Birth of a Theorem

Five months at IAS, two-hundred-fifty pages, and a Fields Medal

By Cédric Villani Published 2015

Claire Calmet

Cédric Villani in the Institute woods in 2009

Cédric Villani, Member in the School of Mathematics in the spring of 2009 and currently Professor at Université Lyon I and Director of the Institut Henri Poincaré, has called his stay at the Institute one of his most productive periods, during which more than 250 pages were written. In his Member report to then-Director Peter Goddard at the end of his stay, Villani wrote of his collaboration with Clément Mouhot from Paris, “Writing up the paper on Landau damping was one of the most intense experiences of my professional life: for three months in a row, we kept unlocking seemingly untractable obstacles on a weekly basis. Our 180-page-long paper solves a fifty-year-old open problem.” A year after his IAS visit, Villani was awarded the 2010 Fields Medal, in part for the work that he did at the Institute on his proof of nonlinear Landau damping. Following are excerpts from Birth of a Theorem, translated by Malcolm DeBevoise (Farrar, Straus and Giroux, 2015), originally published in 2012 as Théorème Vivant (Éditions Grasset & Fasquelle), which describe his fervent, halting, and very human experience in trying to obtain the proof.

Princeton, January 1, 2009

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! [. . .]

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. [. . .]

Right now I’m only thirty-five . . . but with the clarification of the eligibility rule adopted at the last International Congress of Mathematicians, from now on [Fields Medal] 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! 

__________________________

Princeton, January 17, 2009

Saturday evening, dinner together at home.

The whole day was taken up with a trip organized by the Institute for visiting members. A trip to the holiest of shrines for anyone who’s enthralled by the story of life: the American Museum of Natural History in New York.

I recall very well my first visit to this museum, almost exactly ten years ago. The excitement of seeing some of the most famous fossils in the world, fossils whose pictures are found in the guides and dictionaries of dino­saurs that I devoured as a teenager, was indescribable.

Today I went back ten years into the past and left my mathematical cares behind for a few hours. Over dinner, however, they caught up with me.

Claire was rather taken aback, seeing my face contorted by tics and twitches.

The proof of Landau damping still hasn’t come together. My mind was churning. 

What do you have to do, for God’s sake, what do you have to do to get a decay through transfer of regularity with respect to position when the velocities have been composed . . . this composition is what introduces a dependence with respect to velocity—but I don’t want any velocities!

What a mess.

I scarcely bothered to make conversation, responding in as few words as possible, otherwise by grunts.

“Was it ever cold today! We could have gone sledding. . . . Did you happen to notice the color of the flag at the pond this morning?”

“Hmmm. Red. I think.”

Red flag: even if the pond looks frozen, walking on it is prohibited, it’s too dangerous. White flag: go ahead, guys, the water’s frozen solid, jump and shout, dance on the ice if you like.

And to think that I accepted an invitation to present my results at a statistical physics seminar at Rutgers on January 15! How could I have accepted when the proof wasn’t complete? What am I going to tell them?

Well, when I got here at the very beginning of the month, I was completely sure I could finish the project in two weeks—max! Fortunately the talk got pushed back by another two weeks! But even with this reprieve, am I going to be ready?? January 29 isn’t very far away!! I never thought it would be so hard. No way I could have foreseen the obstacles that lay ahead!

The velocities are the problem, the velocities! When there isn’t any dependence with respect to velocity, you can separate the variables by means of a Fourier transform, but when you’ve got velocities, what can you do? In a nonlinear equation, velocities are obligatory—there’s no way I can avoid dealing with them!

“Are you all right? Really, there’s no point worrying yourself sick! Relax, take it easy.”

“Can’t.” 

“You really seem obsessed.”

“Look, I’m on a mission. It’s called nonlinear Landau damping.”

“I thought you were supposed to be working on the Boltzmann equation. That was your big project, wasn’t it? You don’t want to lose sight of what you came here to do, do you?”

“Can’t be helped. Right now it’s Landau damping.”

But Landau damping goes on playing the cold, unattainable beauty. I can’t get next to her.

