Birth of a Theorem: A Mathematical Adventure (11 page)

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 Bambergers 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.

Mathematicians from around the world come and go, explaining their latest discoveries, hoping to attract the attention of the resident faculty members. Invited visitors, whether they stay for a few months or a few years, must think of only one thing while they are here: producing the best theorems possible—and this under the watchful eye of Albert Einstein himself, whose faintly quizzical smile greets them everywhere they go. In sculptures, photographs, and paintings, Einstein is a constant presence at the Institute.

As a guest of the Institute, your every need has been anticipated. If you are a mathematician, you won’t have to worry about anything other than mathematics. If you are accompanied by your family, your children will have been enrolled in school well in advance of your arrival. An army of secretaries stands ready to answer any question and resolve any difficulty. An apartment will be waiting for you only a few minutes from your office. The excellent dining hall will save you the trouble of looking for a restaurant. If you feel like taking a leisurely stroll, you need look no further than the woods that are all around you. Scarcely will you have set foot in the wood-paneled Mathematics–Natural Sciences Library in Fuld Hall than a librarian will introduce herself and help you find the book you are looking for, explaining the card catalogue system that is still in use there, as efficient as it is old-fashioned. The message is unmistakable:
Listen, kid, everything you need is right here. Forget about the outside world—your job is to think about mathematics, mathematics, and nothing but mathematics.

If you should ever happen to visit the Institute in the summer, be sure to go see the modernist Historical Studies–Social Science Library overlooking the pond, across from the mathematics building. At night it’s deserted. You will feel like an explorer discovering a cave filled with treasures from another age, collections of old maps three feet high and wide, massive dictionaries and encyclopedias heavy enough to be used as doorstops.

Then, on coming out of the library, pause at the edge of the pond: on a late June evening it is the most beautiful place in the world. If you’re lucky you’ll hear bellowing deer, you’ll see the ghostly flickering of fireflies, you’ll be mesmerized by the moon’s shimmering reflection on the dark water—and you’ll sense a spectral aura, the aura of some of the most powerful minds of the twentieth century, forming an invisible mist above the pond.

TEN

Princeton

January 12, 2009

Late at night in our new apartment. I’m sitting on the carpeted floor, surrounded by sheets of paper filled with equations and notes. In front of me the big picture window through which the children watch the gray squirrels scurry about outside during the day. Thinking and scribbling away, not saying a word.

In the office, just next door, Claire is watching
Death Note
on her laptop computer. There aren’t many cinemas within walking distance of the Institute, so you’ve got to look for entertainment closer to home. I’ve praised the film version of this diabolical series to the skies. Now it’s Claire’s turn to get hooked on it. An opportunity to listen to Japanese again as well.