Lecture 2. $p$-adic Hodge Theory and Perfectoid Spaces

The second lecture tackles the arithmetic-geometric divide. After introducing Grothendieck's étale cohomology and Fontaine's period rings (such as $B_{dR}$), we will dive into Peter Scholze's perfectoid geometry. We will explicitly detail the algebraic operation of “tilting”, demonstrating how infinite ramification allows us to seamlessly translate complex characteristic $0$ arithmetic problems into computationally friendly characteristic $p$ spaces.