I need a new norm.
A norm, in mathematical jargon, is a special sort of ruler, or measuring stick, designed for the purpose of estimating the size of some quantity one wishes to investigate. If we want to compare the pluviometry of Brest with that of Bordeaux, for example, should we compare the maximum rainfall for a single day in each place or integrate over the whole year? Comparing maximum quantities involves the L∞ norm, usually called the supremum (or sup) norm; comparing integrated quantities involves another norm with an equally lovely name, L1. There are many, many others.
To qualify as a true norm in the mathematical sense, certain conditions must be satisfied. The norm of a sum of two terms, for example, must be less than or equal to the sum of the norms of these terms taken separately. But that still leaves a vast number of norms to choose from.
I need the right norm.
The concept of a norm was formalized more than a century ago. Since then, mathematicians have not stopped inventing new ones. The second-year course I teach at ENS-Lyon is full of norms. Not only the Lebesque norm but also Sobolev, Hilbert, and Lorentz norms, Besov and Hölder norms, Marcinkiewicz and Lizorkin norms, L p, W s,p, H s, L p,q, B s,p,q, M p, and F s,p,q norms—and who knows how many more!
But this time none of the norms I’m familiar with seems to be up to the job. I’ll just have to come up with a new one myself—pull it out of a great mathematical hat somehow.
The norm of my dreams would be fairly stable under composition with elements close to identity, and capable of accommodating the filamentation typically associated with the Vlasov equation in large time. Gott im Himmel! Could such a thing really exist? I tried taking a weighted sup; perhaps I’ve got to introduce a delay.… Clément was saying we need to preserve the memory of elapsed time, in order to permit comparison with the solution of the free transport equation. That’s fine with me—but which one is supposed to be taken as the basis for comparison??
While I was rereading the book by Alinhac and Gérard this fall, one exercise in particular caught my eye. Show that a certain norm W is an algebraic norm. In other words, show that the norm W of the product of two terms is at most equal to the product of the norms W of these terms taken separately. I’ve known about this exercise for a long time, but looking at it again I suspected that it might be useful in wrestling with the Problem.
Maybe so—but even if I’m right, we’ll still have to modify the evaluation at 0 by inserting a sup, or otherwise an integral. But then that’s not going to work very well in the position variable, so we’ll have to use another algebraic norm … perhaps with Fourier? Or else with …
One fruitless attempt after another. Until yesterday. Finally, I think I’ve found the norm I need. I’ve been scribbling away for weeks now, evening after evening, page after page, sending the results to Clément as I go along. The machine is cranked up. Cédurak go!
* * *
Let D be the unit disk in , and W(D) the space of holomorphic functions f on D satisfying
Show that if and if g is holomorphic near the values taken by f on then (Remark that and that W(D) is an algebra; then write
where N is chosen sufficiently large that the series is well defined and converges in W(D).)
[Serge Alinhac and Patrick Gérard, Pseudo-Differential Operators and the Nash–Moser Theorem (chapter 3, exercise A.1.a)]
* * *
Date: Tue, 18 Nov 2008 10:13:41 +0100
From: Clement Mouhot
To: Cedric Villani
Subject: Re: Sunday IHP
I’ve just seen your last emails, will read them carefully, I’m getting a lot of flak for trying to use my trick in a stability theorem for the solution of the transport equation with small analytic perturbation! More soon! clement
Date: Tue, 18 Nov 2008 16:23:17 +0100
From: Clement Mouhot
To: Cedric Villani
Subject: Re: Sunday IHP
A quick comment after having looked at a paper by Tao (actually the summary of it that he gives in his blog) on weak turbulence and the cubic 2d defocusing Schrodinger.
His definition of weak turbulence is: shift of mass to increasingly higher frequencies asymptotically, and his definition of strong turbulence is: shift of mass to increasingly higher frequencies within a finite time. Here’s the conjecture he formulates for his equation: Conjecture.* (Weak turbulence) There exist smooth solutions u(t,x) to (1) such that |u(t)|_{H^s({Bbb T}^2)} goes to infinity as t to infty for any s > 1.
