Justin Asher, March 5, 2024

Abstract.   In his PhD thesis, Peter Scholze introduced perfectoid spaces as a tool to pove Pierre Deligne's weight-monodromy conjecture in mixed characteristic. Deligne had shown the conjecture to be true over local fields of characteristic p, and Scholze used Fontaine's idea of tilting to prove it in mixed characteristic for specific varieties. In particular, he proved it for geometrically connected, proper smooth, set-theoretic complete intersections in toric varieties over local fields of mixed characteristic. Recently, Peter Wear has shown the conjecture to be true for abelian varieties as well.