Birth of a Theorem: A Mathematical Adventure

by Cédric Villani

  The return flight was less glamorous in every respect. Not least because I landed at Charles de Gaulle Airport after 10 p.m. (never book a flight that gets into CDG after 10 p.m., you’re just asking for trouble). Too late to take the RER into Paris. And since I didn’t feel like taking a taxi, almost on principle, I had to wait for the shuttle.… The first one was full even before it reached my stop, the second one too; as for the third one, well, maybe I could have just barely squeezed in—if I had been prepared, as some of the other people waiting in line with me were, to disregard the driver’s instructions and elbow my way on. Finally got into Paris at two in the morning. By chance my old apartment in the city was empty, so I was able to get a couple hours’ sleep before catching the first RER B back to the southern suburbs.

  Going through my email offline, as always. Tons of messages … but ever since Lovász’s phone call in February, and then the official letter that came by regular post a few days later, it’s as though a great weight is being lifted from my shoulders. Not all at once—it will be a while longer before the sense of urgency leaves me—but gradually. Three and a half months from now, I’ll have to deal with another kind of pressure. So I’ve got to try to savor this wonderful feeling of relaxation while there’s still time.

  One of the messages informs me that I’m the only person being considered for position no. 1928, a research appointment at the Université de Lyon-I. Good news. In any case 1928 is my lucky number: it’s the year that the Institut Poincaré was founded! A transfer to Lyon-I would allow me to stay in touch with friends and colleagues in Lyon without preventing ENS-Lyon from hiring a full-time professor; the number of teaching positions there is fixed just now, they’d be in a real bind if I were to stay on.

  A beggar, an older woman, is trying her luck with the few passengers on the train. She approaches me, speaking in a hoarse voice.

  “Coming back from vacation with this big bag?”

  “Vacation? Oh no! My last vacation was at Christmas … and the next one won’t be anytime soon.”

  “Where are you coming from?”

  “I was in Cairo, in Egypt, on business.”

  “Good for you! What is it you do?”

  “I do mathematics.”

  “Ah, good for you. All right, so long. And good luck with the rest of your studies!”

  I can’t help but smile. It makes me really happy to know that people still take me for a student. But after all, she’s right, I still am a student—maybe for as long as I live.…

  * * *

  Today I was in the air and “entertained” myself by taking five minutes to try to feel all the electrical, electronic, electromagnetic, aerodynamic, and mechanical phenomena that are exerted on you in and around an airplane. All these small distinct phenomena that together make up a functioning whole! It’s fascinating to be aware of our surroundings … fascinating!

  Unfortunately, at the controls of an airplane one seldom has more than 5 min. for thinking about this sort of thing.

  Best wishes.

  [From an email sent to me on September 9, 2010, by a complete stranger]

  FORTY-TWO

  Église de Saint-Louis-en-l’Île, Paris

  June 9, 2010

  Just a little too abruptly, I push back the censer that has been swung toward me. Black suit and black ascot as a sign of mourning; green spider pinned to my lapel as a sign of hope. I go up to the casket, touch it, bow in respect. A few inches away lies the body of Paul Malliavin, one of the foremost experts on probability theory during the second half of the twentieth century. Inventor of the famous Malliavin calculus, he did more than anyone else to unite probability with geometry and analysis. My own work on optimal transport follows in the same tradition. “In Malliavin there is Villani,” as I like to remind myself from time to time.

  Malliavin was a complex and fascinating person, a conservative and an iconoclast both, blessed with an exceptional mind. He took an interest in me from the beginning of my career, encouraged me, set me on my way. He also entrusted me with real responsibility as a member of the editorial board of his beloved Journal of Functional Analysis, which he cofounded with two American mathematicians, Ralph Phillips and Irving Segal, in 1966.

  Despite the forty-eight years that separated us in age, we became friends. Our mathematical tastes were similar, and I like to think that my admiration for him was reciprocated. We never went beyond cher ami in addressing each other, but this was not a mere form of politeness: the sentiment was most sincerely felt.

