Perfectoid spaces are a variant of Huber spaces in analytic geometry. The concept was introduced (Scholze 11) in order to generalize the classical theorem of (Fontaine-Winterberger 79) (see also at function field analogy).
This theorem establishes an isomorphism between the absolute Galois groups of an extension of the p-adic numbers and of the perfection of the field of Laurent series of the finite field $\mathbb{F}_p$. In (Scholze 11) this is generalized by the statement that the perfectoid spaces over fields related this way are equivalent. (See Bhatt 14 for a review).
Exposition includes
The concept is due to
motivated by
Review includes
Bhargav Bhatt, What is… a perfectoid space?, Notices of the AMS, volume 61, number 9 (pdf)
Bhargav Bhatt, Lecture notes for a class on perfectoid spaces, (pdf)
Peter Scholze, Perfectoid spaces: a survey, arXiv.