Transfer in the Cohomology of Categories Using Biset Functors

Peter Webb

University of Minnesota

**Abstract:** The cohomology of categories includes as a special case the theory of group cohomology. Defining a transfer on the cohomology of categories is problematic: most attempts to do this in related contexts require induction and restriction functors to be adjoint on both sides, which typically does not happen with categories. We get round this by defining category cohomology as a functor on a biset category where the objects are small categories, extending the usual theory of biset functors when the objects are groups. After summarizing this theory, we show how to make the definition of category cohomology as a biset functor.

Fusion Systems and Brauer Indecomposability of Scott Modules

İpek Tuvay

Mimar Sinan Güzel Sanatlar Üniversitesi

**Abstract:** The Brauer indecomposability of Scott modules is important for obtaining splendid Morita equivalences between the principal blocks of two finite groups whose fusion systems on their Sylow p-subgroups are isomorphic. In this talk, first the connection between the Brauer indecomposability of a p-permutation module and the saturation of the corresponding fusion system will be discussed. Then, recent results along these lines for some special families of 2-groups, including semidihedral and wreathed 2-groups, will be presented. Part of this work is joint with S. Koshitani.

A Functorial Resolution of Units of Burnside Rings

Serge Bouc

Université de Picardie Jules Verne

**Abstract:** Most of the structural properties - prime spectrum, species, idempotents, ... - of the Burnside ring of a finite group have been precisely described a few years after its introduction in 1967. An important missing item in this list is its group of units. After a - non exhaustive - review of this subject, I will present some recent results on the functorial aspects of this group..