  A few years ago we both took part in a conference in Tunisia. Malliavin was already seventy-eight, but he was still so active! At the end, it was my job to summarize the conclusions of the meeting, and in a few words I tried to acknowledge his phenomenal impact on analysis and probability; I don’t know if I called him a living legend, but that was the idea. Malliavin seemed a bit taken aback by my tribute. Later he took me aside and said very quietly, in that deadpan way of his, “You know, the Legend is a little tired.”

  Tired or not, Paul Malliavin went out swinging—doing mathematics right up until the last minute, as his son-in-law reported. He died the same day as Vladimir Arnold, another mathematical giant of the twentieth century, albeit a giant of a completely different style.

  We will have to carry on without you. Vous pouvez compter sur moi, cher ami. The Journal of Functional Analysis is in good hands.

  And … I would have been so proud to tell you about this secret phone call I received in February, I know that you would have been delighted.…

  * * *

  The ceremony’s over—now to rush back to the IHP for the end of the big conference we’ve organized with the Clay Mathematics Institute to celebrate Grigori Perelman’s solution of the Poincaré conjecture. I absolutely have to get there before the last speaker finishes in order to say a few closing words. No choice but to run through the streets as fast as I can, from the Île Saint-Louis to the heart of the fifth arrondissement. If only “Monsieur Paul” could see me now, red-faced, sweating through my suit and puffing like a locomotive, he would permit himself a small smile.…

  Wait, did I remember to bow before the casket the way you’re supposed to? Well, it doesn’t matter, the gesture was heartfelt. That’s what counts.

  * * *

  At the turn of the twentieth century, Henri Poincaré developed an entirely new field of mathematics, differential topology, which seeks to classify the shapes of our physical environment by considering their behavior under deformation.

  Deforming a ring (for example, a bagel or a doughnut) yields a cup, but never a sphere: the cup has a hole, bounded by a handle; the sphere does not. Generally speaking, in order to understand surfaces (the shapes on which a point is picked out within a small area by two coordinates, such as longitude and latitude), you have only to count the number of handles—the number of holes.

  As it happens, we live in a world of three spatial dimensions. To classify such objects, is it really enough to count the number of holes? This is the question Poincaré posed in 1904 by way of conclusion to a series of six articles, notable as much for their untidiness as for their sheer genius, that gave birth to the infant discipline of topology. Poincaré asked, in effect, whether all shapes of dimension 3 that are bounded (finite universes, as they might be called) and have no holes are equivalent. One such shape naturally suggested itself for study: the 3-sphere, a sphere with three coordinates embedded in a space of dimension 4. Poincaré formulated his conjecture thus: A smooth manifold of dimension 3, compact and without boundary, simply connected, is diffeomorphic to the 3-sphere.

  This statement seems perfectly plausible. But is it true? Poincaré answered with these tantalizing words, almost as memorable as Fermat’s famous complaint about the margin that was too narrow: “But this question would take us too far afield.”

  Ages and ages passed.…

  Poincaré’s conjecture became the most celebrated enigma in all of geometry, nourishing research for the whole o
f the twentieth century. In the interval no fewer than three Fields Medals were awarded for steps toward a solution.

  A decisive stage was reached when the American mathematician William Thurston turned his mind to the problem in the early 1980s. Thurston was a visionary. His extraordinary geometric intuition enabled him to see all the shapes—all the “possible universes”—of dimension 3. What he proposed was a sort of zoological, or taxonomic, classification of three-dimensional shapes. Its magnificence dazzled even skeptics: those who still had their doubts about Poincaré nonetheless bowed down before a conception that was so beautiful it had to be true. What is more, proving the so-called Thurston program (or geometrization conjecture) would imply the truth of Poincaré’s conjecture. Thurston himself was able to explore only a part of the vast landscape he had surveyed.

