Birth of a Theorem: A Mathematical Adventure
Page 5
And so I find myself very much in the holiday spirit even with my work. While the children excitedly open their Christmas presents, I’m hanging exponents on functions like balls on a tree and lining up factorials like upside-down candles!
* * *
Donald Knuth is the living god of computer science. As one of my colleagues put it, “If Knuth were to walk into the hall in the middle of a lecture, everyone would bow down before him.”
Knuth took early retirement from Stanford and turned off his email in order to devote himself full-time to completing his major work, The Art of Computer Programming, begun almost fifty years earlier. In the meantime, the three volumes already published by 1976 had revolutionized the subject.
Donald Knuth
In the course of producing these prodigies, Knuth was exasperated by the wretchedly poor graphic quality of the mathematical symbols translated from text file to screen by the software programs that were then commercially available, and resolved to remedy the situation once and for all. Redesigning text editors or fonts wasn’t enough: he was determined to rethink the whole business from the ground up. In 1989 he published the first stable version of the TEX typesetting system. The promise of this new universal medium was to be fully realized a decade later, at the beginning of the twenty-first century, when mathematical communication became massively electronic. By the 1990s, however, it was already the standard format used by mathematicians everywhere to compose and share their work.
Knuth’s language and its offspring are known as free software, since their source code is available to anyone at no cost. Mathematicians exchange text files, which is to say documents consisting solely of ASCII characters, an alphabet that is recognized by computers throughout the world. Such files contain all the instructions needed to reconstruct natural language text and mathematical formulas down to the smallest detail.
With the invention of this typesetting program, Knuth has probably done more than any other living person to change the daily working lives of mathematicians.
Knuth continually worked to improve the original model. The version numbers he assigned to his program are approximations of π, ever more precisely estimated as the program was gradually perfected: after version 3.14 came version 3.141, then 3.1415, and so on. The current version is 3.1415926; according to the terms of Knuth’s will, it will change to π immediately following his death, thus fixing TEX for all eternity.
* * *
Faà di Bruno’s Formula (Arbogast 1800, Faà di Bruno 1855)
… which in TEX is written
[(f circ H)^{(n)} = sum_{sum_{j=1}^n j,m_j = n}
frac{n!}{m_1!ldots m_n!},
bigl(f^{m_1 + ldots + m_n}circ Hbigr),
prod_{j=1}^nleft(frac{H^{(j)}}{j!}right)^{m_j}]
* * *
Date: Thu, 25 Dec 2008 12:27:14 +0100
From: Cedric Villani
To: Clement Mouhot
Subject: Re: parts 1 and 2, almost done
Here you are, another Christmas present in the form of part II. It looks very promising, at last everything works better than one could have hoped on the whole (except that the exponent loss looks to be at least on the order of a cubic root of the size of the perturbation, but there’s no reason why it couldn’t be recovered using a Newton-style iterative method). I’m sending you two files: analytic and scattering, I’ve stopped fiddling with them for the moment. It will be necessary to go over them very carefully, but I think that now our priority should be to make parts 3 and 4 (PDE and interpolation) converge, I suggest that you send me the PDE part once it looks as though it more or less holds together even if it still needs some polishing; that way we can work in parallel on the PDE and interpolation. (I’ll see to putting it into English and proper form…)
And Merry Christmas!
Cedric
Date: Thu, 25 Dec 2008 16:48:04 +0100
From: Clement Mouhot
To: Cedric Villani
Subject: Re: parts 1 and 2, almost done
Merry Christmas and thanks for the presents;)!!
I’m working on the PDE file to make a complete theorem with sup and mixed norms, in fact I have high hopes even for the mixed norm (the scattering is really normed according to your last file so it seems necessary). As for the interpolation file, you’ll find a draft (in French) of the revised Nash inequality we need in the version I sent you earlier, tell me if it needs any more work. More very soon! Best wishes, Clement
Date: Fri, 26 Dec 2008 17:10:26 +0100
From: Clement Mouhot
To: Cedric Villani
Subject: Re: parts 1 and 2, almost done
Hi,
Here’s a preliminary version, in English, of the complete PDE theorem with your mixed norm, starting on page 15 of the file. I’m sending it to you now to give you an idea, even if there are still a few things I have to check regarding the details of the calculations and the indices…, and the time boundary condition which looks rather odd for the moment. In any case the mixed norm seems to fit pretty well with the argument I was making about transfers of normed derivatives without Fourier. At the beginning of section 4, in connection with the theorem I mentioned, I’ve made a few remarks about why it seems to hold up. On the other hand I’m still working with a norm with four indices (even it’s definitely a mixed norm by your definition) and for the moment at least I don’t quite see how to get it down to only three indices …
I’ll think about it some more.
