Singularities in mixed characteristic -- Linquan Ma, May 20, 2020.

Abstract: We introduce mixed characteristic versions of klt/plt singularities and multiplier/adjoint ideals, prove some of their analogous properties, and compare them with the equal characteristic counterparts. The theory relies on André and Gabber's recent work on the existence of weakly functorial perfectoid big Cohen-Macaulay algebras that factor through the absolute integral closure. We discuss some applications, which include a uniform version of the Briançon-Skoda theorem, a klt/plt adjunction for threefolds in mixed characteristic, and families of F-singularities when the characteristic varies. The talk is based on joint work with Karl Schwede, Kevin Tucker, Joe Waldron and Jakub Witaszek.