For this feat he earned the glory of King Oscar’s prize and the money that came with it.
It was understood that the winning paper would appear in Acta Mathematica. The assistant editor responsible for preparing the text for the printer noticed a few rather doubtful things about Poincaré’s solution, but this hardly came as a surprise: everyone knew that Poincaré wasn’t a model of clarity. The editor duly transmitted his queries to the author, with all the deference due to one of the towering figures of the age.
By the time Poincaré realized that a serious error had crept into the proof, the issue containing his article had already been published! An erratum would not suffice, because the results themselves were fatally contaminated.
Mittag-Leffler was unflustered. He managed on various pretexts to get all the copies back, one after another, before anyone spotted the error. Poincaré bore the full cost of pulping the original print run—a sum that exceeded the value of King Oscar’s award!
At this point the story takes an extraordinary turn. Poincaré redeemed his mistake by seizing the opportunity it presented to create a whole new field of scientific investigation. Having remedied the defects of his argument and revised its conclusions accordingly, he found that he had in fact shown the opposite of what he felt sure he had demonstrated: instability was possible after all!
In the corrected and expanded form in which it was republished the following year, in 1890, the article gave rise to the study of dynamical systems, a topic that more than a century later keeps thousands of scientists busy in fields ranging from physics to economics. Quasi-periodic orbits, chaos theory, sensitive dependence on initial conditions, fractals—all these things are found in embryonic form in Poincaré’s revised paper. What could have been a disaster for Acta Mathematica turned out to be a triumph.
The journal’s renown continued to grow, and today it is one of the most prestigious scientific journals in the world, perhaps even the most prestigious of all. To see an article you’ve written take up some of the six hundred pages Acta Mathematica publishes each year is almost enough to assure your future as a professional mathematician.
When Poincaré died, in 1912, he was eulogized in France as a national hero. Four years later, in 1916, Mittag-Leffler’s house outside Stockholm was converted into an international research center where mathematicians from all over the world could gather to ponder old problems and work together on new ones. The Mittag-Leffler Institute, the very first of its kind, is still going strong today. In 1928, the year after Gösta Mittag-Leffler himself passed away, a second such center was founded in Paris, based on the same principles of international fellowship, with special emphasis on postgraduate training. Appropriately, it was named after Henri Poincaré.
Henri Poincaré
Gösta Mittag-Leffler
THIRTY-SIX
Ann Arbor
October 27, 2009
In my hotel room in Ann Arbor. I’m here for a few days at the University of Michigan—a great university with some mathematicians of the first rank.
Clément was really demoralized by Acta Mathematica’s rejection. He wanted to try to convince them to reconsider their decision, to make them understand why our result is so innovative and so important, even if there’s still a little gray area.…
But I’ve had more experience dealing with these people than he has. And as an editor of a rival journal, Inventiones Mathematicae, I know firsthand how tough one must be in judging manuscripts submitted for review. The Acta editors are even thicker-skinned than I am. Nothing will make them change their minds unless a referee can be shown to have acted in bad faith (no hint of this in our case) or unless the proof can be strengthened.
One possibility would be to cut the beast in two in the hope of improving our chances of publishing elsewhere, but I cannot bring myself to do such a thing.… So we’re going to sleep on it.
My talks here have gone well so far, but the same questions keep coming up over and over again. I’ve discussed the matter with Jeff Rauch, a leading expert on partial differential equations who has spent a lot of time working in France. Jeff wasn’t shocked by the fact that the result doesn’t hold for infinite time, but he didn’t like the analyticity assumption. There are others, of course, who would like to see it work in infinite time and who don’t mind the assumption. I could tell myself that it’s no big deal; but since I trust Jeff’s judgment, I find his reservations troubling. So this evening I decided to sit down and write out an argument that would convince him that our proof is as fine a thing as human minds can make it, and that very few, if any, improvements are possible. A good bit of work, but it’s as much for my benefit as for his.
Jeff Rauch
The time went by. Sitting on the bed, scribbling away. I couldn’t find a way to convince myself.… And if I can’t find a way to convince myself, there’s very little chance that I’ll have any luck convincing Jeff!!
What if I went astray somehow, if my estimates weren’t precise enough? Here, though, I haven’t lost anything … there, it would really be surprising if I got it wrong … here, it’s optimal … and there, well, simplifying can only help—unless I’ve been utterly bewitched.…
Like a cyclist inspecting his chain for a weak link, I carefully went through the proof, checking the soundness of the argument at each step.
And there …
There! That’s where I may have been too careless! Jesus, how could I have not seen that the modes are diverging—and that my comparison by summation was too rough!?!? If it’s the sup in relation to the sum, obviously I’m going to lose something!! Well, okay, it got swamped by all the technical detail.…
No time for grumbling, I had to work out the implications.
