TITLE: What are "perfectoid spaces"?
QUESTION [137 upvotes]: This talk is about a theory of "perfectoid spaces", which "compares objects in characteristic p with objects in characteristic 0". What are those spaces, where can one read about them? 
Edit: A bit more info can be found in Peter Scholze's seminar description and in Bhargav Bhatt's.
Edit: Peter Scholze posted yesterday this beautiful overview on the arxiv.
Edit: Peter Scholze posted today this new survey on the arxiv.

REPLY [66 votes]: Here is a completely different kind of answer to this question.
A perfectoid space is a term of type PerfectoidSpace in the Lean theorem prover.
Here's a quote from the source code:
structure perfectoid_ring (R : Type) [Huber_ring R] extends Tate_ring R : Prop :=
(complete  : is_complete_hausdorff R)
(uniform   : is_uniform R)
(ramified  : ∃ ϖ : pseudo_uniformizer R, ϖ^p ∣ p in Rᵒ)
(Frobenius : surjective (Frob Rᵒ∕p))

/-
CLVRS ("complete locally valued ringed space") is a category
whose objects are topological spaces with a sheaf of complete topological rings
and an equivalence class of valuation on each stalk, whose support is the unique
maximal ideal of the stalk; in Wedhorn's notes this category is called .
A perfectoid space is an object of CLVRS which is locally isomorphic to Spa(A) with
A a perfectoid ring. Note however that CLVRS is a full subcategory of the category
`PreValuedRingedSpace` of topological spaces equipped with a presheaf of topological
rings and a valuation on each stalk, so the isomorphism can be checked in
PreValuedRingedSpace instead, which is what we do.
-/

/-- Condition for an object of CLVRS to be perfectoid: every point should have an open
neighbourhood isomorphic to Spa(A) for some perfectoid ring A.-/
def is_perfectoid (X : CLVRS) : Prop :=
∀ x : X, ∃ (U : opens X) (A : Huber_pair) [perfectoid_ring A],
  (x ∈ U) ∧ (Spa A ≊ U)

/-- The category of perfectoid spaces.-/
def PerfectoidSpace := {X : CLVRS // is_perfectoid X}

The perfectoid space project home page is here, and the source code is on github. Information for mathematicians on how to read the code above is here, and an informal write-up is here. Johan Commelin, Patrick Massot and I are in the process of writing up a more formal document explaining what we had to do to construct this type.
This definition of a perfectoid space is completely mathematically unambiguous. All terms are defined precisely, and if you don't know what something means then you can right click on it and look at its definition directly, if you compile the software using the Lean theorem prover. Lean is a computer program which checks that definitions and theorems are formally complete and correct. The definitions and theorems have to be written in Lean's language of course. The definition above says that a perfectoid space is a topological space equipped with the appropriate extra structure (sheaf of complete rings, valuation on stalks etc) and which is covered by spectra of perfectoid rings, and a perfectoid ring is a complete uniform Tate ring  with the usual perfectoid property. The main work is developing enough of the theory of completions of topological rings and valuations to put the appropriate structure on the adic spectrum of a Huber pair; in total this was over 10,000 lines of code.
Computer scientists have been developing and using formal proof verification tools for decades. I want to make the slightly contentious statement that 99.9% of the time they are working on statements about the kind of mathematical objects which we teach to undergraduates (groups, graphs, 2-spheres etc). There have been some huge successes in this area (computer-checked proofs of Kepler conjecture, odd order theorem, four colour theorem etc). My impression is that on the whole mathematicians are either unaware of, or not remotely interested in, this work, which for the most part concentrates on proving tricky theorems about "low-level" objects such as finite groups. I have now seen what these systems are capable of and in particular I believe that if mathematicians, rather than computer scientists, start working with formal theorem provers then we can build a database of definitions and statements of theorems which would be far far more interesting to the mathematical community than what the computer scientists have managed to do so far. Tom Hales is leading the Formal Abstracts project which aims to do precisely this. Furthermore, I now believe that theorem provers will inevitably play some role in the future of mathematical research. I am not entirely clear about what this role is yet, but one thing I am 100 percent convinced of is that AI will not be proving hard theorems that humans can't do, any time soon. However I am equally convinced that computers will soon be helping us to do research -- perhaps by giving us powerful targeted search through databases of theorems which humans have claimed to prove, and perhaps in the future pointing out gaps in the mathematical literature, as human mathematicians begin to formalise computer proofs more and more.