Lorenzo Ramero

Title: "Almost purity and p-adic analytic geometry"

I'll discuss a joint work with O.Gabber, on the foundations of "Almost ring theory". Almost rings provide a natural setup for Faltings' theory of almost etale extensions. First I'll explain what almost rings are and how one can develop "almost commutative algebra". Then I'll talk of our new approach to Faltings' "almost purity" (which is at the heart of his version of p-adic Hodge theory). Our method relies on p-adic analytic geometry and the analysis of ramification in higher rank valuation rings (while Faltings used local cohomology on higher dimensional noetherian rings).