And via the press, an official message of congratulations from the president of France. As expected, Ngô also won a medal. It took me a while to fully appreciate how proud our fellow citizens are of this double victory—to say nothing of the fact that Yves Meyer was awarded the prestigious Gauss Prize for lifetime achievement! People back home are now realizing that for more than three centuries France has been at the forefront of international mathematical research. As of this evening, our country has produced no fewer than eleven of the fifty-two Fields Medal winners!
I fought my way through the crowd and went up to my hotel room. A dull, uninteresting room with nothing of India about it—I might just as well be in Tierra del Fuego! But I’m here to discharge my duty.
For four straight hours I answered calls from journalists, switching back and forth between fixed and mobile phones. Not a moment’s rest. No sooner had one call ended than I checked my voicemail and found new messages. Personal questions, scientific questions, institutional questions. And questions that basically asked the same thing, over and over again: How does it feel to win this award?
Finally I took the elevator back downstairs, looking a little pale, feeling a little hungry—but there are worse things to be endured, after all. I settled happily for a cup of masala chai and then plunged back into the crowd. Throngs of young people, most of them Indian, clamored for my attention. Dazed from signing hundreds of autographs and posing for at least as many pictures, I somehow made it through the end of the day.…
Unlike the other laureates, I came here alone. I thought it would be best if Claire and the children were to stay home in France, far from the noise and the tumult of the ICM. And I was right! In the meantime I had faithfully obeyed my orders and told no one about the medal except my wife. Not even my parents, they learned of it only when journalists phoned them for their reaction!
And … Catherine Ribeiro sent a superb bouquet of roses to my home!
Never for a moment could I have imagined that while I was basking in the limelight in Hyderabad, hordes of shutterbugs snapping away, Michelle Schatzman lay dying back in Lyon. Daughter of the great French astrophysicist Évry Schatzman, Michelle was one of the most original mathematicians it has been my good fortune to know, eager to accept whatever challenge to her abilities the classroom could devise while at the same time exploring frontiers of research where no one else would dare to go, especially the one that lies between algebraic geometry and numerical analysis. Indeed, this was the title she gave to a manifesto she dashed off one day, as though it were something perfectly obvious—Frontières. Michelle was my friend from the moment I came to Lyon in 2000; we went to seminars together, and more than once plotted together to attract a first-rate mathematician to the faculty at the Université de Lyon.
Michelle Schatzman
Michelle never shrank from speaking her mind, even if it meant putting her foot in it, as she not infrequently did. Her scathingly black humor was legendary. For more than five years she battled an incurable cancer, undergoing both chemotherapy and surgeries. With a glint in her eye she told us how good life was now that she didn’t have to spend money on shampoo. A few months ago we celebrated her sixtieth birthday with a workshop in Lyon. The speakers came from near and far. Among them the polymorphous Uriel Frisch, a world-renowned physicist who had been a student of Michelle’s father; and myself, the spiritual son of one of Frisch’s spiritual sons, Yann Brenier, who was also there. Michelle brilliantly suggested a connection between my talk on Landau damping and the “tygers” that Uriel had discussed. Pure elegance!
But then suddenly a few weeks ago her condition began to deteriorate. As proud and forthright in sickness as she had been in health, Michelle refused morphine in order to go on thinking clearly right until the end. She had been impatiently awaiting the results of the Fields Medal competition. On her deathbed she learned that I had won; a few hours later she passed away. Life, as we all know, is filled with joys and sorrows, inextricably entangled.…
* * *
On August 19, 2010, the Hyderabad International Convention Centre in India contained within its walls the greatest concentration of mathematicians in the world. They came from every continent, bringing with them their many and varied talents: experts in analysis, algebra, probability, statistics, partial differential equations, algebraic geometry and geometric algebra, hard logic and soft logic, metric geometry and ultrametric geometry, harmonic analysis and harmonious analysis, the probabilistic theory of numbers and numina; discoverers of models and supermodels, surveyors of macroeconomies and microeconomies, designers of supercomputers and genetic algorithms, processors of images and developers of Banach spaces. Mathematics of the summer, of the fall, of the winter, of the spring: a myriad of specialities that transform their masters into the Great God, Shiva, the god with a thousand mathematical arms.
One after the other, the Fields medalists, together with the winners of the Gauss, Nevanlinna, and Chern prizes, were offered up in sacrifice to Shiva. The high priestess, the president of India, presented the seven terrified laureates to the ecstatic crowd.
This was the beginning of the great festival of the International Congress of Mathematicians, which over a period of ten days or so witnessed a nonstop succession of talks and discussions, cocktail parties and receptions, interviews, photo sessions, and evenings filled with dancing and laughter. Revelers swanning about from one event to the next in luxury limos and romantic rickshaws, everywhere celebrating the unity and diversity of mathematics, its ever-shifting shapes and forms; everywhere overcome with joy at what has so far been accomplished and wonder before all that yet remains to be discovered; everywhere, day and night, dreaming of the unknown.