  It came as no surprise when Poincaré’s conjecture was chosen by the Clay Mathematics Institute in 2000 as one of seven problems for whose solution it was offering a prize of $1 million apiece. At the time it was widely believed that there was a good chance the famous problem would remain unsolved for another century!

  Only two years later, however, in 2002, Grigori Perelman dumbfounded everyone by announcing a solution of Poincaré’s conjecture. A solution on which he had worked alone, in secret, for seven years!!

  Grigori Perelman

  Born in 1966 in Leningrad (alias Saint Petersburg), Perelman had inherited a passion for mathematics from his mother, a college-level teacher. But he was decisively influenced at an early age by two things: the exceptional Russian school of mathematics, led by Andrei Kolmogorov; and his local math club, whose intensely devoted coaches prepared him as a high school student for the International Olympiad. He went on to study with three of the best geometers of the century—Alexandrov, Burago, Gromov—and quickly became a leading expert on the theory of singular spaces with positive curvature. In 1994, his proof of the so-called soul conjecture brought him wide recognition. Seemingly destined for a brilliant career, Perelman then—disappeared!

  From 1995 on there was no sign of him. Far from giving up mathematics, however, he had begun thinking about Richard Hamilton’s seminal work on the Ricci flow, a formula that makes it possible to continually deform geometric objects by stretching their curvature, in the same way that the heat equation spreads out temperature. Hamilton had hoped to be able to use his equation to prove Poincaré’s conjecture, but seemingly insuperable technical problems hampered his progress for many years. The way forward was blocked.…

  Until the famous email message that Perelman sent on November 12, 2002, to a select group of mathematicians in the United States. A message of only a few lines, calling their attention to the preprint of a paper he had just made publicly available on the Internet, which contained, as he put it, a “sketch of an eclectic proof” of Poincaré’s conjecture—and, indeed, of a large part of the Thurston program.

  Borrowing an idea from theoretical physics, Perelman showed that a certain quantity (which he called “entropy” because it resembles Boltzmann’s concept of the same name) decreases when its geometry is deformed by means of the Ricci flow. Thanks to this original insight, whose ramifications have probably not yet been fully appreciated, Perelman was able to prove that the Ricci flow can go on and on without ever “blowing up,” that is, without producing an uncontrollably violent singularity. Or rather: before any such singularity develops, its occurrence can be foreseen and counteracted.

  Perelman then came to the United States to give a series of lectures about his work, and impressed everyone who heard them with his mastery of the problem. He was angered, however, by the inaccuracy of media accounts and the behavior of American academic institutions, and annoyed by how long it took his fellow mathematicians to grasp the arguments he had presented. He went back to Saint Petersburg, leaving it to others to check the soundness of his reasoning. In the event, it took almost four years for three teams of researchers to verify Perelman’s proof and make it complete down to the last detail!

  The enormity of what was at stake, as well as Perelman’s retreat once more from professional life, placed the mathematical community in an unprecedented situation. Yet despite the tensions aroused by controversy over the proper share of credit due to Perelman, it eventually became clear that he had indeed proved Thurston’s grand geometrization conjecture and, along with it, Poincaré’s conjecture. This feat has no equivalent in recent memory, unless perhaps Andrew Wiles’s proof of Fermat’s last theorem fifteen years ago.

  Perelman was showered with honors: the Fields Medal in 2006, followed at once by recognition of his achievement in the journal Science as the “Breakthrough of the Year” (an accolade almost never conferred on a mathematician), and, four years later, in 2010, the Clay Millennium Prize, making him the first recipient of the most richly endowed prize in mathematics. Perelman had no need of awards and turned them down, one after another.

  Editorial writers everywhere fell over themselves in their eagerness to explain his refusal of $1 million, tirelessly harping on the theme of the mad mathematician. All of them missed the point: what was exceptional in Perelman’s case was neither his refusal of money and honors nor his undoubted personal eccentricity—many other such examples can be found in the annals of mankind—but instead the extraordinary force of character and intellectual brilliance required to persevere, through seven long years of courageous and solitary labor, and finally to penetrate the outstanding mathematical enigma of the twentieth century.

