p-adic Hodge Theory and Related Topics

As classical algebraic geometry studies the solutions to polynomials over vector spaces over $\mathbb{C}$, arithmetic geometry studies the solutions to Diophantine equations. We encode related information in p-adic number fields. The attempts to prove Weil conjecture give rise to various cohomology theories over p-adic fields, and these cohomology theories give surprising analogues to classical Hodge theory. We will go over new results about the analogues in the works of Peter Scholze, David Hansen, Shizhang Li and Alexander Petrov and pose some questions. If time allows, we will introduce Scholze and Clausen’s new framework of condensed mathematics.