Once the festival is over, the celebrants will go back to their universities and research centers, to their campuses, business parks, and home offices, and resume once more, each in his or her own way, the great adventure of mathematical exploration. Armed not only with their logical abilities and their appetite for hard work, but also with their imagination and their passion, they will be joined together once more in a common desire to push back the frontiers of human knowledge.
And already they are thinking of the next congress, four years from now, in the lair of the Korean tiger. What will they talk about? Who will be the next sacrificial victims?
Four years from now, thousands of mathematicians will gather in Seoul to pay their respects to the venerable tiger. They will explore its sinuous geometry, axiomatize its implacable symmetry, test its turbulent stochasticity, analyze the reaction–diffusion to which it owes its stripes, perform differential surgery on its powerful paws, measure the curvature of its sharp claws, release it from the potential wells of quantum mechanics, and get high smoking ethereal theories that turn its whiskers into vibrating strings. For a few days the tiger will be a mathematician, from the end of its tail all the way to the tip of its nose.
[My contribution to the Korean edition of Les déchiffreurs: Voyage en mathématiques]
* * *
TYGER PHENOMENON FOR THE GALERKIN-TRUNCATED BURGERS AND EULER EQUATIONS
It is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial Fourier modes having wave numbers in excess of a threshold Kg exhibit unexpected features. The study is carried out for both the one-dimensional Burgers equation and the two-dimensional incompressible Euler equation. At large Kg, for smooth initial conditions, the first symptom of truncation, a localized short-wavelength oscillation which we call a “tyger,” is caused by a resonant interaction between fluid particle motion and truncation waves generated by small-scale features (shocks, layers with strong vorticity gradients, etc.). These tygers appear when complex-space singularities come within one Galerkin wavelength λg = 2π/Kg from the real domain and typically arise far away from preexisting small-scale structures at locations whose velocities match that of such structures. Tygers are weak and strongly localized at first—in the Burgers case at the time of the ap
pearance of the first shock their amplitudes and widths are proportional to Kg−2/3 and Kg−1/3 respectively—but grow and eventually invade the whole flow. They are thus the first manifestations of the thermalization predicted by T. D. Lee [in 1952]. The sudden dissipative anomaly—the presence of a finite dissipation in the limit of vanishing viscosity after a finite time—, which is well known for the Burgers equation and sometimes conjectured for the 3D Euler equation, has as counterpart in the truncated case: the ability of tygers to store a finite amount of energy in the limit Kg → ∞. This leads to Reynolds stresses acting on scales larger than the Galerkin wavelength and thus prevents the flow from converging to the inviscid-limit solution. There are indications that it may be possible to purge the tygers and thereby to recover the correct inviscid-limit behaviour.
[Abstract of article published by Samriddhi Sankar Ray, Uriel Frisch, Sergei Nazarenko, and Takeshi Matsumoto in 2011]
* * *
THE TYGER
Tyger! Tyger! burning bright
In the forests of the night,
What immortal hand or eye
Could frame thy fearful symmetry?
In what distant deeps or skies
Burnt the fire of thine eyes?
On what wings dare he aspire?
What the hand, dare seize the fire?
And what shoulder, & what art,
Could twist the sinews of thy heart?
And when thy heart began to beat,
What dread hand? & what dread feet?
What the hammer? what the chain?
In what furnace was thy brain?
What the anvil? what dread grasp
Dare its deadly terrors clasp?
When the stars threw down their spears,
And water’d heaven with their tears,
Did he smile his work to see?
Did he who made the Lamb make thee?
Tyger! Tyger! burning bright
In the forests of the night,
What immortal hand or eye
Dare frame thy fearful symmetry?
[From William Blake, Songs of Experience, 1794]
FORTY-FOUR
Saint-Rémy-lès-Cheuvreuse
November 17, 2010
Autumn. Everything’s gold, red, and black: golden leaves, red leaves—and shiny black ravens, like the ones in Tom Waits’s November song.
I get off at my station on the dear old RER B line and disappear into the night.
The last three months have been so intense!
The autographs.
The newspaper articles.
The radio interviews.
The television shows.
The documentaries.
My appearance with Franck Dubosc, whom I met for the first time doing a live show on Canal+ … Some critics reproached me for taking part in a “farce,” but where’s the harm in talking to people who think they have no interest whatsoever in mathematics? The next day perfect strangers stopped me in the street: “Hey, I saw you on TV last night!”
And the meetings with politicians, with artists, with students, with industrialists, with business executives, with revolutionaries, with parliamentarians, with senior civil servants, with the President of the Republic …
Questions that run together into one long question: How did you get interested in math why are the French so good at math did the Fields Medal change your life what keeps you interested now that you’ve received the highest honor are you a genius what is the meaning of your spider.…?
Ngô has gone back to the United States, leaving me to face the onslaught alone. I don’t mind. It’s fascinating to get a glimpse of these different worlds—behind the television cameras, inside the newsroom of a big daily paper. I’ve seen first-hand how an interview frequently takes on a life of its own, separate from what the person being interviewed actually says; how an abstract media personality named Cédricvillani comes to be created, someone who’s not really me and whom I can’t really control.
