Mathematics Colloquium

In this talk, I will describe joint work in preparation with Peter Scholze on torsion in the cohomology of certain unitary Shimura varieties. I will explain how the theory of perfectoid spaces and p-adic Hodge theory give new insights into the geometry and cohomology of Shimura varieties. I will also mention applications of these results to elliptic curves over imaginary quadratic fields, joint with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne.