Birth of a Theorem: A Mathematical Adventure (12 page)

Today I spoke to Clément on the telephone. We’ve now gone into high gear. Since I don’t have to give any lectures at the IAS, and since as a government-sponsored researcher he doesn’t have any teaching obligations either, we can work as hard as we want.

The time difference between France and the United States is a help too. With six hours between us, we can work almost round the clock. If I work until midnight in Princeton, three hours later Clément is in his office in Paris, ready to take over.

Clément has latched onto a particular calculation that involves a pretty neat trick, where you cheat on the existence time of the solution. He has high hopes for it. I don’t doubt for a moment that it will be a great help to us going forward.

In the event, Clément’s idea did prove to be of great importance—far greater, in fact, than I could have imagined at the time.

But I simply cannot bring myself to believe that by itself it will be enough to save us. We need another estimate.

A new trick.

*   *   *

Date: Mon, 12 Jan 2009 17:07:07 -0500

From: Cedric Villani

To: Clement Mouhot

Subject: bad news

So, I haven’t had any luck reproducing the transfer of regularity with estimates as good as yours (after conversion in spaces with 3 indices, there’s a snag somewhere). I’ve redone your calculation and found two places where something’s not right: (a) the last index on p. 39, l.8 (before “We use here the trivial estimate”) seems to me it should be \lambda
+
2\eta rather than \lambda
+
\eta; (b) it seems to me impossible that in assumption (5.12) the estimated value does not depend on \kappa (the limits \kappa\to 0 and \kappa\to\infty change the space completely). Conclusion: it seems to me there’s a problem.…

More later,

Cedric

Date: Mon, 12 Jan 2009 23:19:27
+
0100

From: Clement Mouhot

To: Cedric Villani

Subject: Re: bad news

I’ll take a closer look tomorrow afternoon. But I agree about point (a), there surely must be other pbs with indices as well. As for point (b), what I was thinking of using in order to say that (5.12) doesn’t depend on kappa (for kappa in a compact set) is the weak dependence with respect to v of the scattering field $X^{scat}_{s,t}$: since $\Omega_{s,t}$ is near to identity within O(t-s), you’ve got $X^{scat} _{s,t}
=
x
+
O(t-s)$. Whence the fact that any differentiation along v is “flattened” in the O(t-s)?

Talk to you again soon, clement

Date: Sun, 18 Jan 2009 13:12:44
+
0100

From: Clement Mouhot

To: Cedric Villani

Subject: Re: transfer

Hi Cedric,

Since I’m acting as a referee of Jabin’s paper on averaging lemmas (his Porto Ercole course), I checked to see how closely his calculations agree with ours, and I have the impression that in the linear estimate the transfer of regularity has something to do with the averaging lemmas, but expressed in L^1/L^\infty which seems unusual. For example if you try to transfer the regularity of x to v without the gain in x being proportional to (t-s), you’re limited to a gain
<
1 to have integrability in time, which is consistent with a limitation of 1/2 in L^2. Another novelty here in the calculations is that when the gain is proportional to (t-s) there is no longer a limit 1 … Need to see also if this gain proportional to (t-s) might be useful in nonlinear regularity theory (your initial question) … Any news on your end?

Best,

Clement

ELEVEN

Princeton

January 15, 2009

Every morning I go to the common room in the mathematics building to make myself a cup of tea. Einstein’s round, smiling face is nowhere to be seen here. Watching over the mathematicians in his stead is André Weil, whose angular features have been memorialized in the form of a bronze bust.

The common room is sparely furnished. Inevitably there’s a large blackboard, in addition to everything you need to make tea and coffee. As well as chessboards and piles of magazines devoted to chess.

One magazine in particular caught my eye, an issue in memory of Bobby Fischer, the greatest player of all time, who died about a year ago. Powerless to escape the clutches of paranoia, by the end of his life he had become an incoherent misanthrope. But beyond the madness there remain the extraordinary matches of a player whose abilities have never been equaled, before or since.

In mathematics, as in all other fields of creative endeavor, some of the greatest minds have suffered a similarly tragic fate.

Paul Erd
ő
s, who helped found probabilistic number theory, was condemned to a life of restless wandering. Amazingly prolific, Erd
ő
s wrote some fifteen hundred articles (a world record), roaming the length and breadth of the globe in his threadbare clothes, having neither home nor family nor job, only his suitcase, his notebook, and his genius.

Grigori Perelman, after seven solitary years contemplating the mysteries of Poincaré’s famous conjecture, astounded the mathematical world by announcing a solution no one thought possible. Perhaps in order not to spoil the purity of his achievement, Perelman refused the prize of $1 million offered by an American philanthropist—this after having walked away from his post at the Steklov Institute in Saint Petersburg.

Alexander Grothendieck, a living legend, utterly transformed mathematics with the creation of one of the most abstract branches of human thought, and then suddenly resigned from the Institut des Hautes Études Scientifiques, outside Paris. After talks about the possibility of a chair at the Collège de France broke down, he retreated to a small village in the Pyrenees. Once famed for his seductive charm, Grothendieck was fated to pass the rest of his days as a sullen hermit in the grips of madness and a compulsion to write.

