Justin Noel (Regensburg): Connections between equivariant and chromatic homotopy theory

Hopkins, Kuhn, and Ravenel proved that, up to torsion, the Borel-equivariant cohomology of a G-space with coefficients in a height n-Morava E-theory is determined by its values on those abelian subgroups of G which are generated by n or fewer elements. When n=1, this is closely related to Artin's induction theorem for complex group representations. I will explain how to generalize the HKR result in two directions. First, we will establish the existence of a spectral sequence calculating the integral Borel-equivariant cohomology whose convergence properties imply the HKR theorem. Second, we will replace Morava E-theory with any L_n-local spectrum. Moreover, we can show, in some sense, a partial converse to this result: if an HKR style theorem holds for an E_\infty ring spectrum E, then K(n+j)_* E=0 for all j\geq 1. This partial converse has applications to the algebraic K-theory of structured ring spectra.