A quick glance at the little blond darlings. They’re doing fine without me.
“Hmmmm, this calculation’s rather interesting. And what’s the note at the bottom of the page say?… the important thing in the linearized equation is not the spectral analysis, it’s the solution of the Cauchy problem. Well, yeah, I guess so—that’s common sense! Okay, let’s see how they do this … hmmmm … Fourier transform. Smart move, nothing beats good old Fourier’s method. Laplace transform, dispersion relation…”
I’m learning quickly, immersing myself in the stuff, absorbing it like a child soaking up a foreign language. Humbly, without pretending to know much of anything, I’m teaching myself basic concepts that physicists have known for half a century.…
* * *
In the evening I took a break from my studies. Sitting cross-legged in the attic bedroom of the small chalet where we’re staying, I devoured the latest collection of short stories by Neil Gaiman, Fragile Things—just out in paperback last fall, not yet translated into French. I believe Gaiman is right: we owe it to ourselves to tell stories. Like his tale of a brilliant improvisation on double bass. And of a very old, frail woman recalling her past loves. And of a phoenix that is always rising from its ashes and always being cooked and served for dinner …
Closed my eyes, finally, but couldn’t fall asleep. Couldn’t turn on the light, it’s just one room for the whole family. So my brain went haywire. Very old, frail galaxies improvised a story out of Gaiman, the problem kept coming back to life, only to end up being cooked by the mathematicians and served for dinner, over and over again. Stars sprouted inside my head. What exactly was that theorem I wanted to prove?
* * *
“Crawcrustle,” said Jackie Newhouse, aflame, “answer me truly. How long have you been eating the Phoenix?”
“A little over ten thousand years,” said Zebediah. “Give or take a few thousand. It’s not hard, once you master the trick of it; it’s just mastering the trick of it that’s hard. But this is the best Phoenix I’ve ever prepared. Or do I mean, ‘This is the best I’ve ever cooked this Phoenix’?”
“The years!” said Virginia Boote. “They are burning off you!”
“They do that,” admitted Zebediah. “You’ve got to get used to the heat, though, before you eat it. Otherwise you can just burn away.”
“Why did I not remember this?” said Augustus TwoFeathers McCoy, through the bright flames that surrounded him. “Why did I not remember that this was how my father went, and his father before him, that each of them went to Heliopolis to eat the Phoenix? And why do I only remember it now?”
[…]
“Shall we burn away to nothing?” asked Virginia, now incandescent. “Or shall we burn back to childhood and burn back to ghosts and angels and then come forward again? It does not matter. Oh, Crusty, this is all such fun!”
[Neil Gaiman, “Sunbird”]
* * *
Fourier analysis is the study of the elementary vibrations of signals. Suppose that we wish to analyze some quantity that varies with time, such as sound, which arises from slight variations in atmospheric pressure. Rather than examine the complex variations of a signal directly, a scientist and politician named Joseph Fourier had the idea in the early nineteenth century of decomposing it into its constituent sine waves—trains of signal pulses known as sinusoids (along with their twins, cosinusoids)—each of which varies in a very simple and repetitive manner.
Joseph Fourier
Each sinusoid is characterized by the amplitude and the frequency of its variations. In a Fourier decomposition, the amplitude measures the relative strength of the corresponding frequency in the signal being analyzed.
Most of the sounds we hear are the result of the superimposition of a multitude of frequencies. A vibration at 440 pulses per second is the musical note A above middle C: the greater its amplitude, the louder it will sound. Double the frequency to 880 pulses and we hear an A one octave higher; triple the frequency to 1320 pulses and the pitch goes to the fifth (which is to say E) in the next higher octave. In the world around us, however, sounds are never pure. They are always made up of a great many frequencies that jointly determine the timbre. When I was preparing for my master’s examinations I studied all this in a fascinating course called Music and Mathematics.
Fourier analysis is useful for all sorts of things: decomposing sounds and recording them on a CD, for example, or decomposing images and transmitting them over the Internet, or analyzing variations in the level of the sea and predicting tidal waves.…
Victor Hugo delighted in mocking Joseph Fourier, the “little” prefect from the department of Isère whose reputation he felt sure would soon fade—unlike that of the “great” Fourier, the political philosopher Charles Fourier, who he believed would long be remembered by future generations for his utopian socialism. Charles Fourier may not have welcomed the compliment, however. The socialists mistrusted Hugo, and not without reason: he was certainly the greatest writer of his age, but he was no less famous for the changeability of his political opinions; having started out as a monarchist, he became a Bonapartist, an Orléanist, and then a legitimist before exile finally made a republican out of him.
With all due respect to a magnificent author whose works were among my favorites when I was a child, it is indisputable that Joseph Fourier’s influence is now much greater than Hugo’s ever was. Not only is Fourier’s “great mathematical poem” (as Lord Kelvin called it) taught in universities in every country in the world, it is part of the daily lives of billions of people who aren’t even aware of it.
* * *
From a Draft (Dated April 19, 2008)
Formulas will be derived by taking transforms in three variables, x, v, and t. Note that
Note, too, that
(Laplace transform).
For the time being it will be assumed that
Taking the Fourier transform in x of the Vlasov equation, we find
From this, using Duhamel’s formula, we deduce
Integrating over v, we find
(The justification for integrating over v remains to be provided … but we can always assume at the outset that the datum has compact support in velocities, then approximate? or truncate.…)
The first term on the right-hand side of the equation on the second line is none other than (this is the same trick used earlier for homogenization by free transport…).
Making lenient assumptions about f0, we can write, for all
Therefore
Let us assume
(I’m not sure it’s a good idea to put 4π here.…) In certain cases (such as Maxwellian f0), p0 is positive; but generally there is no reason for this to be true. Note that p0 rapidly decays if; it exponentially decays if f0 is analytic, etc. We obtain, finally,
Taking the Laplace transform in we obtain, so long as everything is well defined,
whence we derive
where
FIVE
Kyoto
August 2, 2008
The deafening noise of the cicadas has ceased, but inside the Shugakuin International House the stifling heat lasts late into the night.…
Earlier today here at the University of Kyoto I concluded a series of lectures, sponsored by the mathematical research institute and attended by junior professors and graduate students from some fifteen different countries. Today’s lecture went well. I began at the appointed hour, or within a minute of it, and finished at the appointed hour, not more than a minute later. Failing to respect a timetable is out of the question in this country. I had to be as punctual as the ferry that took me to Hokkaido last week.
On returning to the visitors’ residence this evening I regaled my children with the continuing adventures of Korako, a little Japanese raven. One day, finding himself abandoned by his parents, Korako set off on a long journey through France and Egypt, where he worked in circuses and bazaars, searching for a secret code with his master, a young boy named Arthur. An improvised tale that go
es on and on, what my daughter calls an “imaginary story”—her favorite kind, and also the kind that is the most fun for the storyteller.
Then the children went to sleep, and for once I didn’t delay in following their good example. After the imaginary story about optimal transport that I told to the audience of budding mathematicians and the imaginary story about the little raven that I made up for my children, I had well earned the right to tell an imaginary story to myself. My brain soon embarked on a fantastic journey of its own.
The tale swept me up and away, and the night went racing by. I woke up with a start, a little after 5:30 in the morning, to that wonderful feeling that lasts only a fraction of a second, when you don’t know where you are—not even what continent you’re on! I jumped up from the futon and went over to my computer to make a note of the few fragments of the dream I could still hold on to before they completely melted away in the mind’s morning fog. The complexity and the confusion of the adventure put me in a good mood: I take such dreams as a sign that my brain is in good working order. They’re not as wild or as madly frantic as the ones that David B. records in his comic books, but they are nevertheless convoluted enough that I take great pleasure in trying to remember them.
For several months now I’ve set Landau damping aside. No real progress yet on a proof, but I have succeeded in clearing a major hurdle: now I know what I want to prove. I want to show that the solution of a nonlinear, spatially periodic, close-to-stable-equilibrium Vlasov equation spontaneously evolves toward another equilibrium. Even if this way of stating the problem is quite abstract, the abstraction is firmly rooted in reality, in a set of closely related topics of considerable theoretical and practical importance. And even if the problem is simple to state, it’s probably difficult to prove. What’s more, it asks an original question about a well-known model. So far, so good; I’m very pleased. For the time being I’m keeping Landau damping in the back of my mind. I’ll come back to it when classes resume in September.
Beyond the answer to the question (true or false), I very much hope that the proof will tell us many things! Appreciating a theorem in mathematics is rather like watching an episode of Columbo: the line of reasoning by which the detective solves the mystery is more important than the identity of the murderer.
In the meantime, there are other passions to indulge: I’m adding an appendix to a paper I wrote two years ago, and I’m making headway on an attempt to combine kinetic equations and Riemannian geometry. Between local positivity estimates for hypoelliptic equations and the kinetic Fokker–Planck equation in Riemannian geometry, I’ve got more than enough to keep me busy during these long Japanese nights.
* * *
OPTIMAL TRANSPORT AND GEOMETRY
Kyoto, 28 July–1 August 2008
Cédric Villani
ENS-Lyon & Institut Universitaire de France & JSPS
COURSE OUTLINE (5 lectures)
• Basic theory
• The Wasserstein space
• Isoperimetric/Sobolev inequalities
• Concentration of measure
• Stability of a 4th-order curvature condition
Statements will usually be given, but occasionally elements of proof as well.
GROMOV–HAUSDORFF STABILITY OF DUAL KANTOROVICH PROBLEM
* * *
The Adventures of Korako (cont’d)
When the moment comes, Korako tosses a stink bomb into the compound, another feat of skill he perfected during his years as a circus performer. Soon the terrible smell makes the guards ill, and Hamad and Tchitchoun set to work filling the air vents with sand.