Kurt Gödel, the greatest logician of all time, fatally undermined the foundations of mathematics by showing that no axiomatic system rich enough to accommodate arithmetic is complete: any consistent set of axioms contains at least one statement that is neither true (in the sense of being provable) nor false (in the sense of being disprovable). In the last years of his life, ravaged by a severe persecution complex that led him to believe he was in danger of being poisoned, Gödel gradually starved himself to death.

And John Nash, my mathematical hero, revolutionized analysis and geometry with the proof of three theorems in scarcely more than five years before succumbing to paranoid schizophrenia.

There is a fine line, it is often said, between genius and madness. Neither of these concepts is well defined, however. And in the case not only of Grothendieck but also of Gödel and Nash, periods of mental derangement, so far from promoting mathematical productivity, actually precluded it.

Innate versus acquired, a classic debate. Fischer, Grothendieck, Erd
ő
s, and Perelman were all Jewish. Of these, Fischer and Erd
ő
s were Hungarian. No one who is familiar with the world of science can have failed to notice how many of the most gifted mathematicians and physicists of the twentieth century were Jews, or how many of the greatest geniuses were Hungarian (many of them, but by no means all, Jews). Scientists who worked on the Manhattan Project in the 1940s were fond of saying that Martians really do exist: they have superhuman intelligence, speak an incomprehensible language, and claim to come from a place called Hungary.

Nash, on the other hand, is American through and through, from an old Protestant family. What is more, there was nothing in his ancestry that foretold an exceptional destiny for him. And yet a destiny depends on so many things! The intermingling of genes, the cross-fertilization of ideas, experiences, and chance encounters—all these things have their place in the marvelous, impossibly dramatic lottery of life. Neither genetic inheritance nor environment can explain everything. We should be grateful that this is so.

*   *   *

What happens when you gather 200 of the world’s most serious scholars, isolate them in a wooded compound, liberate them from all the mundane distractions of university life, and tell them to do their best work? Not much. True, a lot of cutting edge research gets done at the celebrated Institute for Advanced Study near Princeton. Due to the Institute’s remarkable hospitality, there is no better place for an academic to sit and think. Yet the problem, according to many fellows, is that the only thing there is to do at the Institute is sit and think. It would be an understatement to call the IAS an Ivory Tower, for there is no more lofty place. Most world-class academic institutions, even the very serious, have a place where a weary bookworm can get a pint and listen to the jukebox. Not so the IAS. Old hands talk about the salad days of the 40s and 50s when the Institute was party central for Princeton’s intellectual elite. John von Neumann invented modern computing, but he is also rumored to have cooked up a collection of mind-numbing cocktails that he liberally distributed at wild fetes. Einstein turned physics on its head, but he also took the occasional turn at the fiddle. Taking their clues from the Ancients, the patriarchs of the Institute apparently believed that men (as they would have said) should be well-rounded, engaging in activities high and low, according to the Golden Mean. But now the Apollonian has so overwhelmed the Dionysian at the Institute that, according to many members, even the idea of having a good time is considered only in abstract terms. Walking around the Institute’s grounds, you might trip over a Nobel laureate or a Fields medalist. Given the generous support of the Institute, you might even become one. But you can be pretty certain that you won’t have adrink or a laugh with either.

[From the liner notes to
Final Report
(1999), a self-produced album by Do Not Erase, the only rock band ever formed at the Institute for Advanced Study]

TWELVE

Princeton

January 17, 2009

Saturday evening, dinner together at home.

The whole day was taken up with a trip organized by the Institute for visiting members. A trip to the holiest of shrines for anyone who’s enthralled by the story of life: the American Museum of Natural History in New York.

I recall very well my first visit to this museum, almost exactly ten years ago. The excitement of seeing some of the most famous fossils in the world, fossils whose pictures are found in the guides and dictionaries of dinosaurs that I devoured as a teenager, was indescribable.

Today I went back ten years into the past and left my mathematical cares behind for a few hours. Over dinner, however, they caught up with me.

Claire was rather taken aback, seeing my face contorted by tics and twitches.

The proof of Landau damping still hasn’t come together. My mind was churning.

What do you have to do, for God’s sake, what do you have to do to get a decay through transfer of regularity with respect to position when the velocities have been composed … this composition is what introduces a dependence with respect to velocity—but I don’t want any velocities!

What a mess.

I scarcely bothered to make conversation, responding in as few words as possible, otherwise by grunts.

“Was it ever cold today! We could have gone sledding.… Did you happen to notice the color of the flag at the pond this morning?”

“Hmmm. Red. I think.”

Red flag: even if the pond looks frozen, walking on it is prohibited, it’s too dangerous. White flag: go ahead, guys, the water’s frozen solid, jump and shout, dance on the ice if you like.

And to think that I accepted an invitation to present my results at a statistical physics seminar at Rutgers on January 15!
How
could I have accepted when the proof wasn’t complete? What am I going to tell them?

Well, when I got here at the very beginning of the month, I was completely sure I could finish the project in two weeks—max! Fortunately the talk got pushed back by another two weeks! But even with this reprieve, am I going to be ready?? January 29 isn’t very far away!! I never thought it would be so hard. No way I could have foreseen the obstacles that lay ahead!

The velocities are the problem, the velocities! When there isn’t any dependence with respect to velocity, you can separate the variables by means of a Fourier transform, but when you’ve got velocities, what can you do? In a nonlinear equation, velocities are obligatory—there’s no way I can avoid dealing with them!