Happy New Year! The publicity in the wake of Grothendieck’s death has left a certain number of non-mathematicians with the question of what it was exactly that he did. I wrote an answer elsewhere that people seemed to find informative, so I’m saving it here for posterity.

This post is as untechnical as I could make it. Grothendieck’s work is incredibly technical, even by modern standards of abstract mathematics, so my description is, if you’re being charitable, highly impressionistic, and if you’re not, wrong in many major details. I also only discussed schemes and the Weil conjectures, which is only part of what Grothendieck is famous for.

Since Descartes, a major topic of mathematics research is understanding the solutions to polynomials equations. Descartes observed that while finding solutions is a matter of algebra, that when you view all of the solutions together, you enter the realm of geometry. For example, the set of solutions to X² + Y² = 1 is a circle.

The set of solutions to one or more polynomial equations is called a variety, and the study of such things is called algebraic geometry.

Originally, algebraic geometry involved solutions in real or complex numbers. (Usually the complex numbers, because that turns out to be much easier, since you can freely take square roots, etc., without having to worry about signs.) But the only things you need for the definitions to work is that you can add, subtract, and multiply. (A set where you can add, subtract, and multiply is called a ring.) There are lots of rings.

So Grothendieck set out to generalize algebraic geometry to arbitrary rings. His generalization of a variety to this setting is called a scheme. Interestingly, if you start with a variety (over the complex numbers), there’s a standard way to associate a ring with it, and in that case Grothendieck’s construction doesn’t give you anything new. It’s for the other kinds of rings that you get something new. So there’s a partial dictionary between varieties and rings, and schemes are missing entries in the dictionary.

Another example of a ring is the integers — you can add, subtract, and multiply integers. Here the idea of schemes captures a weird idea that goes back to the nineteenth century. The scheme for the integers consists of one point for each prime number. So you can picture the integers as points on a straight line at 2, 3, 5, 7, … and nowhere else. (Physicists would put an extra point at 9, and Grothendieck himself would put an extra point at 57.) So schemes are naturally related to number theory, and in fact have helped proved theorems in number theory such as Fermat’s Last Theorem.

On to the Weil conjectures. Think of clockwork arithmetic. You can add, subtract, and multiply hours or minutes on a clock face. In each case, you do the arithmetic with ordinary numbers, and then you throw away multiples of 12 (for hours), or 60 (for minutes). This operation of throwing away multiplies is called the “modulo” operator. So 7 times 2 modulo 12 is 2.

There are a couple of other instances of the modulo operator that you’ve probably used without knowing about it. Taking the last digit in a number is the same as that number modulo 10. So 1234 modulo 10 is 4. Adding up the digits of a number is the same as modulo 9. If you ever learned the trick to check if a number is a multiple of 3 by adding up the digits and checking that, you are actually working modulo 9.

Numbers modulo N give you another ring — you can add, subtract, or multiply modulo N, and that gives you another number modulo N.

What’s nice about numbers modulo N is that there are finitely many of them. They’re also useful in number theory. Let’s say that you want to know there are solutions to some polynomial equation over the integers — say X³ + Y³ = Z³. One easy check is see if there are any solutions modulo N. If there aren’t, then there aren’t any solutions at all. So an interesting question for number theory is how many solutions are there modulo N?

Andre Weil (whose sister was Simone Weil) conjectured a kind of formula for the number of solutions modulo N. He did so via a far-fetched analogy with topology.

Take a disk (a filled-in circle), and consider a continuous map of the disk to itself. One example of a continuous map is a rotation, where you spin the disk around its middle. The point you spin it around is a fixed point — it doesn’t move. You can prove (and it’s a difficult theorem) that every continuous map has to have at least one fixed point. There is a more general formula, called the Leftschetz fixed point formula, that allows you to count the number of fixed points in general (for shapes more complicated than disks).

For the integers modulo N, you can add, subtract, and multiply, but you can’t always divide, and you can’t always do things like take square roots. (Here, x is the square root of y modulo N if x*x is y modulo N. So 3 is the square root of 2, modulo 7. Pretty weird, huh?)

The division problem is easily fixed — just make N be a prime. The root problem is harder to solve, since some numbers don’t have square roots, cube roots, etc. even if N is a prime. The solution is to add “imaginary numbers” modulo N, the same way that we add i, the square root of -1 to get the complex numbers. The complex numbers have an operation defined on them, called conjugation, that sends i to -i. There’s a similar operation modulo N, called the Frobenius automorphism.

Weil said that we pretend that working modulo N was a kind of space, then we could apply the Lefshetz fixed point theorem, and count the number of solutions. This is a completely far-fetched anology, because there’s no geometry here.

That’s where schemes come in. Schemes supply the missing geometry. Grothendieck showed how to generalize the topological techniques to this setting so that a version of the Lefschetz fixed point theorem could be proven to settle the Weil conjectures. The proof is absurdly hard and abstract, but it is related to a relatively concrete question. (Unfortunately, the formula the conjectures give you is it itself a bit hard to use, so I don’t know any easy explanation of what it means, but I think it does have some real-world applications in coding theory and cryptography.)

