Good Shurik Grothendieck

I used to tell people that the title character of the film Good Will Hunting didn’t strike me as very believable — not because of the self-taught janitor’s ability to do cutting-edge research, but because of his contempt for his own work. At one point in the movie, having shown his mentor a proof he’s just written, he sneers “Do you know how easy this is for me?” and sets the proof on fire with his cigarette lighter — at which point his mentor, a world-class mathematician with a Fields Medal to his name, dives onto the carpet not so much to prevent the building from burning down (buildings can be rebuilt, after all) as much as to rescue a proof that the mathematical world will cherish.

“That’s a teenager’s idea of what being a genius is like,” I would tell people.

“Oh, and are you a genius?” one woman once asked me skeptically.

“No,” I answered, “but I know a few.” Which was true: I’d been an undergraduate at Harvard, a visiting student at Cambridge University (where I’d worked closely with John Conway), and a graduate student at U. C. Berkeley before landing a tenure-track position at M.I.T. So I’d gotten a chance to interact with world-class mathematicians at close range, and Will Hunting resembled none of them.

The fact is, school math may have been easy for most of the people who go on to become research mathematicians, but if you really want to make a name for yourself as a researcher, you’re in competition with a lot of other people who were good at school math, and college math, and grad school math, and who also want to make names for themselves; the way to rise in the profession — or, putting things less competitively, the way to do the best work of which you’re capable — is to seek out things that are hard for you, but doable, and then do them. That’s what Andrew Wiles did when he set out to prove Fermat’s Last Theorem; that’s what Yitang Zhang did when he worked on a weakened (but still epochal) analogue of the twin primes conjecture; and a lot of my own research over the course of my career has required hard work. I wouldn’t be able to feel proud of an article I’d written if some effort hadn’t been involved. Plus, in some ways a proof is a story that I tell myself, and part of the fun is encountering surprises along the way; if I know in advance how everything is going to go, it’s less fun to write. I think most mathematicians resemble me in this respect.

But … what if someone was so good at mathematics (or, more realistically, a branch of mathematics or maybe two) that they just saw things that other people didn’t see, and produced breakthrough after breakthrough? Might such a person come to devalue the enterprise of creating new mathematics? Just because I hadn’t met anyone like that didn’t mean that such a person couldn’t exist.

In fact, there was such a twentieth-century mathematician, and his name was Alexandre Grothendieck (he preferred to spell his first name as “Alexander”, and close friends called him “Shurik”). By the time I came of age mathematically, this singular individual who in his younger days had done so much to revolutionize mathematics had turned his back on the field and was even urging others to do the same. You can read about Grothendieck in Rivka Galchen’s excellent article in the May 16, 2022 issue of the New Yorker. For a more mathematical treatment, see the article “Comme Appelé du Néant—As If Summoned from the Void: The Life of Alexandre Grothendieck” by Allyn Jackson, published in the Notices of the American Mathematical Society in 2004 in two installments (part 1 and part 2); Winfried Scharlau’s “Who Is Alexander Grothendieck?”, published in the Notices of the American Mathematical Society in 2008; and the obituary “Alexandre Grothendieck 1928–2014” by Michael Artin, Allyn Jackson, David Mumford, and John Tate, published in the Notices of the American Mathematical Society in 2014 in two installments (part 1 and part 2).

MEASURES OF GREATNESS

Grothendieck had two careers as a mathematician: the first was brilliant, and the second was spectacular. Galchen mentions the first one briefly. She writes: “While at the [University of Montpellier] — which was not an important center of mathematics — Grothendieck independently pursued research on ideas having to do with measures, a field that less gifted students might dismiss as obvious.” Here “measure” is a generic term that refers to things like length, volume, surface area, and analogous notions in higher dimensions. These concepts, at least in three or fewer dimensions, are so intuitive that you might think that there’s nothing much to be said that wasn’t known by the ancient Greeks. But there are paradoxes here to trip the unwary, even in just one dimension.