  In June 2010, the Clay Mathematics Institute and the Institut Henri Poincaré jointly organized a conference in Paris in honor of this remarkable feat. Fifteen months later they announced that the money Perelman had refused would be used to create a very special position at the IHP. The Poincaré Chair, as it was named, is to be held by a succession of extremely promising young mathematicians under ideal conditions, unencumbered by any obligation to give lectures or to reside in Paris for the term of the appointment. The perfect opportunity, in other words, for them to realize their full potential—just as Perelman himself had been allowed to do as a guest of the Miller Institute at Berkeley some fifteen years earlier.

  FORTY-THREE

  Hyderabad

  August 19, 2010

  My name rings out in the immense hall, and my portrait—by the photographer Pierre Maraval, with carmine red cravat and mauve-tinted white spider—is displayed on a gigantic screen. I didn’t sleep last night, and yet I can’t remember ever having felt so alert in all my life. This is the most important moment of my professional career, the moment every mathematician dreams of without daring to admit it. A more or less anonymous young man, number 333 of the one thousand scientists photographed by Maraval, now begins to emerge from obscurity.…

  I get up and make my way to the stage while the citation is read out: A Fields Medal is awarded to Cédric Villani for his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation.

  I mount the steps, not too slowly, not too quickly, and approach the president of India at the center of the stage. Petite though she is, Shrimati Pratibha Pati radiates an aura of power visible in the bearing of her entourage. I stop just in front of her: she bows slightly and I bow in return, much too deeply. Namaste.

  She hands me the medal and I hold it up for all to see. My torso is unnaturally contorted: making a full ninety-degree turn to face the audience would force the chief of state to look at me from the side, so I do my best to split the difference, forty-five degrees for each of them.

  Three thousand people applauding in this vast meeting hall attached to a luxury hotel, site of the 2010 International Congress of Mathematicians. How many were there, eighteen years ago, applauding my welcoming address at the bicentennial ball of the École Normale Supérieure? A thousand, maybe? Those were the days, my friend.… My father was so disappointed he couldn’t film the ceremony; he’d left his camera in a room somewhere and the organizers wouldn’t let him go back and get it. T
he ball in Paris was quite an affair, but small potatoes in comparison with this. There’s a whole army of photographers and cameramen besieging the stage—it’s like the Cannes Film Festival!

  Clutching the medal to my breast, I bow once more to the president, take three steps back, pivot, and walk toward the wings at a measured pace, almost exactly as we had been instructed to do at the rehearsal yesterday evening.…

  * * *

  Not bad. In any case I survived the ordeal better than Elon Lindenstrauss, the first of the four of us to be honored, who made a shambles of all the fine points of etiquette. Completely out of it. As he shuffled off to the side, Stas Smirnov whispered in my ear, “We can’t possibly do worse.”

  The four of us, standing together, were immortalized by the official photographers at the end of the ceremony. After that it’s all a blur. All I know for sure is that many more pictures were taken. Back down on the floor, we held up our medals to the digital swarm—video cameras, camera phones, capturing and registering devices of every imaginable kind. And then there was a press conference.…

  Clément is here, of course, beaming. To think it wasn’t even ten years ago that he first set foot in my office at ENS-Lyon, looking for a thesis topic.… A stroke of luck for him, a stroke of luck for me.

  No computers or cell phones were allowed in the main hall. Soon I discovered three hundred messages in my mailbox. Many more were on the way. Congratulations from colleagues, old friends whom I haven’t seen for ten, twenty, or thirty years, perfect strangers, classmates from way back in elementary school … Some of the messages were very moving. One informed me of the death, several years ago, of a childhood friend. Life, as we all know, is filled with joys and sorrows, inextricably entangled.