Generalizing work of Krasner and Fontaine-Wintenberger on the isomorphism of absolute Galois groups between a mixed characteristic perfectoid field K and its charactersitic p tilt K^flat = prolim_{x -> x^p} K, Scholze introduced a notion of perfectoid adic space and proved an equivalence of category between perfectoid adic spaces over K and over K^flat. The goal of this talk will be to give a model theoretic translation of these results. We will show that, in a well chosen continuous structure, K and K^flat are bi-interpretable and that this immediately yields an equivalence of categories between type spaces over K and K^flat. We will then explain how these result relate to Scholze's results in adic geometry.
This is joint work with Tom Scanlon and Pierre Simon | 
