Birth of a Theorem: A Mathematical Adventure (20 page)

Subject: 38!

Version 38 attached. The modifs:

– 2-3 typos corrected here and there as you can see with diff if you need to

– section 9 is now complete modulo a certain number of formulas, this is the time to summon our courage and see all the calculations through to the end! It’s a rather beautiful thing to behold since all the pieces fit together and lead on to the result. The organization of this section justifies the paper’s overall structure a posteriori (in particular putting the characteristics at the beginning). With a little more tinkering this section ought to be in good shape, and then we’ll be ready to decide about constants (Hello calculations!)

– I’ve cut out most of the old commentary, particularly the remarks concerning regularization.

– But there are still two holes in the way the spatial averages are handled!

* the first one has to do with the need to stratify the estimates on
<
\nabla h^k \circ \Om^n
>
(subsection 9.4). This is tricky, as I explain in the file, we can’t rely on recurrence, and we can’t rely on regularity since \Om^n is highly irregular. The only solution I see is to use the additional Sobolev regularity of the characteristics, which is propagated uniformly over n. Be careful, it’s velocity regularity we need, but that should be OK, Sobolev regularity for the force entails regularity for all variables. We’ve got to gain exactly one derivative, which means that Coulomb’s probably critical here as well.….…

* the second one is the treatment of the zero mode in the estimates in section 6, where for the moment it doesn’t work (constants too large to go on checking the stability criterion). I’m fairly optimistic, counting on being able to recycle my old idea of using the change in the scattering variables, and DIRECT estimates on the characteristics. The first time I tried it I didn’t have the right orders of magnitude in mind, we hadn’t done the stratification yet, in other words we were much less well prepared.

I suggest the following division of labor: first you see about making section 9 converge, forgetting about the two holes above; then you look for a way to plug the first hole. In the meantime I’ll get to work on the second hole. I’m not planning to make any changes to the tex file in the next few days.

As for the Coulomb case: we’ll see later, I think that plugging the holes has to take priority.…

This week is going to be a little hard for me because I’ll be taking care of the kids by myself, and on top of this we’ve got guests at the lab. But it’s pretty much the final sprint.

Best

Cedric

TWENTY-TWO

Princeton

Night of March 15–16, 2009

Sitting on the floor, surrounded by sheets of scribbled notes strewn all over the carpet, I write and type for hours in a state of feverish excitement.…

*   *   *

Made a point of not doing any mathematics during the day. Took the kids along to Sunday brunch at Alice Chang’s house, many famous names in attendance. Alice teaches at the university, a renowned specialist in geometric analysis; a few years ago she was a plenary speaker at the International Congress of Mathematicians in Beijing. It was Alice who invited me to give a series of lectures as part of a program she had organized at Princeton this spring on differential geometry and geometric analysis.

This morning at brunch we talked about a little bit of everything, including the famous Shanghai ranking, the list of the world’s top universities that French politicians and journalists are so fond of citing. When I raised the subject with Alice I wondered how she would react, since she is both Chinese by birth and a member of one of the most celebrated mathematics departments in the world. I thought she might be proud of the reputation this classification has acquired outside her native land. Boy, was I ever wrong!

“Cédric, what’s the Shanghai ranking?”

Alice Chang

When I explained what it was all about, she looked at me as if I were pulling her leg. Cédric, I don’t follow—being on this
Chinese
ranking is considered very prestigious
in France
?? (Are you sure you don’t have it backward, my dear?) How I’d love to introduce Alice to some politicians I know back home.…

It was only much later, well into the evening, in fact, after the children had gone to sleep, that I finally got down to work. And then the miracle occurred. Everything seemed to fit together as if by magic!

*   *   *

Trembling all over, I write out by hand and then type into the computer file the last six or seven pages—all the loose ends of the proof are tied up at last, at least for interactions more regular than the Coulomb interaction.

At 2:30 I go to bed. My head feels like it’s going to explode. I stay awake for a long, long time, eyes wide open.

At 3:30 I finally fall asleep.

At 4:00 my son comes in to wake me up, he’s wet his bed. It’s been years since that’s happened—had to happen tonight, of all nights.…

That’s life. Onward. I get up, change the sheets, remake the bed. The whole routine.

There are times when everything conspires to prevent you from sleeping. So be it!

*   *   *

Every mathematician worthy of the name has experienced, if only rarely, the state of lucid exaltation in which one thought succeeds another as if miraculously.… Unlike sexual pleasure, this feeling may last for hours at a time, even for days.