Best wishes,
Clement
Date: Fri, 26 Dec 2008 20:24:12 +0100
From: Clement Mouhot
To: Cedric Villani
Subject: Re: parts 1 and 2, almost done
What I’ve been calling the “time boundary condition” is the fact that since the loss on the index due to scattering is linear with respect to time (as I put it in the assumption), that meant there had to be a time boundary if the loss wasn’t to be greater than a certain constant. But now it seems to me, having looked at your “analytic” file, that the assumption has to be strengthened, something like a loss
$$
varepsilon , min {1, (t-s) }
$$
which allows the loss to remain small for large $t$ and for $s$ far from $t$ …
v. best, clement
NINE
Princeton
January 1, 2009
It’s pitch-dark; the taxi driver’s completely bewildered. His GPS is pointing in a plainly absurd direction: straight ahead into the trees.
I try appealing to his common sense. We’ve already passed by here once before, obviously the GPS is on the fritz, there’s no choice but to explore the surrounding area. In other words, we’re lost. The only thing that’s certain is that if we follow the machine’s instructions, we’ll wind up getting stuck in the mud and the melting snow!
In back, the children aren’t the least bit worried. My daughter is asleep, worn out by the plane trip and the change in time. My son is watching intently. He’s only eight years old but already he’s been to Taiwan, Japan, Italy, Australia, and California, so getting lost somewhere in New Jersey isn’t about to frighten him. He knows that everything’s going to turn out all right.
We drive around some more, see the twinkling lights of civilization in the distance, and then encounter a human being at a bus stop who gives us directions. A GPS has no monopoly on topographic truth.
Finally, the Institute for Advanced Study—the IAS, as everyone calls it—comes into view. A little like a castle rising up in the middle of a forest. We had to go around a large golf course in order to find it.…
* * *
It is here that Einstein spent the last twenty years of his life. True, by the time he
came to America he was no longer the dashing young man who had revolutionized physics in 1905. Nevertheless, his influence on this place was deep and long-lasting, more so even than that of John von Neumann, Kurt Gödel, Hermann Weyl, Robert Oppenheimer, Ernst Kantorowicz, or John Nash—great thinkers all, whose very names send a shiver down the spine.
Their successors include Jean Bourgain, Enrico Bombieri, Freeman Dyson, Edward Witten, Vladimir Voevodsky, and many others. The IAS, more than Harvard, Berkeley, NYU, or any other institution of higher learning, can justly claim to be the earthly temple of mathematics and theoretical physics. Paris, the world capital of mathematics, has many more mathematicians. But at the IAS one finds the distillate, the crème de la crème. Permanent membership in the IAS is perhaps the most prestigious academic post in the world!
And, of course, Princeton University is just next door, with Charles Fefferman and Andrei Okounkov and all the rest. Fields medalists are nothing out of the ordinary at Princeton—you sometimes find yourself seated next to three or four of them at lunch! To say nothing of Andrew Wiles, who never won the Fields Medal but whose popular fame outstripped that of any other mathematician when he broke the spell cast by Fermat’s great enigma, which for more than three hundred years had awaited its Prince Charming. If paparazzi specialized in mathematical celebrities they’d camp outside the dining hall at the IAS and come away with a new batch of pictures every day. This is the stuff that dreams are made on.…
But first things first: we had to locate our apartment, our home for the next six months, and then get some sleep!
Some people might wonder what there is to do for six months in this very small town. Not me—I’ve got plenty to do! Above all I need to concentrate. Especially now that I can give my undivided attention to my many mathematical mistresses!
First I’ve got to wrestle the Landau damping beast to the ground and break its back. I’ve made good progress so far; the functional framework is firmly established. Two weeks ought to be enough—come on now, time to be done with it! After that I have to finish up the project with Alessio and Ludovic. So far it’s eluded us, the damn counterexample we need to prove that for dimension 3 or greater the injectivity domains of an almost spherical Riemannian metric are not necessarily convex. But we’re going to find it, and when we do it’ll be curtains for the regularity theory of non-Euclidean optimal transport!
If that can be done in two weeks, then I’ll have five months left to devote to my great ambition: proving regularity for the Boltzmann! I’ve brought along all my notes, jotted down in a dozen different countries over the last decade.
Five months might well turn out not to be enough. I was planning to spend two years on the Boltzmann, from last June through the end of my term as a junior member of the Institut Universitaire de France, a five-year appointment during which I have a reduced teaching load in order to give more time to research.
But I keep getting sidetracked. When I began my second book on optimal transport in January 2005, I was determined to limit myself to one hundred fifty pages and to deliver a manuscript sometime in July that same year. In the end it came to a thousand pages, and I didn’t finish until June 2008. More than once I thought of stopping midway through and getting back to work on the Boltzmann. But I decided it would be best to persevere. To be honest, I’m not sure I had a choice: it was the book that decided. It couldn’t have been any other way.
On stories I really like, I’ve sometimes fallen behind … but it doesn’t matter.
As matters stand now, however, I’ve got only eighteen months left with a reduced teaching load and I still haven’t started on what was supposed to be my Big Project. So the invitation to spend a half year in Princeton came at just the right moment. No book to finish, no administrative responsibilities, no courses to teach—I’m going to be able to do mathematics full-time. The only thing I’m required to do is show up for lectures now and then and take part in seminars on geometric analysis, the special theme this year at the IAS School of Mathematics.
