# TTT55: Sheffield

Incorporating the inaugural Sheffield GATA lecture.

Department of Pure Mathematics, the University of Sheffield.

This was a working meeting with the usual time for participant discussion.

## Date

6 June 2006

## Speakers

Assaf Libman (Aberdeen)

Minami type splitting and some exotic p-local finite groups

Johann Sigurdsson (Sheffield)

Some applications of parametrised homotopy theory

GATA Lecturer, Bertrand Toen (Toulouse)

Stacks and derived categories I

## GATA lectures

The Sheffield GATA lectures are a principle activity of the Sheffield GATA Centre (directed by Professor Tom Bridgeland). The GATA lecturer is invited to spend an extended period in Sheffield and to give a series of lectures on a current topic of broad interest.

## Abstracts

### Assaf Libman

I will present a new proof of Minami's stable splitting of classifying spaces of finite groups which depends on Peter Symonds's resolution of Webb's conjecture and the Bredon cohomology groups of certain spaces. I will show how the method applies to obtain a stable splitting of the classifying spaces of certain exotic p-local finite groups.

### Johann Sigurdsson

I will give an introduction to the basic techniques of parametrised homotopy theory and then apply them to derive well known classical results such as Poincar\'e duality and the Wirthm\"uller isomorphism.

### Bertrand Toen

The purpose of these two talks is to report on recent works which use stack theory to study derived categories. In the first talk I will discuss the problem of constructing a reasonable moduli space for compact objects in a given triangulated category (or rather a triangulated "dg-category"). I will explain some motivations coming from algebraic geometry and representation theory (eg the contruction of moduli spaces of complexes of sheaves on an algebraic variety, the definition of "Hall algebras" for derived categories).

The second part will be devoted to present a solution to this problem using a notion of "derived \infty-stack": the main theorem states that the (derived \infty-) stack of compact objects in a given "saturated" dg-category is algebraic. Some corollaries and possible future applications will be discussed.

## Further information

The meeting was partially supported by a Scheme 3 grant from the London Mathematical Society.