[André Weil]

TWENTY-THREE

Princeton

March 22, 2009

In the end it turned out that my solution didn’t quite work. It took me almost a week to convince myself of this. The better part of the proof survives intact, but the abominable zero mode continues to taunt us. Still, we came close!

Clément is in Taiwan, giving the first public talks about our work. He pondered my ideas, combined them with his own, and expressed the result in his own way. Then I reworked it and expressed it in mine.

The new version is much simpler than my first attempt—and it works! We’ve been slaving away at the proof for exactly a year now, and for the first time it really seems to hang together!

The timing couldn’t be better: I’m due to announce the result in Princeton two days from now.…

*   *   *

Date: Sun, 22 Mar 2009 12:04:36
+
0800

From: Clement Mouhot

To: Cedric Villani

Subject: Re: finishing touches

Okay, I think I finally understand what you had in mind for the spatial average!! And I think it has to be combined with the idea I mentioned to you on the phone (in fact the two are complementary), here’s the plan:

(1) I think that the calculation you had in mind, using the best stratified regularity on the background, is the calculation at the beginning of subsection 6 (pages 65–66): in this case (no scattering), one can in fact use the margin of regularity on the background “for free” to create growth (independently of the level of regularity on the force field).

(2) It’s a question then of reducing to this case by means of the idea I mentioned on the phone (the “remainder” that we talked about isn’t trivial, it’s got to be treated by (1)):

a. we replace $F[h^{n
+
1}] \circ \Omega^n _{t,\tau} \circ S^0 _{\tau,t}$ by $F[h^{n
+
1}] \circ S^0 _{\tau,t}$, the remainder has the right time decay thanks to the estimates on $\Omega^n - Id$

» therefore we’re left with

\int_0 ^t \int_v F[h^{n
+
1}] \cdot
<
((\nabla_v f^n) \circ \Omega^n)
>
(x-v(t-\tau),v) \, d\tau \, dv

b. now we make the key move, changing the variable to replace \Omega^n by \Omega^k in \nabla_v f^n (for any k between 1 and n): the pb we used to have applying \Lambda no longer exists since we no longer compose \Omega^n X with (\Omega^n)^{-1} \Omega^k, now we have only (\Omega^n)^{-1} \Omega^k, which we already have estimates on.

c. once again we get rid of the application (\Omega^n)^{-1} \Omega^k which has been transferred to F[h^{n
+
1}] by the same trick as in step a., which creates a nice new remainder term that steadily diminishes over time,

» therefore we’re left with

\sum_{k
=
1} ^n \int_0 ^t \int_v F[h^{n
+
1}] \cdot
<
((\nabla_v h^k) \circ \Omega^k) (x-v(t-\tau),v)
>
\, d\tau \, dv

d. only now do we invert the gradient in v with composition by scattering:

<
(\nabla_v f^n) \circ \Omega^k
>
=
\nabla_v (
<
f^n \circ \Omega^k
>
)

+
remainder with steady decay into \tau

» therefore we’re left with

\sum_{k
=
1} ^n \int_0 ^t \int_v F[h^{n
+
1}] (x-v(t-\tau),v) \, \nabla_v U_k (v) \, d\tau \, dv

with functions U_k (v) of regularity \lambda_k, \mu_k.

e. At this level we finally apply calculation (1) on pages 65–66 for each k, which ought to give a uniform stratified estimate.

Tell me what you think and if you get the same results for the calculations …

Best regards, Clement

TWENTY-FOUR

Princeton

March 24, 2009

The first of my three seminar talks at Princeton. Before a distinguished audience of tough-minded mathematical physicists, none of them tougher than Elliott Lieb, cordial but implacable.

Clément is still in Taipei. A time difference of thirteen hours—very nearly the optimal difference for efficient collaboration at a distance! With the added advantage that this way we split up the world: he spreads the good word in Asia, I do the same in the United States.

This time I was ready to take the plunge. It wouldn’t be anything like my wobbly performance at Rutgers back in January, I was sure of that: the proof is 90% correct or better, and all the major aspects of the problem have been identified. I was confident, prepared to submit to questioning and to explain the argument.

While the results made quite an impression, Elliott wasn’t convinced by the assumption of periodic boundary conditions, which seemed to him wholly unwarranted.

“If it isn’t true in the space as a whole, it’s meaningless!”

“Elliott, in the space as a whole there are counterexamples. There’s no choice but to set boundaries!”