Not everyone in the mathematics laboratory at ENS-Lyon was happy about this. They were all counting on me to take over as director of the lab starting in January 2009, exactly the moment I chose to take a leave of absence. Too bad—there are times when one has to put one’s own interests first. I’ve worked for years to help strengthen our group. Once the Princeton interlude is over, I shall be more than happy to work on behalf of the general interest once again.
And then there’s the Fields Medal!
The prize whose name no one who covets it dares speak. The highest award there is for mathematicians in their prime, given out every four years on the occasion of the International Congress of Mathematicians to two or three or four mathematicians under the age of forty.
Of course, it’s not the only swell prize to be won in mathematics! Indeed, the Abel Prize, the Wolf Prize, and the Kyoto Prize are all probably harder to win than the Fields Medal. But they don’t have the same impact or give the same exposure; and since they come at the end of a mathematician’s career, they don’t serve the same purpose, of recognizing early promise and encouraging continued achievement. The Fields is far more influential.
One tries not to think of it. Thinking of it, trying to win it, would only bring bad luck.
One doesn’t even refer to it by name. I’m careful not to mention it in conversation at all. In correspondence I speak simply of the “FM.” Whoever I’m writing to knows what I’m talking about.
Last year I won the prize awarded by the European Mathematical Society. In the eyes of many of my colleagues, this was a sign that I was still in the running for the FM. Perhaps the biggest thing in my favor is my range of interests, unusually broad for someone of my generation: analysis, geometry, physics, partial differential equations. And it doesn’t hurt that the young Australian prodigy Terry Tao is no longer a candidate, having won the medal at the last ICM in 2006, in Madrid, just after his thirty-first birthday.
But my accomplishments are not entirely immune to criticism. The conditional convergence theorem for the Boltzmann equation, in which I take such pride, assumes regularity; for the theorem to be perfect, regularity would have to be proved. My work on the theory of Ricci bounds in the weak sense is still in its early stages. The general criterion we’ve proposed for curvature-dimension is not yet unanimously accepted. And even the great advantage of my versatility carries with it the disadvantage that probably no one mathematician is qualified to judge my achievement as a whole. To have a chance, and also for the sake of my own peace of mind, what I need to do soon—very soon—is to prove a difficult theorem on a significant physical problem.
Then there is the age limit of forty. Right now I’m only thirty-five … but with the clarification of the eligibility rule adopted at the last ICM, from now on candidates must be under the age of forty on January 1 of the year of the congress. The moment the new rule was officially announced, I understood what it meant for me: in 2014 I will be too old by three months, so the FM will be mine in 2010—or never. The pressure is enormous!
Since then not a day has gone by without the medal trying to force its way into my mind. Each time it does, I beat it back. Political maneuvering isn’t an option, one doesn’t openly compete for the Fields Medal; and in any case the identity of the jurors is kept secret. To increase my chances of winning the medal, I mustn’t think about it. I must think solely and exclusively about a mathematical problem that will occupy me completely, body and soul. And here at the IAS, I’m in the ideal place to concentrate, following in the footsteps of the giants who came before me.
Just think of it—I’m going to live on Von Neumann Drive!
* * *
When the stock market crashed in 1929, Louis Bamberger and his sister Caroline Bamberger Fuld could consider themselves lucky. They had amassed a fortune from their chain of department stores in Newark, New Jersey, then sold the business six weeks before the stock market collapsed. At a time when the economy lay in ruins, the B
ambergers were rich. Very rich.
There is no point being wealthy if one does not put one’s wealth to good use. The Bambergers wished to serve a worthy cause, to change society for the better. Their first thought had been to endow a dental school, but soon they were persuaded that their fortune would be best used to establish an institute of theoretical science. Theory was relatively inexpensive. And with all the money at their disposal, why not aim to create the world’s foremost institute of theoretical science, an institute whose influence would extend beyond the seas and across the oceans?
In mathematics and theoretical physics, even if researchers don’t see eye to eye on everything, they do agree about who the best people are. And once the best people have been identified, well, surely they’ll want to pitch in and help make this dream a reality!
After several years of patient negotiation the Bambergers succeeded in luring away the very best, one after another. Einstein came in 1933. Then Gödel. Weyl. Von Neumann. And many more … As the political climate in Europe became increasingly unbearable for Jewish scientists and their friends, the world’s scientific center of gravity shifted from Germany to the United States. By 1939 the Bambergers’ dream had assumed concrete form with the dedication of the Institute’s first building, Fuld Hall. On eight hundred acres of land! Adjacent to Princeton University, a prestigious institution almost two hundred years old, itself the beneficiary of another family of philanthropists, the legendarily wealthy Rockefellers. Permanent members of the IAS could look forward to an even more comfortable life than their counterparts at Princeton: no courses to teach, no administrative duties—and extremely generous salaries!
The Institute evolved over the years. Today the School of Natural Sciences is home not only to theoretical physics in all its forms (astrophysics, particle physics, quantum mechanics, string theory, and so on) but also to theoretical biology. Schools of social science and of historical studies came to be added as well, both carrying on the same tradition of excellence.