What Did Grothendieck Do?

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.)

9 thoughts on “What Did Grothendieck Do?

  1. Nice post, thanks!
    A historical remark : “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”
    Kyhayyam in 1070 had already written about “geometric algebra”
    (Incidentally he also had written about “Pascal’s” binomial theorem)

  2. ain’t bad. what needs to be understood quickly about Alex Grothendieck is that his work shows an extreme (divine?) vision in unifying what seems to have nothign to do with anything (schemes do that.)
    on a secondary point WEIL (IAS) *whose* sister was the religious thinker Simone Weil (& ~ Weil who’s sistert etc.) best for 2o15

  3. Regarding

    “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.”

    I’ve seen a bunch of these “schemes for the layman” things since Grothendieck died, and I think it’s *really* worth saying that this ring is just the ring of coordinate functions on the variety. The way it is now this just sounds like some arbitrary, inscrutable algebraic invariant, whereas in reality it’s something very simple. Most people will be able to make a mental picture of things like the latitude and longitude functions on a sphere. (Okay, technically one of those isn’t an actual coordinate function, but it at least gives a morally correct picture.)

    Also it’s probably worth making the point that Grothendieck didn’t just say “Let’s associate geometric-ish objects to arbitrary rings and see what happens.” He knew that representable functors had better categorical properties than arbitrary ones, that certain functors “ought” to be representable, and that the prime spectrum of a ring was what controlled the associated hom-functor. I think it’s really more accurate to say that Grothendieck’s primary insight here was that instead of taking the objects you’re interested in and dealing with whatever category they form, you should build a good category including them and then understand the new objects.

  4. I wouldn’t say Grothendieck’s work is “incredibly technical” ; I’d save that for something like Andrew Wiles’ proof of Fermat’s Last Theorem. I’d say it was “incredibly novel”, bringing in simple new concepts that take time to get used to because they require that we change our idea of what mathematics is about… and lots of them!

    He broke his proofs down into small steps so that each step is “trivial” if you remember all the new concepts and all the previous steps. But developing an intuition for all the new concepts takes real work. You have to rewire your brain.

  5. Great essay! By the way, at one point you write:
    “(for shapes more complicated that disks)”
    Typo: “that” -> “than”

    Also I must have missed the reference – what do you mean by a ring punishing Moby for his sins?

  6. palma and Avi: Thanks for the corrections. I’ve updated the post. (The bit about Moby was a joke for the person I originally wrote this for, and forgot to delete.)

  7. Roadmaps are a keystone species in the STEAM literature, and in mathematics a type specimen of a Grothendieck-era roadmap is The Mathematical Sciences; a Report (1968, National Academy of Sciences pub. #1681).

    After 47 years, this 1968 roadmap’s discussion of the significance of Grothendieck’s work stands up pretty well:

    Homological Algebra and Category Theory Modern mathematics is characterized by an ever-increasing range of applications of algebra to other mathematical subjects. A particularly striking example is topology, a branch of geometry concerned with qualitative rather than quantitative aspects of shapes of geometric figures. In the early 1920’s it was recognized, under the influence of Emmy Noether especially, that the methods used by topologists are basically algebraic. But just as everu science that uses mathematics not only exploits the existing mathematical theories but reshapes them to its own needs, so topologists developed algebraic tools suitable for their needs. The next step was a return to pure algebra. Algebraic methods created for the needs of topology have been analyzed, codified, and studied for their own sake. This led to two new subdivisions in algebra: category theory and homological algebra. They are perhaps the most abstract specialities in algebra. Categories provide a language for discussing \emph{all} algebric systems of a given type. The result is, as is so often the case in mathematics, a wide variety of applications to diverse mathematical fields, in this case from logic to such “applied” fields as the theory of automata.

    Algebraic Geometry An oustanding problem was whether, given $k-n$ independent algebraic equations in $k$ unknowns, it is possible to represent all solutions of this system by a smooth geometric figure. (More precisely, can one transform the variety defined by this system into a smooth figure by using so-called birational transformations?). […] Recently the general case (any $n$) was settled affirmatively by Hironaka.

    Another achievement, of a quite different nature, is the systematic rebuilding of the foundations of algebraic geometry now led by Grothendieck in France. His work, which also leads to solutions of important concrete problems, has influenced many young mathematicians, including those working in different fields.

    Nowadays engineers regard Hilbert space as the “smooth figure” of the above observations, which arises as the algebraically birational (and computationally efficient) resolution of an underlying varietal state-space.

    Quantum dynamical trajectories computationally “unravel” (in the sense of Howard Carmichael) upon these algebraic state-spaces, in service of innumerably many objectives of modern system engineering. Mark Murcko’s recent video essay Accelerating Drug Discovery (reference below) surveys the astounding pace of development of the resulting mathematical and computational capabilities  capabilities that (as it seems to me) are grounded entirely in Grothendieck’s transformational vision.

    Who authored this foresighted assessment from 1968? One candidate is lattice theorist Robert P. Dilworth (whose thesis advisor Eric Temple Bell and student Juris Hartmanis will be familiar to many students of mathematics and theoretical computer science). And it is impressive too (as it seems to me) that this early assessment of Grothendieck’s work could receive the public imprimatur of a committee so diverse as the collection of mathematicians listed below.

    Conclusion  Not every STEAM roadmap committee gets it right … this one did!

    @book{NAS:1968, Title = {The mathematical
    sciences; a report}, Year = {1968} Address =
    {Washington}, Author = {{Committee on Support of
    Research in the Mathematical Sciences of the
    National Research Council for the Committee on
    Science and Public Policy of the National Academy
    of Sciences}}, Publisher = {National Academy of
    Sciences}, Series = {Publication number 1681},
    Annote = {Committee members: Lipman Bers, T. W.
    Anderson, R. H. Bing, Hendrick W. Bode R. P.
    Dilworth, George E. Forsythe, Mark Kac, C. C.
    Lin, John W. Tukey, F. J. Weyl, Hassler Whitney,
    C. N. Yang }}

    @inproceedings{Murcko:2014aa, Title =
    {Accelerating Drug Discovery: The Accurate
    Prediction of Potency}, Author = {Mark Murcko},
    Note = {URL:
    , Booktitle = {Advances in Drug Discovery and
    Development}, Month = {24 September},
    Organization = {Chemical \&\ Engineering News
    (Virtual Symposium)}, Year = {2014}}

  8. I think Alexandre Grothendieck is much overrated when people claim that he was one of the greatest mathematicians of the 20th century.

    The most important work in the last century was done in foundations:

    – Gödel’s Incompleteness Theorems;
    – Computable functions in the sense of Turing, Church, Kleene and others; especially Turing’s work is the origin of computing and today’s computers.

    Oh – and did any algebraic geometrist help win a war like Turing did?

Comments are closed.