Title: Finiteness of local etale fundamental group in mixed characteristics

Abstract: It is conjectured by Kollar and proved by Braun that a klt singularity of a complex variety of dimension greater 2 has finite local (topological) fundamental group. In this talk, I will explain a mixed characteristic analogue of this theorem, namely the local etale fundamental group of a BCM-regular local ring is finite. The key tool is to use the perfectoidization functor and Faltings' normalized length function. This is joint work with Seungsu Lee, Linquan Ma, Karl Schwede and Kevin Tucker.