All the while continuing to do the job I’m paid to do as director of the IHP. The same day I appeared with Dubosc on Canal+, I’d already done an interview with RTL, attended a meeting at the Hôtel de Ville on university housing, had a long conversation with the chairman of my board of directors, and recorded a show for Des Mots de Minuit.
A lot of my time has been taken up guiding a joint effort to obtain major funding through the government’s Investments for the Future program (aka “The Big Loan”), a complicated business that requires coordinating the interests of the four national and international institutes of mathematics in France: the Institut Henri Poincaré (IHP), the Institut des Hautes Études Scientifiques (IHÉS) in Bures-sur-Yvette outside Paris, the Centre International de Rencontres Mathématiques (CIRM) in Luminy, and the Centre International de Mathématiques Pures et Appliquées (CIMPA) in Nice.
The IHÉS is the French version of the Institute for Advanced Study in Princeton: a magnificent rural retreat where the autumn air crackles with the sound of chestnuts falling to the ground, where the fantastically brilliant Grothendieck produced the better part of his incomparable work, and where talented young people can accelerate the pace of their research through contact with some of the best mathematicians in the world. CIRM, with its weeklong conferences, is the French counterpart to the institute in Oberwolfach, except that here the austere beauty of the Black Forest has been replaced by the deep and rocky inlets around Marseille, no less splendid in their way. CIMPA, for its part, is a thoroughly international organization devoted to supporting the study and use of mathematics, mainly in developing countries but anywhere, really, that its assistance is both needed and welcome.
The governing bodies of these four institutions are very different. Getting them to agree to collaborate on this project took hours and hours of negotiation. After a year at the helm of the IHP, with a few bureaucratic skirmishes under my belt, I felt ready to step forward and take responsibility for coordinating the joint initiative. Our group is to be called CARMIN, for Centre d’Accueil et de Rencontres Mathématiques Internationales (Reception Center for International Mathematics Meetings).
In my spare time I composed and delivered two public lectures on mathematics, part of an ongoing series, and, appropriately enough, wrote a long paper on the subject of time for a theoretical physics seminar. On top of all this I had to take on extra administrative duties to help the Institute get through a rough patch when, by a sort of curse, several staff members fell sick at the same time. Fortunately for me, everyone else pitched in and worked twice as hard as well!
These three months have worn me out. There were times I had to plan my sleeping schedule several days in advance. Hasta que el cuerpo aguante!
Thinking back on this exhausting autumn as I walk home along the dirt path from the RER station … now I come to the dark part of my journey.
To my left, a forest, with foraging foxes and deer; to my right, a field, cows peaceably slumbering; in front of me, for the next three hundred yards, complete darkness. No public lighting, not the least speck of luminous pollution.
Nothing is more precious than an unlit path! When the moon is hidden, you can’t see even ten feet ahead. You walk a bit faster, your heart beats a little more quickly, your senses are in a heightened state of alert. The slightest noise makes your ears prick up. You tell yourself that the way home seems longer than usual. You imagine a robber lying in wait. You try not to run.
This gloomy tunnel is a little like the one you pass through when you begin work on a new mathematical problem. A mathematician is a blind man in a dark room looking for a black cat that isn’t there … (as Darwin may or may not have said). Total obscurity. Bilbo in Gollum’s tunnel.
A mathematician’s first steps into unknown territory constitute the first phase of a familiar cycle.
After the darkness comes a faint, faint glimmer of light, just enough to make you think that something is there, almost within reach, waiting to be discovered.… Then, after the faint, fa
int glimmer, if all goes well, you unravel the thread—and suddenly it’s broad daylight! You’re full of confidence, you want to tell anyone who will listen about what you’ve found.
And then, after day has broken, after the sun has climbed high into the sky, a phase of depression inevitably follows. You lose all faith in the importance of what you’ve achieved. Any idiot could have done what you’ve done, go find yourself a more worthwhile problem and make something of your life. Thus the cycle of mathematical research …
For the moment I’m making my way through the darkness, literally, recollecting the events of a day of great emotion. Ngô, Meyer, and I met with the president of the National Assembly, in whom we recognized a comrade in arms the moment we became aware of his scientific background; then, just before question time got under way, we were acclaimed by the whole Assembly. Earlier, in the library, I had been allowed to see an indescribable treasure, a massive piece of furniture specially designed to hold the works composed by the scientists, engineers, and scholars who accompanied Napoleon on the expedition to Egypt. Works by Monge, Fourier, and so many others whose findings revolutionized our understanding of natural history, physical geography, archaeology, ancient technology, you name it. The beauty of the illustrations, drawn by hand, using improvised tools; the majesty of these extraordinary oversized volumes, which normally only highly qualified conservators are allowed to handle—all of this deeply moved me, imbued me with a warm inner glow.
And yet, in the back of my mind, a tiny seed of doubt has grown little by little over the past few months into a nagging worry. Still no word from Acta! Still no word from the referees! Impartial review by experts whose anonymity is carefully protected: this, and only this, can confirm or disconfirm our results.
Birth of a Theorem: A Mathematical Adventure Page 19