But of course … the modes diverge, the weight shifts, if you put them all together the loss is huge!! That means they’ve got to be kept separate!!!
This was my moment of illumination, the moment when the light bulb went on in my head. I jumped up from the bed and feverishly paced up and down, pencil in one hand, a sheet of paper covered with cabalistic formulas in the other. I kept on staring at them. This time it wasn’t a question of fixing an error; it was a question of improving our results.
How are we going to get from here to there?
I don’t know. But now at last we’re on our way. We’re going to take the whole thing apart and put it back together. We’ve finally figured out how to lay these two infernal objections to rest, once and for all.
* * *
Since γ = 1 is the most interesting case, it is tempting to believe that we stumbled on some deep difficulty. But this is a trap: a much more precise estimate can be obtained by separating modes and estimating them one by one, rather than seeking an estimate on the whole norm. Namely, if we set
then we have a system of the form
(7.15)
Let us assume that . First we simplify the time dependence by letting
Then (7.15) becomes
(7.16)
(The exponential for the last term is right because (k + 1)(kt/(k + 1)) = kt.) Now if we get a subexponential estimate on Φk(t), this will imply an exponential decay for ϕk(t).
Once again we look for a power series, assuming that Ak is constant in time, decaying like e −ak as k → ∞; so we make the ansatz with ak,0 = e −ak. As an exercise, the reader can work out the doubly recurrent estimate on the coefficients ak,m and deduce
whence
(7.17)
This is subexponential even for γ = 1: in fact, we have taken advantage of the fact that echoes at different values of k are asymptotically rather well separated in time.
Therefore, as an effect of the singularity of the interaction, we expect to lose a fractional exponential on the convergence rate: if the mode k of the source decays like then ϕk, the mode k of the solution, should decay like More generally, if the mode k decays like A(kt), one expects that φk(t) decays like Then we conclude as before by absorbing the fractional exponential in a very slow expo
nential, at the price of a very large constant: say,
[From my 2010 Luminy Summer School lecture notes on Landau damping]
THIRTY-SEVEN
Charlotte
November 1, 2009
In transit between Palm Beach and Providence. An impersonal airport in North Carolina; it could be anywhere, really. Removing all metal items when you pass through the security checkpoint is a bit of a chore for most people, but when you also wear cuff links and a fob watch, and on top of that carry one or two USB flash drives and a half dozen pens in your pockets …
In Florida, at the international workshop on geometric inequalities that Emanuel Milman helped organize in Boca Raton, the living was easy! From downtown it was only a few steps to the beach. And the ocean was like a warm bath. Even at night the temperature was ideal, no one around, no need for a bathing suit … really, it was like swimming in a bathtub—a bathtub with tides and soft sand! And all this in November!
But now the fun’s over, back to the cold. It’s going to happen so fast, thanks to the unnatural swiftness of modern air travel!
In Boca Raton I was able to forget Landau damping for a day or two, but now once again it occupies my every waking moment. I’m beginning to see what needs to be done to make the proof stronger overall, how to exploit the flash of inspiration that struck me in Ann Arbor. Still, it’s going to be a huge amount of work! Waiting for the connecting flight to Providence, I wasn’t sure if I really had enough confidence to talk about the new plan of attack at Brown without having yet worked out all the details. It’s a very important talk: Yan Guo, the one by whom the Problem came, is going to be in the audience.
I took out a blank piece of paper and began to sketch the new plan, redoing calculations etc., when all of a sudden it leapt to my eyes—there’s something wrong, a contradiction.
I can’t possibly prove an estimate that sharp.…
After a few minutes I’d convinced myself that there must be a mistake somewhere in one of the especially involved parts of the proof. Or could it be that—it’s all wrong? The airport began to pitch and roll around me.…
I pulled myself together.
Cédric, the error can’t be very serious. Everything else holds together too well. The error has to be local, it’s got to be right here, in this passage somewhere. The reason for it has to be that the calculation is obscured by these two miserable little shifts—the double time-shift you introduced after coming back from the museum!
But Clément definitely showed that we can do without them!! So be it, we’ll have to get rid of them after all, they’re too dangerous. In a proof this complex, the least source of obscurity must be ruthlessly eliminated.
Even so, if I hadn’t come up with this double shift we might have been stymied for good. The double shift was the thing that gave us hope, that allowed us to move forward again. Later on, we saw that it wasn’t necessary. And so what if in the end it turns out to be wrong!? Fine. We’ll rewrite everything, without even mentioning it.
For the moment I need to decide what to say tomorrow at Brown. I’ll have to say that I’ve found a way to improve the proof, and then explain why it’s important, because it will answer the two criticisms of our result that have been made over and over again. But I mustn’t cheat—no bluffing this time!
