(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) , dtau , 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) > , dtau , 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) , dtau , 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!”
“Yes, but the result has to be independent of boundaries—otherwise it’s not physics!”
“Elliott, Landau himself used boundary conditions, and he showed that the result depended very strongly on boundaries. You don’t mean that he wasn’t a physicist, do you?”
“But it’s meaningless!”
Elliott had gotten up on his high horse. And he wasn’t my only critic. Greg Hammett, a physicist at the Princeton Plasma Physics Laboratory (the PPPL, as everyone calls it), didn’t much like my assumption of stability in the case of plasmas. Too strong, he felt, to be realistic.
The triumphal reception I was hoping for turned out to be anything but. A real letdown!
* * *
Elliott Lieb is one of the most famous and formidable figures in mathematical physics today. A professor of both mathematics and physics at Princeton, he has devoted a part of his life to finding an explanation for the stability of matter: What is it that forces atoms to come together rather than keep their distance, standing apart from one another in serene isolation? Why is it that we are physically coherent creatures? Why don’t we just melt away into the surrounding universe? Freeman Dyson, one of the twentieth century’s greatest physicists (and now an emeritus professor at the IAS), was the first to pose this problem in mathematical terms. In clearing the way for future investigation, Dyson inspired a younger generation of researchers to carry on the quest.
Elliott Lieb
Elliott gave himself up to it completely. He looked everywhere, seeking a solution in physics, in analysis, in the calculation of energies. Over a long career he has trained many other scientists, created influential schools of thought, and published spectacular proofs that have changed the face of mathematical analysis.
To Elliott’s way of thinking, nothing beats a good inequality in trying to understand a problem. An inequality expresses the domination of one term in an equation by another, of one force by another, of one entity by another. Not only has Elliott profoundly improved a number of celebrated statements, including Hardy–Littlewood–Sobolev inequalities, Young inequalities, and Hausdorff–Young inequalities, he has given his own name to two more groups of fundamental relations: Lieb–Thirring inequalities and Brascamp–Lieb inequalities, used today by scientists throughout the world.
Now almost eighty years young, Elliott is still as active as ever. His trim figure testifies to his physical fitness; his barbed wit, dreaded by all, to his mental sharpness. His eyes light up when he speaks of Japan, inequalities, and gourmet cooking (as it happens, the Japanese word for haute cuisine, kaiseki, also means mathematical analysis).
TWENTY-FIVE
Princeton
April 1, 2009
The first day of April—the day of fishes and fools!
This afternoon watched an episode of Lady Oscar with Claire and the children. Marie-Antoinette, Axel de Fersen, and Oscar de Jarjayes spinning round in a whirl of fine phrases and noble sentiments amid the lengthening shadows of the French Revolution.
And this evening, before going to sleep, we watched a YouTube video of Gribouille singing “Le Marin et la Rose.” Simply marvelous! There’s some great stuff on the Internet.
During the past week I’ve learned so much from lecturing on Landau damping.
After my first talk, once his irritation had subsided, Elliott shared some valuable insights into the conceptual difficulties of the periodic Coulomb model.
At the second talk I laid out the main physical idea of the proof. Elliott very much appreciated the mixture of mathematics and physics; he seemed not only engaged but genuinely supportive.
By the time of the third talk I’d come up with an answer to Hammett’s objection, and I was able to formulate almost optimal assumptions regarding the stability condition and perturbation length.
I’d taken a risk, presenting completely new results that were still only half-baked, but the gamble paid off: their criticisms enabled me to make much faster progress than I could have otherwise! Once again I had to put myself in a vulnerable position in order to become stronger.
And … the connection with KAM finally became clear to me!
The ability to detect hidden connections between different areas of mathematics is what has made my reputation. These connections are invaluable! It’s a bit like a game of Ping-Pong: every discovery you make on one side helps you discover something new on the other. The connections make it possible to see more of the landscape on both sides.
My first important result, with the Italian mathematician Giuseppe Toscani, came in 1997, when I was twent
y-four years old: the unsuspected link between Boltzmann entropy production, the Fokker–Planck equation, and entropy production for plasmas.
