Birth of a Theorem: A Mathematical Adventure
Page 8
I was on the edge of my seat, listening intently. Then, after making two or three general remarks, he abruptly stopped.
“I’m sorry, I’ve got to go now. I use public transportation, just now with the snow it’s very slippery and my balance isn’t very good, it’s a rather long walk home from the bus stop, and…”
Once he’d finally gone through all the reasons why he had to leave right away, the monologue came to an end. The time we’d actually spent discussing mathematics was no more than five minutes, from which I learned nothing. To think that this was the same man who is responsible for the most amazing theorem in all of fluid mechanics! Living proof that having a superior mind is no guarantee of being able to communicate.
I returned to Joel’s office and told him about my meeting with Scheffer. I said that I was sorry it had been so brief.
“Cédric, five minutes with Vlad is about as much time as anyone in the department has spent talking with him in the last five years.”
An experience that will remain forever engraved in my memory.… But now it was time to go back to thinking about Landau damping.
On the way home, my doubts returned once more.
Finally I realized the proof isn’t going to work.
* * *
The talk at Rutgers marked a turning point in my quest. To announce results you have not yet proved is a grave sin, a violation of the bond of trust that unites a speaker and his audience. My back was to the wall: if I was ever to make up for my transgression, I had to prove the result I had just announced.
John Nash, my mathematical hero, is said to have regularly put himself under fantastic pressure by announcing results that he did not yet know how to prove. There is no question he did this at least once, in the case of the isometric embedding theorem.
On my way home that day I began to feel something like the pressure Nash must have felt. The terrible sense of urgency that now hung over me was not going to go away until I succeeded, one way or another, in completing this proof. I had to complete it—or else forever be disgraced!!
* * *
* * *
Imagine you’re walking through the woods on a peaceful summer’s afternoon. You pause at the edge of a pond. Everything is perfectly calm, not the slightest breeze.
Suddenly the surface of the pond becomes agitated, as though seized by convulsions; a few moments later, it is sucked down into a roaring whirlpool.
And then, a few moments after that, everything is calm once more. Still not a breath of air, not even a ripple on the surface from a fish swimming beneath it. So what happened?
The Scheffer–Shnirelman paradox, surely the most astonishing result in all of fluid mechanics, proves that such a monstrosity is possible, at least in the mathematical world.
It is not based on an exotic model of quantum probabilities or dark energy or anything of that sort. It rests on the incompressible Euler equations, the oldest of all partial differential equations, used by mathematicians and physicists everywhere to describe a perfectly incompressible fluid without any internal friction.
It has been more than two hundred fifty years since Euler derived his fundamental equations, and yet not all of their mysteries have been penetrated. Indeed, they are still considered to mark out one of the most treacherous regions of the mathematical world. When the Clay Mathematics Institute set seven “millennium problems” in 2000, offering $1 million apiece for their solution, it did not hesitate to include the regularity of solutions to the Navier–Stokes equations. It was very careful, however, to avoid any mention of Euler’s equations—a far greater and more terrifying beast.
And yet at first glance Euler’s equations seem so simple, so innocent, utterly devoid of guile or cunning. No need to model variations in density or to grapple with the enigmas of viscosity. One has only to write down the classical laws of conservation: conservation of mass, quantity of motion, and energy.
But then … suddenly, in 1993, Scheffer showed that Euler’s equations in the plane are consistent with the spontaneous creation of energy! Energy created from nothing! No one has ever seen such bizarre behavior in fluids in the natural world! All the more reason, then, to suspect that Euler’s equations hold still more surprises in store for us. Big surprises.
Scheffer’s proof is a stunning feat of mathematical virtuosity, as obscure as it is difficult. I doubt that anyone other than its author has read it carefully from beginning to end, and I am certain that no one could reconstruct its reasoning, unaided, in every detail.
There was more to come. Four years later, in 1997, the Russian-born mathematician Alexander Shnirelman, renowned for his originality, presented a new proof of this staggering result. Shortly afterward Shnirelman proposed a physically realistic criterion for solutions to Euler’s equations that would prohibit pathological phenomena of the sort Scheffer had discovered.
Alas! A few years ago, two brilliant young mathematicians, Camillo De Lellis, an Italian, and László Székelyhidi, a Hungarian, proved a general and still more shocking theorem that showed, among other things, that Shnirelman’s criterion was powerless to resolve the paradox. Additionally, using the techniques of convex integration, they were able to develop a new method for producing these “wild” solutions, an elegant procedure that grew out of earlier research by a number of mathematicians, including Vladimir Šverák, Stefan Müller, and Bernd Kirchheim. Thanks to De Lellis and Székelyhidi, we now realize that even less is known about Euler’s equations than we thought.
And what we thought we knew wasn’t much to begin with.
* * *
Excerpt from the 2008 Bourbaki Seminar Talk
Theorem [Scheffer (1993), Shnirelman (1997)]. There exists a nonzero weak solution of the incompressible Euler equations in dimension 2,
without forcing (f ≡ 0), with compact support in space-time.
