Tom Bachmann (MIT)

"Power operations in normed motivic spectra"

In joint work with M. Hoyois, we established (the beginnings of)  a theory of ”normed motivic spectra”. These are motivic spectra with some extra structure, enhancing the standard notion of a motivic E_\infty-ring spectrum (this is similar to the notion of G-commutative ring spectra in equivariant stable homotopy theory). It was clear from the beginning that the homotopy groups of such normed spectra afford interesting *power operations*. In ongoing joint work with E. Elmanto and J. Heller, we attempt to establish a theory of these operations and exploit them calculationally. I will report on this, and more specifically on our proof of a weak motivic analog of the following classical result of Würgler: any homotopy  commutative ring spectrum with 2=0 is generalized Eilenberg-
MacLane.