The next one came eighteen months later, with my German coauthor Felix Otto: the hidden link between the logarithmic Sobolev inequality and Talagrand’s concentration inequality. Two other proofs have been proposed in the years since.… This is how I got started exploring the field of optimal transport. Thanks to our paper I was invited to give a graduate-level course at Georgia Tech, which in turn gave birth to my first book.
During my thesis defense in 1998, Yves Meyer marveled at the “miraculous” relations I had brought to light. “Twenty years ago people would have laughed at your work. No one believed in miracles then!” But I believe in miracles—and I shall uncover more of them.
In my thesis I recognized four spiritual fathers: my thesis director, Pierre-Louis Lions; my tutor, Yann Brenier; and Eric Carlen and Michel Ledoux, whose works opened up the fascinating world of inequalities to me. In addition to the joint influence of these four teachers, I incorporated other elements and created my own mathematical style, which then evolved—as it pleased chance to bring me into contact with new friends and new ideas.
Three years after my defense, with my longtime collaborator Laurent Desvillettes, I discovered an apparently improbable link between Korn’s inequality in elasticity theory and the production of entropy for the Boltzmann.
After that I developed the theory of hypocoercivity, based on a new analogy between problems associated on the one hand with regularization, and on the other with convergence to equilibrium, for degenerate dissipative partial differential equations.
There was also the hidden link between optimal transport and Sobolev inequalities, which I had detected earlier with Dario Cordero-Erausquin and Bruno Nazaret—a connection that astounded many analysts who thought they really understood these inequalities!
In 2004, as a visiting research professor in the Miller Institute at Berkeley, I had the good fortune to meet another future coauthor, the American mathematician John Lott, then a guest of the MSRI. Together we showed that insights from the study of optimal transport in economics could be used to tackle various problems in nonsmooth non-Euclidean geometry, which together make up the problem of so-called synthetic Ricci curvature. The theory that came out of our collaboration, now called the Lott–Sturm–Villani theory, has had the effect of breaking down a few more barriers between analysis and geometry.
In 2007, once again sensing the presence of a preexisting harmony, I managed to demonstrate a strong relationship between the geometry of the tangent cut locus and the curvature conditions necessary for optimal transport regularity—another connection that seemed to come from nowhere, which I proved with Grégoire Loeper.
Each time, a personal encounter set everything in motion. It was as though I had acted as a catalyst! But I also firmly believe in the importance of searching for preexisting harmonies—after all, Newton, Kepler, and so many others have already shown us the way. Everywhere you look, the world is filled with unsuspected connections!
Nope, didn’t no one ever suppose
They’d anything at all in common
Him, the sailor who was in Formosa
And her, who was the rose of Dublin
And only a finger on the lips …
Nor did anyone ever imagine for a moment that Landau damping and Kolmogorov’s theorem had the slightest thing in common.
Except for Étienne Ghys. Tricked, perhaps bewitched by some mischievous sprite, Étienne had somehow divined that there was a connection between the two. Almost one year to the day after our conversation back in Lyon last April, I’ve finally figured out what it is!
Hmmm … a loss of regularity in a perturbative context, due to resonance phenomena, is made up for by a Newton scheme exploiting the completely integrable character of the system that is disturbed.…
Amazing that this idea ever occurred to me at all! Who would ever have imagined that something so weird could be real? Landau damping, to begin with—who would ever have believed that it’s fundamentally a question of regularity?
* * *
THE SAILOR AND THE ROSE
Once upon a time there was a rose
Aye, a rose there was and a sailor
The sailor, he was in Formosa
The rose, she was a Dubliner
Didn’t never see each other, nope,
Far too great a distance between them
Him, he never left his sailing boat
Her, she never left her fair garden
Over the chaste rose, high in the sky,
Chased swirls of birds in swooping rings
And after them waves of clouds swam by
And after them waves of suns and springs
Over the fickle sailor passed shrouds
Of dreams, each one the same as the ones
Before, dreams of springs and dreams of clouds
Birds of the mind and imagined suns
The sailor, he died in September
And the rose, on the very same day
Wilted in a room free from slumber
Where wept a girl love had thrown away
Nope, didn’t no one ever suppose
They’d anything at all in common
Him, the sailor who was in Formosa
And her, who was the rose of Dublin
And only a finger on the lips
Lovely as only lightning can be
Now the sun draws near to its eclipse
An angel casts petals down upon the sea
TWENTY-SIX
Princeton
Night of April 8–9, 2009
Version 55. The tedious process of rereading and fine-tuning. Then, suddenly, a new hole opens up.