Young Grothendieck. From https://www.ams.org/publications/journals/notices/201603/rnoti-p242.pdf

As a graduate student, the comparatively untrained Grothendieck ended up reproducing a famous result of the mathematician Henri Lebesgue in the theory of measures. Not discouraged when he learned he’d been scooped, Grothendieck went on to study the field of topological vector spaces, and tackled fourteen problems that his mentor, Laurent Schwartz (a future Fields Medalist) had proposed; in his doctoral thesis Grothendieck solved all of them, prompting Schwartz to say it might be time for him to stop teaching Grothendieck and starting learning from him instead.

If Grothendieck had wanted, he could have continued in this line of work and been among the top dozen mathematicians of his generation, an expert mathematician respected around the world. Instead, at age 27 he switched to algebraic geometry and became, in a manner of speaking, a seer, though he would have described himself as a builder. Mathematician David Ruelle has said that in this phase of his life, Grothendieck (metaphorically speaking) built the entire ground floor of a vast cathedral. But Grothendieck did not do it alone.

THE ODD COUPLE

One member of Grothendieck’s circle in the 1950s and 1960s was Pierre Cartier (who incidentally came up with the application of Galois fields to coding theory that I described in one of my earlier essays). In 2010 Cartier wrote an article entitled “A country of which nothing is known but the name: Grothendieck and “motives””. A few weeks after Grothendieck’s death in 2014, Cartier gave an interview to mathematician Sylvie Paycha. In both the article and the video, Cartier describes the role played by some of Grothendieck’s colleagues, especially Jean Dieudonné.

The renowned Institut des Hautes Études Scientifiques, France’s equivalent of the Institute for Advanced Study, didn’t start out big; originally it had only two salaried employees, Grothendieck and Dieudonné. Dieudonné had made the hiring of Grothendieck a precondition of his coming to the newly-founded I.H.E.S. The two were an odd couple; the former was an establishment-type, politically conservative and happy to work within the system as it was, while the latter was a political radical, prone to gestures like fasting one day a week to protest the ongoing war in Vietnam. Yet Dieudonné, for all that he was a world-class mathematician, saw that young Grothendieck, twenty years his junior and in many ways his opposite, was truly exceptional, and set about to become a conduit for the younger mathematician’s insights.

Shaved Grothendieck. From https://www.ihes.fr/en/professeur/alexander-grothendieck-1/

Both were part of a semi-secret mathematical collective called Bourbaki. (They didn’t keep their existence a secret at all; they published dozens of books. They did try to conceal who the members of the collective were, but they didn’t try very hard.) Here’s how the collaboration between Grothendieck and Dieudonné worked: Grothendieck would write his ideas in the form of sketches, feverishly writing all night, and presenting his notes to the older mathematician at five in the morning. At that point, Dieudonné would spend several hours fleshing out Grothendieck’s notes and writing them up in a form that was suitable for sharing with the other members of Bourbaki.

Cartier describes an incident in which a frustrated Dieudonné, in a move more reminiscent of young Will Hunting than his mentor, dramatically tore up what he’d presented to the group, at which point two of the other mathematicians dove to the floor to rescue it (a step that proved to be unnecessary as Dieudonné had prudently made a second copy).

Earlier in this essay I asked “What if someone was so good at mathematics that they just saw things that other people didn’t see?” But to say that Grothendieck was good at math misses an important point. Grothendieck himself, though not prone to false modesty, would be the first to say that his colleagues Jean-Pierre Serre and Pierre Deligne had more raw talent than he did. Two things made him more productive than the others during this period. First, he worked incredibly hard (I mentioned his serial all-nighters); never would Grothendieck have claimed, Will-Hunting-ishly, that the work was easy. Second, Grothendieck was different. Some of that difference came from having been self-taught, so that he had to find his own idiosyncratic ways of doing things. (Most of the time, being self-taught is a disadvantage; not so in Grothendieck’s case.) But some of that difference may have been inborn. Cartier describes Grothendieck as a kind of eagle, soaring up to great heights to get a panoramic view of its prey before attacking in one blinding strike. Mathematician Michel Demazure had a different way of describing the Grothendieck difference: “He did math upside down.” Most of us start with examples, and build up the intuition that lets us grasp generalities. Grothendieck started with the generalities and somehow managed not to lose his footing among the clouds.

