Schemes in Lean Kevin Buzzard Introduction The beginnings. The project Schemes in Lean starts. What is a scheme? Simple type Kevin Buzzard, Kenny Lau, Chris Hughes, Ramon theory / FOL rant slide Fernandez Mir First success. Imperial College London Equality and cheating. Schemes 2.0 AITP , 11th April 2019. The future.

Schemes in Lean Overview of the talk Kevin Buzzard Introduction The beginnings. [ note: links in this pdf are clickable.] The project • A scheme (for the purposes of this talk) is a technical starts. object in modern algebraic geometry. What is a scheme? • I’m an algebraic number theorist. Simple type theory / FOL • I’m going to tell the story of how me and a team of rant slide First success. mathematics undergraduates defined the notion of a Equality and scheme in Lean. cheating. • Lean is an ITP , it uses dependent types and the calculus Schemes 2.0 of inductive constructions. The future. • The weirdest part of the story is that none of us had a clue what an ITP was two years ago.

Schemes in Lean Lean Kevin Buzzard Introduction The beginnings. • I got interested in interactive theorem proving just under The project starts. two years ago. What is a scheme? • Why? For reasons unrelated to this talk. Simple type • I learnt some Haskell and I played around with Coq for two theory / FOL rant slide weeks. First success. • I am a formalist and found these new (to me) toys very Equality and cheating. appealing to use. Schemes 2.0 • Then I watched Tom Hales’ talk in the 2017 Big Proof The future. conference, and he mentioned Lean, so I thought I’d give Lean a go.

Schemes in Lean Kevin Buzzard • I am interested in learning and interested in teaching. Introduction • Summer 2017: I began to teach myself Lean. The beginnings. • Learning an ITP from scratch with no background is hard. The project starts. • However there is an incredibly helpful, polite, responsive What is a and useful live Lean chat at Zulip – scheme? leanprover.zulipchat.com/. Simple type theory / FOL • Without this chat I would never have been able to start. rant slide First success. • October 2017: I began to teach my “introduction to proof” Equality and course at Imperial College. cheating. Schemes 2.0 • I also started doing the problem sheets from my own The future. course in Lean. • I very quickly realised how difficult this formal proof verification malarkey was.

Schemes in Lean Kevin Buzzard Introduction The beginnings. First part of the first question on the first problem sheet: The project starts. Q1) True or false: x 2 − 3 x + 2 = 0 = ⇒ x = 1. What is a scheme? Me) “False: set x = 2”. Simple type theory / FOL Lean) OK so now you have to prove 2 2 − 3 × 2 + 2 = 0 and that rant slide 2 � 1. First success. Equality and Me) . . . cheating. Schemes 2.0 The future.

Schemes in Undaunted by the facts that Lean (a) I could not prove the first question on the first problem Kevin Buzzard sheet, and Introduction (b) the vast majority of my colleagues had no interest in, and The beginnings. no understanding of what I was doing, The project I did three more things: starts. What is a • I started a blog – xenaproject.wordpress.com. The spin: scheme? Xena is a “virtual undergraduate” taking my course. Can Simple type theory / FOL she (i.e. can I) do the problem sheet questions? Can she rant slide pass the exam? First success. Equality and • I got an account at github and started a completely chaotic cheating. “xena” repository. Schemes 2.0 • I started a club – the Xena project club. Meets on The future. Thursday evening in the computer room at Imperial College maths department. • It was the blind leading the blind, at first. • And then some people started bubbling to the top.

Schemes in • 5th November 2017 : Kenny Lau (1st year Imperial maths Lean Kevin Buzzard UG) had a PR accepted into Lean’s maths library. Subspaces of vector spaces; linear maps, and kernels etc. Introduction The • Six weeks earlier, Kenny Lau had never heard of theorem beginnings. provers. The project starts. • My response: I immediately got Kenny to start doing What is a scheme? commutative ring theory in Lean. (note to self: write vague Simple type intro to rings on the blackboard) theory / FOL rant slide • January 2018: I had my first PR accepted – the complex First success. numbers! (an indication of the state of Lean’s maths Equality and library in early 2018). cheating. Schemes 2.0 • Feb 2018 Chris Hughes (another 1st year Imperial maths The future. UG) had his first PR accepted. • Fast forward to now – Chris has had 162 commits accepted and is now a maintainer. He has proved quadratic reciprocity, Sylow’s Theorems, the fundamental theorem of algebra,. . . all in Lean.