. . . Still, there’s that little calculation I did on getting home from the museum—doesn’t that give some reason for hope? But man, is it ever complicated! I added two more parameters to the norm. Our norms used to have five regularity indices, which already was the world record—now they’re going to have seven! But so what, applying the two indices to a function that doesn’t depend on velocity leaves you with the same norm as before, there’s no inconsistency. . . . I’ve got to check the calculation carefully. But if I try to do it right now, it’s going to turn out wrong, so let’s wait until tomorrow! I’m going to have to do the whole damn thing over again, this time with norms that have got seven indices. Good Lord.

Seeing how glum I looked, Claire felt sorry for me. Or at least felt she had to do something to cheer me up.

“Look, tomorrow’s Sunday. You can spend the day at the office if you like; I’ll take care of the little lambkins.” 

At that moment, nothing in the world could have pleased me more.

Princeton, January 21, 2009

Thanks to the rabbit I pulled out of my hat on my returning from the museum the other evening, I’ve been able to get back on track. But today I’m filled with a strange mixture of optimism and dread. Got around one major roadblock: made a few explicit calculations and eventually figured out how to manage a term that had gotten too big—that much gives me hope. At the same time, the complexity of the mathematical landscape that’s now opened up makes my head spin if I think about it for more than a few moments.

Could it really be that Vlasov’s splendid equation, which I thought I was beginning to get a handle on, operates only by fits and starts? On paper, at least, it looks as though sometimes the response to external perturbations suddenly occurs very, very quickly. I’ve never heard of such a thing; it’s not in any of the articles and books that I’ve read. But in any case we’re making progress.

__________________________

Princeton, February 27, 2009

A bit of a party atmosphere at the Institute today now that the five-day workshop on geometric partial differential equations is coming to an end. Very fine casting, with many stars—all the invited speakers agreed to participate. 

In the seminar room I found a place to stand all the way at the back, behind a large table. Sometimes an audiovisual control board is set up on it, but not today, so I could spread out my notes on top. There’s no better place to be. I was lucky to get there before Peter Sarnak, a permanent professor at the Institute who likes it as much as I do. You can always be sure of staying awake, for one thing. If you’re sitting in a chair you’re more likely to drift off—and you’ve also got to settle for writing on a small fold-down tablet. 

I like to be able to pace back and forth in my stocking feet when I’m listening to a lecture, ideal for stimulating thought. 

At the break I rushed outside without bothering to put my shoes back on and ran upstairs to my office. Quick phone call to Clément.

“Clément, did you see my message from yesterday with the new file?”

“You mean the new scheme you got by writing down the characteristic equations? Yeah, I looked at it and I began to do the calculations. Looks like a bear to me.”

Monster, beast, bear—these words occur over and over again in our conversations. . . . 

“I have a feeling we’re going to run into problems with convergence,” Clément continued. “I’m also worried about Newton’s scheme and the linearization error terms. There’s another technical detail, too—you’re always going to have scattering from the previous step, and it won’t be trivial!”

I was a bit annoyed that my brilliant idea hadn’t convinced him.

“Well, we’ll see. If it doesn’t work, too bad, we’ll stick with the present scheme.”

“It’s pretty wild—we’ve got more than a hundred pages of proofs by this point and we’re still not done yet!! Do you really think we’ll ever finish?”

“Patience, patience. We’re almost there. . . .”

The intermission in the seminar room was over. I hurried back downstairs to hear the concluding talks.

Princeton, March 1, 2009

I read the message that had just appeared on my computer screen, and then read it again. Couldn’t believe my eyes.

Clément’s come up with a new plan? He wants to give up on regularization? Wants to forget about making up for the loss of regularity encoded in the time interval?

Where did all this come from? For several months now we’ve been trying to make a Newton scheme work with regularization, as in Nash-Moser—and now Clément is telling me that we need to do a Newton scheme without regularization? And that we’ve got to estimate along the trajectories, while preserving the initial time and the final time, with two different times??

Well, maybe he’s right, who knows? Cédric, you’ve got to start paying attention, the young guys are brilliant. If you don’t watch out, they’re going to leave you in the dust! 

Okay, there’s nothing you can do about it, the next generation always ends up winning . . . but . . . already?

Save the sniveling for later. First thing, you’ve got to try to understand what he’s getting at. What does this whole business of estimating really amount to, when you get right down to it? Why should it be necessary to preserve the memory of the initial time?

In the end, Clément and I will be able to share the credit for the major innovations of our work more or less equally: I came up with the norms, the deflection estimates, the decay in large time, and the echoes; he came up with the time cheating, the stratification of errors, the dual time estimates, and now the idea of dis­pensing with regularization. And then there’s the idea of gliding norms, a product of one of our joint working sessions; not really sure whose idea that was. To say nothing, of course, of hundreds of little tricks . . . [. . .]