Remains to be seen whether this can also be shown for the solutions that they’re trying to construct (for free transport, the derivatives in x really blow up). As in our case they need confinement through the torus apparently in order to be able to see this phenomenon without dispersion in the real variable x getting in the way. On the other hand one thing I don’t understand is that he argues the phenomenon is nonlinear and isn’t observed in linear cases. In what we’re looking at it does seem to be found at the linear level …
More later, clement
Date: Wed, 19 Nov 2008 00:21:40 +0100
From: Cedric Villani
To: Clement Mouhot
Subject: Re: Sunday IHP
Okay, here’s what I’ve done today. I’ve added a few comments to the Estimates file, deleted the first section (which was obsolete, really) and reorganized various estimates that were dispersed in various files, so that now pretty much everything is in a single file.
I don’t think we’re clear yet about the norm we should be working in:
– since the equation on rho in the case of a homogenous field is integral only in time (!) we have to work in a fixed norm, which therefore must be _stable_ under composition by Om.
– Fourier seems unavoidable if we’re going to be able to convert the analytic into exponential decay. I don’t know how to do the exponential convergence directly without Fourier, obviously it must be possible.
– since the change of variable is in (x,v) and the Fourier transform of rho is a Dirac in eta, it looks like what we need is an analytic norm of the L^2 in k and L^1 in eta type.
– but the composition will certainly never be continuous in an L^1 space, so that can’t be right, probably we’ll have to be fairly devious and begin by “integrating” the etas. That would leave an L^2 analytic norm in the variable k.
Conclusion: We’ll have to go on being devious.
More later,
Cedric
Date: Wed, 19 Nov 2008 00:38:53 +0100
From: Cedric Villani
To: Clement Mouhot
Subject: Re: Sunday IHP
On 19/11/08, 00:21, Cedric Villani wrote:
> Conclusion: We’ll have to go on being devious.
Right now I have the impression that in order to find a way around this difficulty we’ll need the theorem on continuity of composition by Omega for the L^2 analytic norm in Fourier (without loss of generality…), treating eta as a parameter. Talk to you tomorrow:-)
Date: Wed, 19 Nov 2008 10:07:14 +0100
From: Cedric Villani
To: Clement Mouhot
Subject: Re: Sunday IHP
After a good night’s sleep I see now that it’s UNREALISTIC: composing by Omega will NECESSARILY force us to lose a bit on lambda (this is already the case when Omega = (1-epsilon) Id). So we’re just going to have to deal with it somehow.
More later.…
Cedric
Date: Wed, 19 Nov 2008 13:18:40 +0100
From: Cedric Villani
To: Clement Mouhot
Subject: update
Updated file attached:
I’ve added subsection 3.2 in which I examine what appears to be a fundamental objection having to do with something we talked about on the phone, the problem of the loss of functional space due to the change of variable. The conclusion is that it isn’t lost, but we’ll have to be very precise in our estimates of the change in variable.
Cedric
Date: Wed, 19 Nov 2008 14:28:46 +0100
From: Cedric Villani
To: Clement Mouhot
Subject: update
New addition at the end of section 3.2. Things now seem rather promising.
Date: Wed, 19 Nov 2008 18:06:37 +0100
From: Cedric Villani
To: Clement Mouhot
Subject: Re: update
I’m pretty sure that section 5 in its present form is wrong!! The problem arises after the phrase “In assigning powers and factorials”: the line that follows seems OK, but in the formula a bit further on the indices don’t match up
(N_{k-i+1}/{k-i+1}! ought to yield N_k/k!
and not N_k/(k+1)!)
The result seems much too strong. It would mean that in composing by an approximation of the identity the same index for the analytic norm is preserved. I think we’ve got to aim instead at something like
|fcirc G|_lambda leq const.
|f|_{lambda |G|} |G|
or something along these lines.