Palm Beach to Providence, more turbulence than expected …
* * *
SUMMARY OF YOUR WEST PALM BEACH–PROVIDENCE TRIP
Flight details: Sunday 1 November 2009
Travel time: 6 hrs 39 mins
Depart: 03:00 PM, West Palm Beach, FL (PBI)
Arrive: 04:53 PM, Charlotte, NC (CLT)
US Airways 1476 Boeing 737-400 Economy Class
Depart: 07:49 PM, Charlotte, NC (CLT)
Arrive: 09:39 PM, Providence, RI (PVD)
US Airways 828 Airbus A319 Economy Class
* * *
Coulomb/Newton (most interesting case)
Coulomb/Newton interaction and analytic regularity are both critical in the proof of nonlinear Landau damping; but the proof still works for exponentially large times “because”
• the expected linear decay is exponential
• the expected nonlinear growth is exponential
• the Newton scheme converges bi-exponentially
Still, it seems possible to go further by exploiting the fact that echoes at different spatial frequencies are asymptotically rather well separated.
[From notes for my talk at Brown University, November 2, 2009]
THIRTY-EIGHT
Saint-Rémy-lès-Cheuvreuse
November 29, 2009
Sunday morning, scribbling away in bed. One of the special moments in the life of a mathematician.
I’m rereading the final version of our article, crossing out, correcting. More relaxed than I’ve been in months! We’ve rewritten the whole thing. Completely eliminated the treacherous double shift. Succeeded in exploiting the asymptotic time separation of the echoes, recast the main part of the proof, substituted a mode-by-mode study for the original aggregate-level treatment, relaxed the analyticity condition, and, last but not least, included the Coulomb case in infinite time, the thing that everyone had been complaining about for so long.… Everything revised, everything simplified, everything checked once more, everything improved, everything gone over one last time.
All of this could easily have taken three months, but in our present state of feverish excitement three weeks turned out to be enough.
More than once, going through the argument with a fine-toothed comb, we’ve wondered how in the world we could have come up with this little trick or that little piece of cleverness.
The result is now much stronger. In the process we also managed to solve a problem that has long intrigued specialists such as Guo, technically known as the orbital stability of homogeneous nonmonotonic linearly stable equilibria.
We added a few passages, but simplifying has reduced the length elsewhere, so that now the manuscript is scarcely longer than the one we submitted earlier.
New computer simulations came in as well. When I saw the first batch of results last week, I was staggered: the numerical calculations that Francis had performed using an extremely precise formula seemed to completely contradict our theoretical results! But I didn’t buckle. I told Francis I was skeptical, and he reran everything using another method that is considered to be even more accurate. When the second batch came in, the results were consistent with the theoretical prediction. Phew! Just goes to show that computer simulations are no substitute for qualitative insight.
Tomorrow we’ll be ready to make the new version available via the Internet. And at the end of the week we’ll be able to resubmit the paper to Acta Mathematica, with a much greater chance of success this time around.
In my idle moments I can’t help but think of Poincaré himself. One of his most famous articles was, well, not rejected by Acta, but nevertheless withdrawn, and then corrected and finally republished. Perhaps the same thing is going to happen to me? It’s already been my Poincaré year: I won the Henri Poincaré Prize, and I’m head of the Institut Henri Poincaré.
Poincaré … careful, Cédric, beware delusions of grandeur!
* * *
Paris, 6 December 2009
From: Cédric Villani
École Normale Supérieure de Lyon
& Institut Henri Poincaré
11, rue Pierre & Marie Curie
F-75005 Paris
FRANCE
[email protected]
To: Johannes Sjöstrand
Editor, Acta Mathematica
IMB, Université de Bourgogne
9, avenue A. Savarey, BP 47870
F-21078 Dijon
FRANCE
[email protected]
Re: Resubmission to Acta Mathematica
Dear Professor Sjöstrand:
Following your letter of October 23, we are glad to submit a new version of our paper, “On Landau dampin
g,” for possible publication in Acta Mathematica.
We have taken good note of the concerns expressed by some of the experts in the screening reports on our first submission. We believe that these concerns are fully addressed by the present, notably improved, version.
First, and maybe most importantly, the main result now covers Coulomb and Newton potentials; in an analytic setting this was the only remaining gap in our analysis.
Analyticity is a classical assumption in the study of Landau damping, both in physics and mathematics; it is mandatory for exponential convergence. On the other hand, it is very rigid, and one of the referees complained that our results were tied to analyticity. With this new version this is not so, since we are now able to cover some classes of Gevrey data.
Birth of a Theorem: A Mathematical Adventure Page 16
