University of Chicago
Department of Mathematics

In recent work with Radu Stancu we proved a bijection between saturated fusion systems on a finite p-group and certain idempotents in its p-localized double Burnside ring. The Segal conjecture relates the double Burnside ring to stable maps between classifying spaces, and so we can regard saturated fusion systems as objects in stable homotopy theory. In the talk I will show how, adopting this point of view, we can answer a long-standing question on stable splittings of classifying spaces, and give a (best possible?) generalization of the Adams--Wilkerson criterion for recognizing rings of invariants in the cohomology of an elementary abelian p-group. Time permitting, I will present a new, simplified model for p-local finite groups.