If Clément is right, the last great conceptual obstacle has just been overcome. On this first day of March our undertaking has entered into a new phase, less fun, but also more secure. The overall plan is in place, the period of free-ranging, open-ended exploration is over. Now we’ve got to consolidate, reinforce, verify, verify, verify. . . . The moment has come for us to deploy the full firepower of our analytical skills!

Tomorrow I’m taking care of the kids; there’s no school on account of the snowstorm. But come Tuesday, the final push begins. One way or another the Problem simply has got to be tamed, even if it means going without sleep. I’m going to take Landau with me everywhere—in the woods, on the beach, even to bed. Time now for him to watch out!

__________________________

Princeton, night of April 8–9, 2009

Version 55. The tedious process of rereading and fine-tuning. Then, suddenly, a new hole opens up.

Hopping mad, I’ve just about had it.

Had it up to here with this whole business! Before it was the nonlinear part. Now it’s the linear part that seemed to be under control and then came apart at the seams!

We’ve already announced our result more or less everywhere: last week I gave a presentation in New York, tomorrow Clément’s doing the same in Nice. There’s no excuse for the slightest error at this point—the thing has to be completely correct!!

But there’s no getting around it, there is a problem. Somehow this damn Theorem 7.4 has got to be fixed. . . .

The children are asleep, Claire’s away again. Working in front of the big picture window, looking out into the dark night. The hours go by. Sitting on the sofa, lying on the sofa, kneeling in front of the sofa, I turn my bag of tricks inside out, scribbling away, page after page. To no avail.

I go to bed at four o’clock in the morning in a state close to despair.

__________________________

Princeton, June 26, 2009

My last day in Princeton. Rain, nothing but rain the last few weeks—so much, in fact, it was almost comical. At night the fireflies transform the oaks and red maples into Christmas trees, romantic, impossibly tall, decorated with innumerable blinking lights. Enormous mushrooms, a small furtive rabbit, the fleeting silhouette of a fox in the night, the startling noises of stray rutting deer . . . 

In the meantime a lot has happened on the Landau damping front! We were finally able to get the proof to hang together, went through the whole thing one last time to be sure. What a wonderful feeling, finally to post our article online! As it turned out, it was in fact possible to control the zero mode. And Clément discovered that we could completely do without the double time-shift, the trick I came up with on my return from the Museum of Natural History back in January. But since we didn’t have the courage to go over the whole thing yet again, and since we figured it might come in handy dealing with other problems, we left it where it was. It’s not in the way, not interfering with anything. We can always simplify later if we have to.

By this point I’ve given quite a few talks about our work. Each time I was able to improve both the results and the exposition, so the proof is now in very good shape. There may still be a bug somewhere, of course. For the moment, however, all the pieces fit together so well that if an error is discovered, I’m confident we’ll be able to fix it. [. . .]

The state of mathematical grace in which I had been living almost from the beginning of my stay in Princeton lasted until the very end. Once the Landau damping problem was solved I immediately went back to my other major project, the collaboration with Ludovic and Alessio. Here again, just when the proof seemed to be in jeopardy, we were able to overcome all the obstacles facing us and everything began to click, as if by magic. Our good fortune included one true miracle, by the way, an enormous calculation in which fifteen terms recombined to constitute a perfect square—an outcome as unhoped for as it was unexpected, since ultimately what we succeeded in demonstrating was the opposite of what we had set out to demonstrate!

With regard to Landau damping, it must be admitted that Clément and I didn’t manage to solve quite everything. For electrostatic and gravitational interactions, the most interesting cases, we were able to show that damping occurs on a gigantic time scale, but not an infinitely long one. And since we were stymied on this point, we were also stymied on regularity—we couldn’t find a way to get out from the analytic framework. Very often at the end of my talks somebody would ask one of these two questions: In the case of Coulomb or Newton interaction, does one also have damping in infinite time? Can one do without the analyticity assumption? In either case my response was the same, that I couldn’t say anything without consulting my lawyer first. Honestly, I don’t know whether there’s a profound mystery here, or whether we simply weren’t clever enough to work out the answer. [. . .] 

Published in The Institute Letter Spring 2015