Birth of a Theorem: A Mathematical Adventure

by Cédric Villani

  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.

  * * *

  Oh, here’s a young woman walking alone, like me, one of the participants in the mathematics program. It turns out she heard my talk on optimal transport and found it a good introduction to the subject. We walk together the rest of the way back to Simonyi Hall, talking of mathematics in the fading spring light of a Princeton evening.

  Need to go up to my office one last time. It’s empty—except for an enormous stack of handwritten notes and rough drafts containing all my attempts to deal with one problem or another, successful and aborted ones alike, together with all the intermediate versions I had carefully composed, then compulsively printed out and furiously corrected.

  I would love to take them back to France with me, but it’s just too much to deal with at this point. We’ve got more than enough luggage to check at the airport as it is! There’s no choice but to throw it all out.…

  Seeing me standing there, staring at this huge pile of paper, my young companion immediately comprehends the minor tragedy of having to do away with the work of so many months, all of it fraught with so much emotion. She kindly offers to help me stuff it into the wastepaper basket.

  We stuff it into the basket—and pile the rest all around on the floor. There was enough to fill at least four baskets!

  My stay in Princeton is now well and truly at an end.

  * * *

  WELCOMING REMARKS BY THE DIRECTOR OF THE INSTITUT HENRI POINCARÉ, NOVEMBER 6, 2009

  For a long time I found the idea of bringing together young female mathematicians puzzling—until earlier this year when I participated myself, as a speaker, in the annual Program for Women and Mathematics at the Institute for Advanced Study in Princeton. The liveliness and enthusiasm of everyone who took part made an unforgettable impression on me. It is my sincere hope that the atmosphere of the lectures and discussions awaiting you in this ninth “Forum des Jeunes Mathématiciennes” at the Institut Henri Poincaré will be no less relaxed, and no less intense, than the one it was my good fortune to discover for the first time a few months ago. Welcome to the Maison des Mathématiciennes!

  THIRTY-THREE

  Lyon

  June 28, 2009

  How strange to be back home again, after being away so long!

  You haven’t really settled in after an extended stay abroad until you’ve gone out food shopping. You’re reunited at long last with your favorite grocers, you find the breads and cheeses you’ve been missing, you’re astonished to hear everyone speaking French. A glass of raw milk, my first in six months, brought tears to my eyes. The ciabatta’s soft crumb and the baguette’s crispy crust require no comment.

  But now that I’m back in my element, nothing is quite the same anymore. Remodelers were hard at work while we were gone, you can scarcely recognize the apartment.… But that’s nothing compared to the inner transformation I’ve experienced as a result of the work I did in Princeton. I feel like a mountain climber who has come back down to earth, his head filled with vivid images of the heights he has scaled. Chance redirected the course of my research to a degree I wouldn’t have thought possible six months ago.

  In the 1950s, the Metropolis–Hastings algorithm brought about a scientific revolution by showing that, if a space is too rich in possibilities, you are often better off moving around at random rather than exploring it in a systematic way or selecting probability distribution samples in a perfectly haphazard manner. Since then the algorithm has spawned an entire field of research, so-called Markov chain Monte Carlo (MCMC) methods, whose unreasonable effectiveness in physics, chemistry, and biology has yet to be satisfactorily explained. Exploration in the MCMC sense is neither wholly deterministic nor completely aleatory; it is something in between, a compromise between choosing the best possible path at every step and choosing each successive step at random. That way you are free from time to time to choose a path that, although it may not seem the best one at first, promises to lead to increasingly better options down the road.

  But there’s really nothing new about this. We do the same thing in everyday life: passing more or less at random from one situation to another enables us to explore many more possibilities than we could otherwise. The world of mathematics is no different: unpredictable encounters lead us from one problem to another, sometimes even from one scientific continent to another.

  And now, just when everything has finally been put back in its place, a new home is waiting to be furnished. My personal belongings are already packed in boxes; soon the movers will come and take away all the things I’ve lived with for so many years. The futon that my mother was amazed anyone would want to sleep on (like reinforced concrete, she said). The music system that after fifteen years of intensive use has more than made good on its promise of high fidelity. The hundreds of CDs that nearly ate up my salary as an undergraduate research assistant, the scavenged cassettes and the second-hand vinyl records. The massive wooden partners desk, the great colonial bookcases that hold vast numbers of books, the heavy armchair carved from a single piece of solid wood that I found in London, sculptures bought in the Drôme, my grandfather’s paintings—all these things are going to accompany me on my new adventure. Three days from now I take up my duties as director of the Institut Henri Poincaré in Paris: my predecessor vacates his office on June 30, I move in July 1. No time to prepare; I’ll just have to learn the job as I go along. The beginning of a new period in my life.

  A new stage in my personal MCMC.