As is well-known, the lattice of submodules of a module is modular. What I did not know is that the converse is not true, and that lattices of submodules must satisfy a stronger property, the arguesian law.

The Arguesian law is a lattice-theoretic analogue of Desargues’ theorem in projective geometry. I read the statement of the theorem several times and I have no intuition about what it means.

There is a kind of converse to this result: a complemented lattice can be embedded into the lattice of submodules of a module if and only if it is arguesian. (I found the result in Gratzer’s book on lattice theory, which is viewable in Google Books.)

Chandan Singh Dalawat has a nice survey article about K₂. It just gives the highlights of the theory, without proofs, so it’s closer to a teaser trailer than it is to full-length movie. But sometimes you just want a teaser trailer to tell you if you want to invest the time in the movie.

I came across a number theory paper Twists of X(7) and Primitive Solutions of x² + y³ = z⁷ that I find completely fascinating. I find it fascinating because a) the question is so easy, b) the answer is so hard, and yet c) someone was able to answer it.

An earlier expository paper, Faltings plus epsilon, Wiles plus epsilon, and the Generalized Fermat Equation, talks about the general question of finding solutions to the “generalized Fermat eqution”, x^p + y^q = z^r.

Justin Curry has written an excellent introduction to cosheaves. Cosheaves are the dual notion to sheaves, but many specific properties of sheaves of sets do not dualize, so they have a somewhat different flavor. The introduction includes some applications of cosheaves in networks.

Simon Willerton has thoughts on PERT charts as copresheaves.

As part of my new program of bringing you 2009’s internet to you today, I was fiddling around with Translation Party, which repeatedly translates a sentence in English into Japanese and back, until it finds a fixed point. Once I got tired of song lyrics, I tried various mathematical statements paraphrased into plain English. Most of the time, it converges right away on something close to the original sentence, or on gibberish, but I did find one example where it converted a true mathematical statement into an intelligible, but false, mathematical statement: that a matrix group is a group algebra.

I like to read the Low-Dimensional Topology blog, despite the fact that I know almost nothing about the subject. (It’s possible I like to read it because I know nothing about the subject.)

Over the past year, several posts convey palpable excitement over a series of preprints that prove two conjectures: the virtually Haken conjecture and its generalization the virtually fibered conjecture. These were apparently the outstanding open conjectures after the proof of geometrization. This post in particular describes the techniques involved in the proof. To see how fast things changed over the past year, this post on the Wise conjecture (an important ingredient of the proof) makes it clear that from the perspective of March of this year it was very much an open question which way the result would most likely turn out.

I’d been meaning to learn more about the subject, just to have a better idea of what happened. (For example, I still don’t really understand what a Haken manifold is, even though I’ve read the definition. Fortuitously, Erica Klarreich has written a long
general-audience article that gives at least some of the flavor of what’s going on.

Nate Eldridge has written a program, Mathgen, to randomly generate a nonsense math paper. (It’s based on an older program, SCIgen, that generates random computer science papers.) While they don’t make any sense, the Mathgen papers capture the typical style of mathematical writing pretty well. The main quirk that gives it away is that a real math paper would repeat terminology, Mathgen creates new mathematical terms every sentence. (This an inevitable consequence that the algorithm used is context-free.)

Apparently it doesn’t give it away for everyone, though. A Mathgen-generated paper was submitted to a journal, Advances in Pure Mathematics, where it was accepted with revisions. I’ve never heard of this journal, so I would assume that it’s like the mathematical version of a vanity publisher that makes money from publication fees. But what’s amazing is that the paper was peer reviewed! The suggested revisions are of the form “please make this make sense”, but still, out there somewhere there’s a person who read this paper, and tried to make constructive comments. Who was this person?

As probably most of you have heard, Shinichi Mochizuki has announced a proof of the abc conjecture. At some point I decided to stop posting about announcements of solutions to famous unsolved problems, after several high-profile retractions. This time, it’s been long enough to wonder if the proof will hold up. The papers are of daunting technical complexity, so it sounds like it will be some time before we hear the verdict.

PolyMath has the definitive round-up of links on the abc conjecture and Mochizuki’s work, including this nice expository article from the Notices.

This article from the New York Times has a startling statistic: sales of the print edition of the Encyclopedia Britannica have dropped from 120,000 in 1990 to 12,000 today. (The article says 8,000, but a later article says the whole print run of 12,000 sold out.) I knew that Wikipedia had seriously hurt the sales of encyclopedias, but I had no idea it was on the order of 90%.

When I was a kid, I had the Funk & Wagnalls encyclopedia. (They sold it at the supermarket, one letter at a time.) I remember vividly reading the article on algebra, where it had a big table of axioms, like “the commutative axiom,” and “the associative axiom.” I was fascinated to find out that someone had isolated a list of properties of numbers, and that these properties had names.