Description:

Title: Motivic Power Operations at the Characteristic

Abstract: Motivic power operations acting on the mod-ℓ motivic cohomology of smooth F_p-schemes were constructed by Voevodsky and played a key role in his proof of the Milnor and Bloch–Kato conjectures. In this talk, I describe joint work with Elden Elmanto, in which we extend Voevodsky's operations on mod-p motivic cohomology from characteristic 0 to characteristic p, thereby obtaining the long-sought-after motivic power operations at the characteristic. These operations satisfy all expected properties (except possibly generating, together with the Bockstein, all the endomorphisms of HF_p). If time permits, I will also discuss how to extend these operations to the motivic cohomology of singular Fp-schemes, as recently defined by Elmanto–Morrow and Kelly–Saito. Additionally, I may mention other applications, such as defining obstructions to lifting motivic cohomology classes to algebraic cobordism classes, solving the motivic Steenrod problem for singular varieties at the characteristic, and identifying algebraic cycles that are not smoothable at the characteristic.