# Peter Scholze

Perfectoid spaces and the almost purity theorem

**Abstract**:

We will first explain some almost mathematics, to properly formulate (a generalization of) Faltings's almost purity theorem. Using the theory of perfectoid spaces, we will sketch a proof of this result. Afterwards, we will explain some applications of the almost purity theorem, indicating some crucial steps in the proof of the Hodge-Tate comparison theorem.