  * * *

  Having survived a difficult period in the 1970s and 1980s, the “Home of Mathematics” embarked on a new chapter of its history in 1990. The substantial costs of the Institute’s physical renovation were mainly assumed by the state, under the terms of a four-year agreement with the Université Pierre et Marie Curie, the governing body of the new IHP, with additional funding from the Centre National de Recherche Scientifique.

  The new administrative structure was put in place under the guidance of the mathematician Pierre Grisvard, who died prematurely in 1994, a few months before the Institute’s official reinauguration by the minister for higher education and research. Grisvard was succeeded as director by Joseph Oesterlé (Université Pierre et Marie Curie), who was followed by Michel Broué (Université Denis Diderot) in 1999, and then by Cédric Villani (École Normale Supérieure de Lyon) in 2009.

  [From a brief history of the Institut Henri Poincaré]

  THIRTY-FOUR

  Prague

  August 4, 2009

  Prague! If ever there was
a mythic city in Europe, this is it. The legend of the Golem, Messia’s song, Mairowitz’s biography of Kafka—thoughts of these and many more things filled my head as I strolled through streets with thousand-year-old clocks and strip clubs cheek by jowl, and dance clubs with lines of students wearing devil horns and superhero capes waiting to get in.

  A few weeks ago, in the little town of Oberwolfach, passersby stared wide-eyed at my costume, but in Prague I could almost pass for an accountant.

  Yesterday was the opening of the International Congress of Mathematical Physics, held every three years under the auspices of the international association of the same name, the IAMP. I was one of four who were honored with great pomp and ceremony as winners of the Henri Poincaré Prize. In addition to the Austrian Robert Seiringer (recognized, as I was, in the junior category), the laureates included the Swiss Jürg Frölich and the Russian Yasha Sinai. Specialists in classical and quantum mechanics, statistical physics, and dynamical systems—all of them friends of the farsighted Joel Lebowitz, who long ago invited them to serve on the editorial board of his Journal of Statistical Physics. I was both pleased and flattered to find myself in such good company.

  Winning the Poincaré Prize meant I was entitled to give a plenary address to the Congress even though I hadn’t been selected as a speaker beforehand. Although I had received the prize for my work on Boltzmann, I chose instead to talk about Landau damping: an unexpected opportunity to communicate my results to the most distinguished audience of mathematical physicists imaginable.

  Three minutes before I was scheduled to begin, my heart was beating wildly, adrenaline coursing through my veins. But once I began to speak, I felt calm and sure of myself.

  “It so happens that I was recently appointed director of the Institut Henri Poincaré, and now I have been awarded the Henri Poincaré Prize. Just a coincidence, of course—but I like it.…”

  I had meticulously prepared my speech. Everything went as planned, I finished right on time.

  “To conclude, let me note a nice coincidence. In order to treat the singularity of the Newton interaction, you use the full power of the Newton scheme. Newton would be delighted! Again, this is just a coincidence—but I like it.…”

  Thunderous applause. On certain faces in the audience I thought I detected looks not only of astonishment and admiration, but also, to some extent, of fear. To be honest, even I find the proof intimidating!

  And then there were the Czech girls. Before my turn came to speak they didn’t pay much attention to me, but afterward it was an entirely different story: they crowded around, congratulated me on the clarity of my exposition; one young woman rather emotionally recited a little speech in shaky French.

  Unsurprisingly, my results once more prompted the same two questions. Can the analytic regularity assumption be relaxed? For a Newton interaction, isn’t it possible for the damping to go on and on, in infinite time? None of this mattered in the least to my Portuguese friend Jean-Claude Zambrini. Amid the clamor outside the lecture hall he exclaimed, “Since you attract coincidences, Cédric, let’s hope that you’ll receive an invitation from the Fields Institute next!”

  The Fields Institute for Research in Mathematical Sciences—which plays no role in the awarding of the Fields Medal—is located in Toronto, and regularly sponsors colloquia on all sorts of topics.

  Jean-Claude and I had a good laugh. But only a month and a half later, by pure coincidence, the invitation arrived.

  * * *

  Date: Tue, 22 Sep 2009 16:10:51 -0400 (EDT)

  From: Robert McCann

  To: Cedric Villani

  Subject: Fields 2010

  Dear Cedric,

  Next fall I am involved in organizing a workshop on “Geometric Probability and Optimal Transportation” Nov 1-5 as part of the Fields Theme Semester on “Asymptotic Geometric Analysis”.

  You will certainly be invited to the workshop, with all expenses covered, and I hope you will be able to come. However, I also wanted to check whether there is a possibility you might be interested in visiting Toronto and the Fields Institute for a longer period, in which case we would try to make the opportunity attractive.

  Please let me know,

  Robert

  THIRTY-FIVE

  New York

  October 23, 2009

  Back home, about a week ago, the children made the acquaintance of a baby wild boar their uncle had captured with his bare hands. How I wish I could have been there!

