Birth of a Theorem: A Mathematical Adventure (40 page)

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is the French counterpart to the institute in Oberwolfach, except that here the austere beauty of the Black Forest has been replaced by the deep and rocky inlets around Marseille, no less splendid in their way.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, faint 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.

After all of the honors I have received, what will I say if our findings turn out to be wrong? The Fields committee must have taken care to check the explanation of Landau damping, knowing full well what was at stake—but as usual I have no idea what is really going on. What if, after long and patient scrutiny, an error were to be detected, or if a favorable report were to be contradicted by a second, still more thorough round of review?

Cédric, you’re a father—ritual suicide is not an option
.

Seriously, though, everything is going to work out just fine. And besides, I’m almost through the tunnel of darkness. There it is, all the way at the end, a faint, faint, flickering glimmer—the light of the digicode panel. I punch in my security number on the keypad at the front gate. Made it!

It feels so good every time I make it home safely through this black forest, there’s nothing like it! I push open the heavy wrought-iron gate, cross the courtyard, unlock the front door, turn on the lights, go upstairs to my office, plug in my laptop, and download my emails. What, only eighty-eight new messages in the past twelve hours?! Slow day …

But there, buried in the middle of the list, one name immediately attracts my notice:
Acta Mathematica
! I feverishly open the message from Johannes Sjöstrand, the editor handling our paper.

The news about your paper are good.

He should have written “is,” of course:
news
, like
mathematics
, is singular despite the final
s
. But who cares? I don’t need to read any further. I forward the message at once to Clément, adding two words:
Gooood news
.

Our theorem has truly been born at last.

*   *   *

Theorem (Mouhot and Villani, 2009)

Let d
≥
1
be an integer, and let
be a locally integrable, even periodic function whose Fourier transform satisfies

Let f
⁰
=
f
⁰
(
v
)
be an analytic distribution
such that

for some
λ
₀
>
0
, where
designates the Fourier transform of f.