Grothendieck had other, gentler ways of describing his style of doing math. One metaphor he liked was soaking a nut in water and letting it slowly soften rather than trying to crack it open. Another was the metaphor of a slowly rising sea: nothing dramatic happens, but the land (that is, the problem) that seemed so imposing before has somehow disappeared. He also claimed that at least one of his ideas (the notion of a “scheme”) came not from soaring, as in Cartier’s image, but from stooping lower than anyone else had dreamed of stooping. For more on Grothendieck’s views of his own style, see Colin McLarty’s “The Rising Sea: Grothendieck on Simplicity and Genius”; for the story of schemes, see McLarty’s “How Grothendieck Simplified Algebraic Geometry” (from the March 2016 issue of the Notices of the American Mathematical Society).

BEING GOOD

I’ve never much liked the title of Good Will Hunting. Sure, Will is good at math, but as I recall, he only progresses from being a total jerk to being a partial jerk; he never becomes an actively good person, unselfishly striving to make the world a better place.

Middle-aged Grothendieck. From https://www.nytimes.com/2014/11/16/world/europe/alexander-grothendieck-math-enigma-dies-at-86.html

Grothendieck, in contrast, was very much concerned with being good, but he tended to focus on the good of the world as a whole, not on the good of people around him. In this he reminds me of the old joke about the married couple in which the wife takes care of all the unimportant decisions (where to live, what car to buy, how to raise the kids, etc.) and the husband takes care of all the important decisions (what the president should do about the economy, what the president should do about the environment, etc.). Grothendieck married several times and had several children, but he was a neglectful husband and father, and he was sometimes cruel in a high-minded way. This dual tendency is highlighted in a story that mathematician Barry Mazur tells about a time when he and his wife Gretchen had dinner with Grothendieck and his wife-at-the-time Mireille. Mazur went to extraordinary pains to prepare a vegetarian meal for the Grothendiecks since Alexandre was a principled vegetarian. After profusely thanking Barry and Gretchen for the wonderful spread, Alexandre turned to Mireille and harshly told her “See how easy it is to make a vegetarian meal!” This streak of intolerance would eventually grow to the point where it alienated every single one of Alexandre’s friends.

Cartier describes one of the more momentous ruptures in Grothendieck’s career, when he and Dieudonné parted ways as a result of a dispute about where and how Grothendieck could distribute political pamphlets during a mathematical conference. He abruptly resigned from the I.H.E.S. in 1970, and incredibly, soon thereafter he stopped publishing mathematics. As Ruelle says: after building the ground floor of his cathedral with his bare hands, Grothendieck turned his back on it and walked away. He began to claim that mathematicians and physicists were the most dangerous people in the world because they gave politicians access to tools that could destroy the human race. He became a hermit in the Pyrenees, with no phone and no postal address.

THE ABSOLUTE GALOIS GROUP

Galchen’s article discusses the mathematicians Leila Schneps and Pierre Lochak, who in the 1990s decided to track down the hermit. Schneps had become captivated by his work, especially his strange memoir Récoltes et Semailles (“Reapings and Sowings”), and she was eager to meet the mind that had created it.

One mathematical creation of Grothendieck’s that had enchanted Schneps was a subject to which Grothendieck had given the curious name “dessins d’enfants“, or “children’s drawings”, in a 1984 research proposal that he never published but which was informally circulated for over a decade before Schneps and Lochak published it in 1997. Following the proposed research program, one would be able to catch glimpses of the absolute Galois group, one of the most complicated mathematical objects ever studied, through what might seem the narrowest of windows: simple drawings of lines and curves on a piece of paper joining black and white dots.

