Birth of a Theorem: A Mathematical Adventure (39 page)

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
K
_g
exhibit unexpected features. The study is carried out for both the one-dimensional Burgers equation and the two-dimensional incompressible Euler equation. At large
K
_g
, 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
λ
_g
=
2
π
/
K
_g
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 appearance of the first shock their amplitudes and widths are proportional to
K
_g
^−
2/3
and
K
_g
^−
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
K
_g
→
∞
. 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.…?