Schemes in Lean Kevin Buzzard Introduction The beginnings. The project By Feb 2018 Kenny Lau had written a robust API for starts. localization of rings. “localization” means “adding a controlled What is a scheme? amount of division”. Simple type theory / FOL I proposed to Kenny that we should collaborate on defining a rant slide scheme in Lean. Our source was Johan de Jong’s stacks First success. project stacks.math.columbia.edu/ Equality and cheating. Schemes 2.0 The future.

Schemes in Lean Schemes – a bridge between Kevin Buzzard geometry and algebra. Introduction The beginnings. The project starts. What is a scheme? Simple type Schemes were invented by Alexandre Grothendieck around theory / FOL rant slide 1960, to put algebraic geometry back on a rigorous footing. First success. They are a brilliant example of a mathematical definition . Equality and cheating. (board explanation of affine schemes) Schemes 2.0 The future.

Schemes in Lean Kevin Buzzard Introduction C [ X ] is the ring of polynomials in one variable. The beginnings. √ Example of an element: f ( X ) = 3 X 2 + π X + − 1. The project starts. You can think of f ( X ) as a function C → C . What is a scheme? What about 1 / X though? This is not a polynomial. Simple type theory / FOL However it is still a function – not on all of C , but on C − { 0 } . rant slide First success. Localization of rings is the theory of adding “controlled Equality and denominators” to rings. This is what Kenny had made, and this cheating. Schemes 2.0 is what Grothendieck needed to define functions on his The future. schemes.

Schemes in Lean Technical interlude (lasts for one slide only). A scheme is: Kevin Buzzard • A topological space X ; Introduction The • a ring (the “ring of functions”) associated to each open beginnings. subset of X ; The project starts. • a bunch of axioms ( X is a locally ringed space and locally What is a affine). scheme? Simple type Discussion point: Attaching a ring to every open subset of a set theory / FOL rant slide or type seems to me to be hard to do in HOL light, First success. Isabelle-HOL, etc., and impossible to do in first order logic 1 . Equality and cheating. FOL / Simple type theory is not well suited to MSc/beginning Schemes 2.0 PhD level geometry. For me, as an arithmetic geometer, this is The future. a big turn-off . This sheaf (the ring attached to each open set) is a dependent type which shows up naturally in what mathematicians regard as “basic mathematics”. 1 It was pointed out to me in the talk that of course you can just add the axioms of ZFC and do it there.

Schemes in Lean Kevin Buzzard Introduction • Kenny and I managed to define schemes in Lean by The beginnings. March 2018. The project • I then started asking about comparing our work with starts. What is a formalisations in other theorem provers. It turned out that scheme? as far as I could see, there were no other formalisations of Simple type theory / FOL a scheme in a theorem prover. rant slide • Discussion point: 99% of mathematicians are not going to First success. Equality and be interested in formal proof verification systems until they cheating. can see the kind of objects that they think about Schemes 2.0 formalised in these provers. And many mathematicians The future. think about schemes.

Schemes in Lean Kevin Buzzard Introduction The Our announcement of the formal definition of a scheme led to beginnings. enthusiastic discussion on the Zulip chat. The project starts. Q) “Did you formalise any examples?” What is a Me) No... scheme? Simple type Q) “Well how do you know you have not made an error?” theory / FOL rant slide Me) . . . First success. I told the chat that it would probably not be too difficult to prove Equality and cheating. that an affine scheme was a scheme. I was told that this was Schemes 2.0 an important part of the process. So off we went again. The future.

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