I’ll try give you a bit of the flavor of the absolute Galois group.  Mathematicians distinguish between, on the one hand, numbers like the square root of two and the imaginary number i which are defined by algebraic equations (x² = 2 and x² = –1, respectively), and, on the other hand, numbers like e and π which can’t be defined in this way.  Numbers of the former type are called algebraic; numbers of the latter type are called transcendental. The set of all algebraic numbers is often represented by the symbol ℚ, not to be confused with the symbol ℚ, which represents the much smaller set of all rational numbers.

Gal(ℚ/ℚ) (“gal-q-bar-over-q”), the absolute Galois group, is an infinite collection of mathematical operations that you can perform on ℚ, of which only two can be concretely described. One of them is the operation called conjugation, that (for instance) turns 3+4i into 3–4i and vice versa. If you perform conjugation twice, you get back what you started with; this do-nothing operation is the other graspable elements of Gal(ℚ/ℚ). The rest exist, but in a much more tenuous way; you can’t really get your hands on any of them, because to specify any single one of them requires that you make an infinite sequence of choices, some of which are arbitrary (Buridan’s ass comes to mind) and others of which are forced by earlier choices, in a garden of forking paths from which there is no exit). Moreover, although the set ℚ can be squashed into the plane, the squashing does violence to its essential nature by marring some of its symmetry; the place ℚ really “wants” to live is an infinite-dimensional space, where the absolute Galois group can be seen as the group of symmetries of an impossibly intricate sort of crystal.

The absolute Galois group casts various shadows that are easier to understand: the (relative) Galois groups Gal(K/ℚ), where K is a number field intermediate between ℚ and ℚ. (I wrote a bit about towers of number fields in an earlier essay.) We can try understand the absolute Galois group by way of its shadows, but the absolute Galois group itself still remains beyond our grasp.

I hope I’ve convinced you that Gal(ℚ/ℚ) is an extremely abstract thing, so that you can appreciate how amazing it is that Grothendieck gave the world a new way to think about Gal(ℚ/ℚ) using the most homely of tools: childish-looking drawings. One might go so far as to compare dessins with Will Hunting himself: an unlikely source of deep mathematical insight. Moreover, some of the dessins bear a striking (albeit coincidental) resemblance to some diagrams Will drew on a blackboard in the hallway.

Will puts down his mop to work on a graph theory problem. From https://www.empireonline.com/movies/reviews/good-will-hunting-review/

Some dessins d’enfants.

Schneps was especially charmed by the way Grothendieck drew actual pictures to explain his ideas.  The sage usually stayed up in the clouds of abstraction, letting others figure out specific cases. Yet what it came to dessins d’enfants, he wasn’t content merely to lay the foundations of a general theory; he also worked out specific examples.

In their search for Grothendieck, Schneps and Lochak met with success, of a kind; they did eventually meet their hero. But he wasn’t always friendly, and he never, ever discussed math with them. That part of his life was over.

Old Grothendieck. From https://www.heidelberg-laureate-forum.org/laureate/alexander-grothendieck.html

(For another account of a Grothendieck fan tracking the man down, see Roy Lisker’s diary of his quest for Grothendieck: “Visiting Alexandre Grothendieck”.)

After Grothendieck’s death, Schneps and others were determined to preserve the thousands of pages of writing that Grothendieck had created during his decades living off-the-grid. Many of these writings are difficult or impossible to classify using standard categories of scientific writing, aubiography, or literature. For details of the work of the people dedicated to preserving Grothendieck’s mathematical and post-mathematical legacy, see the Grothendieck Circle webpage.

So, if you ask me nowadays about the movie Good Will Hunting, I won’t say that there couldn’t be such a character. Because in a way, there once was. (But his mentor? No way! Nobody gets an office that big.)

Thanks to Leila Schneps.