory should be to constructible higher sheaves what the homotopy theory of spaces is to locally constant higher sheaves.

In this talk, I will present a bridge between these perspectives through the existence of a particularly well-behaved semi-model structure on poset-stratified spaces. Most classical geometric examples are bifibrant; all cofibrant objects, which are precisely the retracts of stratified cell complexes, admit the exodromy correspondence; and weak equivalences between cofibrant objects are exactly those stratified maps that induce equivalences between the associated categories of constructible higher sheaves, i.e., induce equivalences on infinity categories of exit paths.

Ulrich Pennig, 27 October

Title: Units of K-theory and Their Applications in Operator Algebras

Abstract: Complex topological K-theory arises from a commutative symmetric ring spectrum KU, which in turn has an associated spectrum of units gl_1(KU). The first group of the resulting cohomology theory classifies twists of K-theory. In a similar way, any strongly self-absorbing C*-algebra D gives rise to an analogous ring spectrum KU^D with its own spectrum of units gl_1(KU^D). In several joint papers with Bianchi, Dadarlat, Evans, Giron-Pacheco and Izumi we have shown how this spectrum and its variants are closely intertwined with twisted K-theory, as well as with the lifting obstructions for cocycle actions and G-kernels on strongly self-absorbing C*-algebras. In this talk, I will highlight the most recent developments in this project.