Hopping mad, I’ve just about had it.
Had it up to here with this whole business! Before it was the nonlinear part. Now it’s the linear part that seemed to be under control and then came apart at the seams!
We’ve already announced our result more or less everywhere: last week I gave a presentation in New York, tomorrow Clément’s doing the same in Nice. There’s no excuse for the slightest error at this point—the thing has to be completely correct!!
But there’s no getting around it, there is a problem. Somehow this damn Theorem 7.4 has got to be fixed.…
The children are asleep, Claire’s away again. Working in front of the big picture window, looking out into the dark night. The hours go by. Sitting on the sofa, lying on the sofa, kneeling in front of the sofa, I turn my bag of tricks inside out, scribbling away, page after page. To no avail.
I go to bed at four o’clock in the morning in a state close to despair.
* * *
Date: Mon, 6 Apr 2009 20:03:45 +0200
From: Clement Mouhot
To: Cedric Villani
Subject: Landau version 51
I’m sending you what I’ve got, after going over everything line by line for 120 pages. I’m exhausted, going to take the night off. Here’s version 51, which should incorporate (having now carefully reviewed your messages) all your modifs and email requests (figures, remarks, dependence of constants…), as well as your rewritten section 10 (from the last version you sent me, version 50) and the new section 12.
For my part, I’ve reread everything up through section 9 (in other words up through page 118). There are a fair number of NdCMs that you’ll have to look at, but also a bunch of minor corrections that don’t seem to me to require any discussion. Only two of my comments raise concerns about the proofs (but in neither case do they call the result into question): section 7 page 100 and section 9 page 116.
Here’s what I suggest we do in the next couple days: how about if you work from this version 51 and go over sections 1 to 9, looking at all my comments and deleting each one in turn once you’ve dealt with it, in what let’s call version 51-cv, and I’ll carefully reread sections 10-11-12-1
3-14? (can we both aim to finish up by tomorrow evening or Wednesday morning?)
Best regards, Clement
TWENTY-SEVEN
Princeton
Morning of April 9, 2009
Uhhhhh … man, is it hard to wake up! Finally, with the greatest difficulty, I manage to sit up in bed.
Huh?
I hear a voice in my head. You’ve got to bring over the second term from the other side, take the Fourier transform, and invert in L2.
Unbelievable!
Scribble a note to myself on a scrap of paper, holler at the kids to get dressed, go down to make them breakfast. Hurry up or you’ll be late for school. Scampering out the door, skipping over the wet grass and beating me to the bus stop, they take their place in line behind the other children, patiently waiting to board a beautiful yellow-and-black bus—just like the ones in American movies!
Remarkable how many sons and daughters of top scientists take this bus to Littlebrook Elementary School every morning. Look, there are Ngô Bảo Châu’s kids! Ngô left Paris several years ago to accept a five-year visiting appointment at the IAS. He’s already famous in the mathematical community for his spectacular solution of a long-standing problem called the Fundamental Lemma, now being checked and verified. The branch of mathematics in which Ngô works, algebraic geometry, is renowned for its difficulty—and totally foreign to me. Everyone considers him an odds-on favorite for one of the next Fields Medals!
At Littlebrook the students are pampered. Private English lessons, instruction in other subjects tailored to their individual needs, etc. Great efforts are made to give them a sense of self-confidence—something American teachers can certainly be trusted to do. This afternoon the kids will come back from school in good spirits, actually looking forward to their homework. Lucky thing that the hatred of homework hasn’t yet reached the United States, at least not Princeton.
Birth of a Theorem: A Mathematical Adventure Page 11