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Perfectoid spaces and their applications

Alessandro Maria Masullo
Friday, September 25, 2015 - 11:00

I will give a fairly detailed and at the same time motivational view on the theory of Perfectoid Spaces, introduced by Peter Scholze a few years ago, building on some explicit examples. I'm going to briefly sketch the main deep ideas which led to the birth of Perfectoid Spaces, mainly due to Tate, Faltings and Fontaine. Once introduced the notion of an affinoid perfectoid space, avoiding few technicalities, I will state the main deep theorems which, roughly, explain how Perfectoid Spaces serve as a bridge between the mixed-characteristic (0,p) world, p a prime, and the characteristic p world, in terms of spaces and their étale cohomology. I will explicitly solve an instructive exercise proving that every connected affinoid Noetherian rigid space over Q_p is a K(pi, 1), which will only need the basics, and will at once show the power of the theory. Time permitting, I will sketch one more advanced topic leading to results of mine which are intended to "geometrize" Iwasawa Theory by means of the natural geometric framework provided by adic spaces and their "perfectoid covers". Lecture notes will be provided shortly after the seminar.

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