  But I’d convinced myself that during their fall break from school my time would be better spent lecturing in the United States—a grueling tour that will take me through half the country in just ten days. Already I’ve been to Cambridge, paying visits to MIT (in the footsteps of Wiener and Nash!) and Harvard. Now I’m in New York. I console myself with the thought that once I return to France I’ll be able to go see the little piglet and take him for a walk in the woods.

  On opening my email this evening, my heart skipped a beat: a message from Acta Mathematica, considered by many to be the most prestigious of all mathematical research journals. This is where Clément and I submitted our one-hundred-eighty-page monster for publication. No doubt that’s what they’re writing me about.

  But … it’s not even four months since we submitted it! Considering the size of the manuscript, it’s much too soon for the editorial board to have decided in our favor. Only one possible explanation: they’re writing to say that the article has been rejected.

  I open the message, skim the cover letter, feverishly scroll through the attached referees’ reports. Lips pursed, I read through them again, this time more carefully. Six reports, very positive on the whole. Not a word to be said against us, except … sure enough, always the same misgiving: it’s the analyticity assumption that bothers them—that and the limit case in large time. Always the same two questions. I’ve already had to respond to them dozens of times over the past few months—and now it looks like they’ve sunk our hopes of publication! The editor who signed the letter isn’t persuaded that our results are definitive, and because the manuscript is so long he feels obliged to hold it to a stricter standard of review than is customary.

  But this simply isn’t right!! After all the ingenuity we’ve displayed, all the new ground we’ve broken, all the technical obstacles we’ve managed to overcome, all the days of hard work that stretched on late into the night, night after night—and still it’s not good enough for them?? Give me a break, for Chrissake!

  But then … I noticed another message, informing me that I’d just won the Fermat Prize. Named after the great Pierre de Fermat, whose mathematical enigmas infuriated the whole of seventeenth-century Europe; the prince of amateurs, as he was known, who revolutionized number theory and the calculus of variations, and laid the foundations of modern probability theory. Since 1989 the prize established in his name has been awarded every two years to one or two mathematicians under the age of forty-five who have made major contributions in one of these domains.

  The news comes as a great consolation. Even so, it isn’t enough for me to get over the frustration of seeing our article turned down. To get over that, I’d need a big long hug. At a minimum.

  * * *

  In 1882, Gösta Mittag-Leffler persuaded a group of fellow mathematicians in his native Sweden and the other Scandinavian countries to create a review devoted to publishing research of the highest quality. It was named Acta Mathematica, with Mittag-Leffler as its first editor-in-chief.

  Mittag-Leffler was in touch with the top mathematicians in the world. Blessed with unfailingly good taste, and an unusually large appetite for risk, he rapidly succeeded in attracting the most exciting new work of the day. His favorite author was the brilliant and unpredictable Henri Poincaré, whose long, trail-blazing manuscripts Mittag-Leffler published without the least hesitation.

  The most famous episode in the history of Acta Mathematica is also one of the most famous ep
isodes in Poincaré’s career. In 1887, on Mittag-Leffler’s advice, King Oscar II of Sweden announced an international mathematics competition. Poincaré accepted the challenge, and from a short list of topics selected the stability of the solar system, an open problem since Isaac Newton first posed it two hundred years earlier! Newton had written down the basic equations for planetary movement in the solar system (the planets are attracted by the sun and attract one another), but he was unable to show that these equations entail the stability of the solar system itself, or to determine whether, on the contrary, they contain the seeds of a preordained catastrophe—the collision of two planets, for example. Every mathematical physicist of Poincaré’s time was familiar with the problem.

  Newton thought that the solar system was inherently unstable, and that the stability we perceive is due to a divine helping hand. Later, however, the results obtained first by Laplace and Lagrange, and then by Gauss, made it clear that Newton’s system is stable over a huge period of time, perhaps as long as a million years—much longer than Newton himself believed. This meant that the behavior of the sun and the planets had been qualitatively predicted on a time scale far greater than the whole of recorded human history!

  One question nonetheless remained unanswered: Once such a huge period of time has passed, is catastrophe likely—even inevitable? If we wait not one million but one hundred million years, are Mars and Earth in real danger of colliding? This turned out to be no ordinary problem, for it concealed fundamental questions about physics itself.

  Poincaré didn’t propose to treat the motions of the entire solar system. Way too complicated! Instead he considered an idealized model of the system on a reduced scale, taking into account only two bodies turning around the sun, one of which he assumed to be minuscule by comparison with the other—rather as though one were to neglect all the planets except Jupiter and Earth. Poincaré studied this simplified problem, and then simplified it still further, until he had finally gotten to the very heart of the matter. Devising novel methods to suit his purpose, he proved the eternal stability of this scaled-down system!