Topology Seminar: The spectrum of equivariant stable homotopy theory

In this talk, I will discuss the spectrum of the G-equivariant stable homotopy category, for G a finite group.  In joint work with P. Balmer, we were able to describe this space, as a set, for all finite groups and gain a lot of information about its topology, obtaining a complete answer for groups of square-free order. We also reduced the problem of understanding the topology for arbitrary finite groups to understanding a specific question about the topology for p-groups.  Understanding this unresolved question for p-groups boils down to understanding an interesting phenomenon --- that the Tate construction performs a chromatic "blue shift".  Recently, T. Barthel, M.  Hausmann, N. Naumann, T.  Nikolaus, J. Noel and N. Stapleton have clarified such behavior, showing that an earlier "blue shift" conjecture of ours was too naive, and have thereby succeeded in computing the spectrum for all finite abelian groups. The task that remains is to find and prove a correct "blue shift" conjecture for nonabelian p-groups.