Theorem [De Lellis and Székelyhidi (2007, 2008)]. Let Ω be an open interval of and a uniformly continuous function Ω×]0, T [→]0,+∞[, with Then for all η > 0 there exists a weak solution (v, p) of the Euler equations, without forcing (f ≡ 0), such that
(i) ;
(ii) v(x, t) = 0 if ; in particular
(iii) for all and almost all
(iv)
Moreover,
(v) in L2(dx dt),
where each (vk, pk) is a pair of C ∞ functions with compact support—the classical solution of the Euler equations with a well-chosen forcing in the sense of distributions.
SIXTEEN
Princeton
Morning of February 25, 2009
Plenty of peace and quiet here! The woods, the gray squirrels, the pond, biking.
And good food! The other day in the dining hall for lunch we had a velvety pumpkin soup, just like at home, a grilled swordfish filet, very tender and well seasoned, a dessert with mulberries and cream that melted in your mouth.…
In the afternoon we’ve only just gotten back to work in our offices when the bell in the clock tower atop Fuld Hall chimes three o’clock: time to go drink tea in the common room and eat the freshly baked cakes that change every day. The madeleines in particular are irresistible, every bit as scrumptious as the ones I used to make for the boys and girls in my dormitory fifteen years ago.
Bread is a real weak point: the crispy French-style baguette is hard to find in Princeton. An even more serious deficiency, as far as products of the highest necessity are concerned, is how scandalously poor the cheese is. The fruity Comté, the delicate Rove, the fragrant Échourgnac, the smooth Brillat-Savarin, the soft Navette, the spicy Olivia, the indestructible Mimolette—I can’t find any of them anywhere. My entire family has been suffering since we got here!
Earlier this month I made a lightning visit to the West Coast, to the Mathematical Sciences Research Institute at Berkeley. The MSRI is the world’s foremost sponsor of mathematical programs and workshops, welcoming hundreds of visitors every year in addition to a smaller number of research fellows invited for extended stays. It was an
emotional moment for me, being back in Berkeley, where I lived for five months in 2004.
Naturally I made a special point of visiting the Cheeseboard, my favorite place in town, a cooperative run on socialist principles (just as you’d expect—this is Berkeley, after all!) offering a selection of cheeses that would put most shops in France to shame.
I loaded up. There was even some Rove, which I knew would make my kids happy. They can’t eat enough of it. And when I complained to the people working there about how hard it is to find good cheese in Princeton, they told me to check out Murray’s the next time I go to Manhattan. Can’t wait!
The French equivalent of the MSRI is the Institut Henri Poincaré in Paris, founded in 1928 with the help of the Rockefeller Foundation and the Rothschild family. Two months ago the governing board of the IHP informed me that I had been selected as the next director of the institute—unanimously, I was told. But I didn’t accept at once. I set a number of conditions, and deliberations dragged on and on.
I was first approached about the directorship four months ago. Once the initial surprise wore off, I decided it would be an interesting experience and agreed to be considered as a candidate. I didn’t tell my colleagues at ENS-Lyon, fearing they would take it badly. Why should I want to be director of an institute when I’d refused to be director of our laboratory? Why leave Lyon for Paris when I have flourished in Lyon? And who in this day and age really wants to be head of a major scientific organization, weighed down with administrative duties and having constantly to comply with new government regulations and legislative mandates?
How naïve I was to imagine that my candidacy could remain a secret! Not in France …
My colleagues in Lyon quickly learned of it—and they were amazed. Why would a mathematician my age seek to be appointed to a position with such burdensome responsibilities? I must be hiding something, they whispered among themselves. There must be some personal secret, some private agenda.
There’s no secret. And no agenda beyond a sincere desire to do something new and challenging. But only under the right circumstances! The news wasn’t very encouraging. In fact, there wasn’t any news at all. Deliberations at the IHP seemed to have gotten bogged down.…
Would we be pushing off to Paris, then, or going back to Lyon? Perhaps neither one. Cheese or no cheese, life here is very agreeable, and I have an offer to stay at the IAS for a year, maybe longer if things go well, with a handsome salary and other benefits. And now that Claire’s been able to get on with her own research again, this is a good place for her to be too. She’s part of a team in the Department of Geosciences at Princeton that’s analyzing what may turn out to be the oldest known animal fossils—an extraordinary discovery! The leader of the team is urging her to apply for a postdoc. As it is, by coming with me to Princeton she lost her teaching position in Lyon, and by now it’s too late for her to be considered for the next round of faculty assignments.
None of this makes Claire really want to go back. Staying on here would certainly be simpler for her, and more satisfying as well. And so it’s difficult to resist the allure of Princeton. To be sure, I can’t see myself settling permanently in a place where good bread is so hard to find.… But for a few years, why not? And if the IHP can’t be bothered to come up with an attractive offer, well, there’s nothing I can do about it!
Anyway, I’d been mulling all this over for several weeks and just last night decided to send a letter to France declining the job.
But this morning, when I went to open my email, there it was, a message from the IHP saying that all my conditions had been accepted! Okay to more money, okay to no teaching duties, okay to continued research funding. All of which would have been approved in the United States as a matter of course, but in France it’s quite unheard of. Claire was reading the message over my shoulder.
“If they can be counted on to do everything they say they will, you ought to accept.”
Exactly what I was thinking. And so it’s decided: we will say goodbye to Princeton and go back to France at the end of June!
Now to tell my new colleagues here the news. No doubt some of them will understand and offer their encouragement. (Give it all you’ve got, Cédric! It’s going to be a tremendous experience, etc.) Others will be worried for me. (Cédric, have you really given this enough thought? Running an institution like that will leave you no time for your own work, etc.) One or two, I’m willing to bet, will be terribly upset with me. In any event, my diplomatic skills are going to be tested right away—in the United States rather than France!
In the midst of all this confusion, one thing is certain: nothing is more important right now than the work I’m doing with Clément.
* * *
The Institut Henri Poincaré (“Home of Mathematics and Theoretical Physics”) was founded in 1928 to put an end to the state of isolation in which French science found itself following World War I. Soon it was renowned not only as an institution of scientific training and research but as a cultural forum as well. Einstein lectured on general relativity there, Volterra on the use of mathematical methods in biology. The IHP was home to the first French institute of statistics and the site of the first French computer project. It was also, and not least, a place where artists mingled with scientists. Some of the surrealists found inspiration there, as the photographs and paintings of Man Ray attest.
Later incorporated as a branch of the mathematics faculty of the University of Paris, the IHP was moribund for two decades after 1968, then reestablished in the early 1990s as a department of Université Pierre et Marie Curie (UPMC) and an organ of national scientific policy supported by the Centre National de la Recherche Scientifique (CNRS).
As an integral part of a very large university, the IHP is assured of financial stability during uncertain times and benefits from the expertise of a sizable staff of administrators and technical specialists that a private institution of its size could not afford to maintain. Affiliation with the CNRS provides additional financial and administrative resources, as well as direct access to a national network of scientific research organizations.
The IHP does many things. Above all it serves as a meeting place for mathematicians and physicists from France and around the world. In addition to offering a graduate-level curriculum, it sponsors lectures and tutorials for students and visiting researchers on selected topics, and hosts a great many conferences and seminars every year, welcoming an endless stream of invited speakers. The thirty members of the IHP’s governing body, elected in part by national ballot, include representatives of all the major scientific institutions in France; the twelve seats on the scientific council, which is totally independent of the board of directors, are occupied by researchers of the first rank. The Institute’s historic location, its outstanding reference library, its commitment to both teaching and research, and its close partnership with learned societies and other associations devoted to the advancement of mathematics and physics all contribute to its great and enduring influence.
[From the mission statement of the Institut Henri Poincaré]
SEVENTEEN
Princeton
Afternoon of February 25, 2009
The children are back from school, merrily building a playhouse. The squirrels are prancing on the lawn outside.…
I’m on the phone with Clément. He’s feeling rather less merry.
“We can resolve some of the problems I mentioned using stratified estimates. But there are still loads of other problems.…”
“Well, at least we’re making progress.”
“I’ve been studying Alinhac–Gérard, and there’s a serious problem with the estimates: you’d need some leeway regarding regularity to get convergence to zero in the regularization term. And it gets worse: regularization could eliminate the bi-exponential convergence of the scheme.”
“Damn! I missed that! You’re sure that Newton’s rate of convergence is lost? Well, we’ll figure out something.”
“And the regulari
zation constants in the analytic are monsters!”
“They’re a big problem, absolutely. But we’ll find a way to deal with them.”
“And then in any case the constants are going to blow up way too fast to be killed off by the convergence in Newton’s scheme! Since the background has got to be regularized in order to cope with the error created by the function b, it’s the inverse of the time, but there’s a constant, and this constant has got to make it possible to control the norms that come from the scattering.… Don’t forget, these norms grow in the course of the scheme, since we want the losses to be summable on lambda!”
“Okay, okay. I don’t know yet how we’re going to handle all of this. But I’m sure we’ll find a way!”
“Cédric, do you believe—really believe—that we’ll be able to handle all of this by means of regularization?”
“Yes, of course. These are all technical details. Look at what we’ve done overall. We’ve made a hell of a lot of progress. We’ve figured out how to deal with resonances and the plasma echo, we’ve worked out the principle of time cheating, we’ve got good scattering estimates, good norms—we’re almost there!”
Clément must think that I’m a pathological optimist, one of those people you’re crazy to have anything to do with, the sort of person who goes on hoping when there’s no hope left to be had. This latest difficulty does indeed look formidable, I admit. But still I feel sure we’ll find a solution. Already three times in the last three weeks we’ve found ourselves at an impasse, and each time we’ve found a way around it. It’s also true, however, that some obstacles we thought we’d put behind us have come back to block our way in another form.… Nonlinear Landau damping is a monster, no doubt about it—with as many heads as the Hydra! Nonetheless, I remain convinced that nothing can stop us. My heart will conquer without striking a blow.