End of the hunt: the defenses having been breached, the compound is destroyed, Hamad overpowers everyone … (long apocalyptic description). Arthur’s father has been rescued at last, along with his companion in misfortune, a fellow Egyptologist specializing in hieroglyphics. Their abductors had tried to make them talk about a confidential document—an ancient papyrus containing a secret method for bringing mummies back to life.
The bandits have all been taken prisoner. They are brought to the Madman and told that they are going to be tortured and killed if they don’t confess who their leader is. Interrogations follow. Korako is puzzled by the reaction of Arthur’s father, who seems to feel at ease; in fact, he seems to know the place—as though he used to live here. Korako secretly listens in on an interrogation and finds his suspicions confirmed: the Madman and Arthur’s father already know each other. The next morning he is going to see Arthur and tell him the disturbing news.
[From a summary of the story written up afterward]
* * *
Notes on a Dream (August 2, 2008)
I am an actor in a period film, as well as a member of a ruling family. The historical part of the dream involves both the film and a story in which I am a character, with several simultaneous levels of narration. The prince has absolutely no luck at all. He is constantly being hounded by the crowd, the press. There’s a lot of pressure. The king = father of the princess is hatching a plot, something to do with money and a son in hiding. Freedoms are not fully guaranteed. I curse the editorial on the first page of Le Monde. They have also committed political blunders. But there is grave international concern about the rise in raw materials prices; the Nordic countries, a significant share of whose revenues come from transport, are suffering, particularly Iceland and Greenland. No improvement in sight in any case. I comment on the chances of going to Paris, for example, or at least of meeting famous athletes, they’re the real celebrities. I stick holograms containing images of my children on their backs. But a collective suicide has been ordered. Now that the hour has come, I wonder if everyone is present and accounted for. Vincent Beffara isn’t here. He played one of the children, but he’s no longer suited for the role. The filming has gone on for a long while, and Vincent has grown up in the meantime; instead the same actor is used twice, he doesn’t have much to say at the end, there’s no problem using a child. I’m very moved, the operation is going to be launched soon. I contemplate the paintings and posters on the walls, which depict the persecution of certain orders of nuns long ago. Nuns belonging to two distinct orders let down their hair before going to their death, despite the general belief that only nuns under the rule of one particular order would die in this condition, that only those nuns had to let down their hair. There is also a painting called In Praise of Dissidence or something like that, in which monsters/policemen seize demonstrators who are also vaguely protesters. I give Claire a last kiss. We are very moved. It’s almost five o’clock in the morning, the whole family is reunited. I’m going to have to call the agency in charge of transport, disguise my voice, explain that we need explosives and that they can send them here; when they speak of precautions or whatever, I will say (in English): Thank you, I’ve just been released from a psychiatric hospital (i.e., with these explosives in hand I’ll be dangerous). The person at the other end will think that it’s a practical joke, he’ll go ahead and send everything, and so everything’s going to be blown up. Everything is set for 5:30 a.m. I wonder if I’m really going to live my life in an alternative reality, trying to go in another direction, or whether I’ll be reborn as a baby, finding myself in limbo for years until my consciousness reemerges … I’m rather anxious.… Wake up at 5:35—real time!
SIX
Lyon
November 20, 2008
And the days and the nights
passed
in the company of the Problem.
In my sixth-floor walk-up apartment, at the office, in bed asleep …
In my armchair, evening after evening, drinking one cup of tea after another after another, exploring paths and subpaths, meticulously noting every possibility, crossing off dead ends from my list as I go along.
One day in October a Korean mathematician, a young woman who had studied under Yan Guo, sent me a manuscript on Landau damping to be considered for publication in a journal of which I am an editor. It was titled “On the existenc
e of exponentially decreasing solutions of the nonlinear Landau damping problem.”
For a moment I thought that she and her coauthor had proved the result that I would so dearly like to prove myself, by constructing solutions to the Vlasov equation that spontaneously relax toward an equilibrium! I wrote at once to the editor in chief, saying that I was faced with a conflict of interest and could not in good faith handle the manuscript.
On taking a closer look, however, I realized that they had not come close to doing what I have in mind. They proved only that some damped solutions exist—whereas what needs to be proved is that all solutions are damped! If you know only that some solutions are damped, there’s no way of telling whether you’re going to come across one of them or not.… As it happens, two Italian mathematicians published an article ten years ago proving a fairly similar result, but the authors don’t seem to be aware of this earlier work.
No, the Problem hasn’t been cracked yet. Besides, it would have been a real disappointment if the solution had turned out to be so simple! An article of thirty pages or so that doesn’t resolve any major difficulty is unlikely to do the trick, however good it may be otherwise. Deep down I am convinced that the solution will require completely new tools, which will allow us to look at the problem in a new way.
Birth of a Theorem: A Mathematical Adventure Page 3
