TTT121

University of Sheffield

Date Tuesday 17th December 2024

Speakers Anja Meyer (Loughborough), Ulrich Pennig (Cardiff) and Neil Strickland (Sheffield).

Programme

All talks will be in J11 on J floor of the Hicks Building, the tea/coffee break will be in I15 common room on I floor of the Hicks Building.

12:00 - 13:00 Neil Strickland

lunch break

14:15 - 15:15 Anja Meyer

tea/coffee break

16:00 - 17:00 Ulrich Pennig

Titles and abstracts

Anja Meyer (Loughborough)

Finite Matrix Groups; Cohomology and Stable Elements

Abstract: In their 1956 book Cartan and Eilenberg show results that tell us that the modular cohomology of a finite group G is equal to the set of stable elements in the modular cohomology of a Sylow p-subgroup of G. In this talk we will look at the groups SL_2(Z/p^n) for n>1. Their cohomology is not yet known, however there is a way to obtain the cohomology, using a combination of tools from homological algebra, profinite group theory, and fusion systems. We will introduce the concepts used and show how they can facilitate the explicit computations.

Ulrich Pennig (Cardiff)

Topological 2-groups and Symmetries of C*-algebras

Abstract: A G-kernel is a group homomorphism from a (discrete) group G to Out(A), the outer automorphism group of a C*-algebra A. There are cohomological obstructions to lifting such a G-kernel to a group action. In the setting of von Neumann algebras, G-kernels on the hyperfinite II_1-factor have been completely understood via deep results of Connes, Jones and Ocneanu. In the talk I will explain how G-kernels on C*-algebras and the lifting obstructions can be interpreted in terms cohomology with coefficients in crossed modules. G-kernels, group actions and cocycle actions then give rise to induced maps on classifying spaces. For strongly self-absorbing C*-algebras these classifying spaces turn out to be infinite loop spaces creating a bridge to stable homotopy theory. Not only does this make the invariants computable, it also gives rise to equivariant refinements. The first part is a joint project with S. Giron Pacheco and M. Izumi, the second with my PhD student V. Bianchi.

Neil Strickland (Sheffield)

Topological Hochschild homology of the dual circle

Abstract: Let A be the function spectrum F(S^1_+,S), and let R be the topological Hochschild homology spectrum of A. This plays an important role in the disproof by Burklund, Hahn, Levy and Schlank of Ravenel's Telescope Conjecture. In this talk I will explain some ideas related to that. In particular, R has a large and interesting monoid of ring-endomorphisms, which help to illuminate the key properties of R required for the telescope paper.