More later,
Cedric
Date: Wed, 19 Nov 2008 22:26:10 +0100
From: Cedric Villani
To: Clement Mouhot
Subject: good news
In the attached version I’ve gotten rid of the problematic section 5 (we can always put it back in if we need to) and in its place I’ve put calculations on composition, still using the same analytic variants, which this time seem to work beautifully as far as composition is concerned (the formula I had suggested won’t do, the right one turns out to be even simpler, though still of the same kind).
More later, Cedric
Date: Wed, 19 Nov 2008 23:28:56 +0100
From: Cedric Villani
To: Clement Mouhot
Subject: good news
New version attached. I’ve checked to see that the usual calculation can be done with the norm suggested by the composition rule (section 5.1). It’s only slightly more complicated but it seems to give more or less the same result. That’s all for today.
Cedric
SEVEN
Bourgoin-Jallieu
December 4, 2008
Headlights suddenly loom out of the darkness at the exit from the parking lot. Momentarily blinded, I approach the driver. It’s my third try.
“Excuse me, are you going to Lyon?”
“Uh … yes.”
“Would you be kind enough to give me a lift? The trains have stopped running!”
The driver hesitates for a split second, glances at her passengers, invites me to take a seat in back. I get in.
“Thanks so much!”
“So you were at the concert?”
“Of course! Wasn’t it terrific?”
“Really great, yeah.”
“I wouldn’t have missed the Têtes Raides’ twentieth anniversary tour for anything—but I hate to drive, so I came down by train, thinking I wouldn’t have any problem hitching a ride back.”
“It’s no trouble. I brought along my son and his friend, who’s sitting next to you in back.”
Hello everybody.
“The pogoing wasn’t out of control, there was a lot of room, you weren’t getting stepped on all the time. It was pretty relaxed.”
“The girls had no reason to complain.”
“Oh, some of them love it when it really gets wild!”
Fond memories of one ravishing punk chick in particular, pierced, incredibly full of energy, whom chance threw into my arms one night while dancing at a Pigalle concert in Lyon.
“Nice spider.”
“Thank you. I always wear one, it’s part of who I am. I have them custom-made in Lyon. Atelier Libellule.”
“Are you a musician?”
“No!”
“An artist?”
“A mathematician!”
“A mathematician?”
“Yes—mathematicians do exist!”
“What do you work on?”
“Hmmm. Do you really want to know?”
“Sure, why not?”
“Yeah, go ahead!”
A deep breath.
“I’ve developed a synthetic notion of Ricci curvature lower-boundedness in complete, locally compact metric-measure spaces.”
“What!?!”
“You must be kidding.”
“Not at all. I wrote an article about it that made a pretty big splash in the community.”
“What’s it about again? Sounds cool.”
“Okay, one more time: a synthetic theory of lower-boundedness of Ricci curvature in metric-measure spaces that are separable, complete, and locally compact.”
“Wow!”
“What’s it good for, anything?”
The ice is broken, we’re off. I start talking, explaining, demystifying. Einstein’s theory of relativity. Curvature—the cornerstone of non-Euclidean geometry—and the deflection of light rays: if the curvature is positive, the rays get closer the farther they travel; if the curvature is negative, the farther they travel the more they diverge. Curvature is usually expressed using the language of optics, but it can also be expressed using the language of statistical physics: density, entropy, disorder, kinetic energy, minimal energy—that’s the discovery I made, with the help of a few other mathematicians. But how can one speak of curvature in a space that’s covered with spikes, like the bristles on the head of a hedgehog? Then there’s the problem of optimal transport, the subject of my thousand-page book. You encounter it everywhere, economics, engineering, meteorology, computer science, geometry.…
I go on and on. The miles fly by.
“We’re coming into town now. Where can I drop you off?”
“I live in the first arrondissement—the intellectuals’ neighborhood! But anywhere that suits you is fine, really. I’ll manage.”
“It’s no problem at all. Just tell me how to get there and we’ll take you home.”
“That would be wonderful. Can I give you some money for the toll?”
“No, absolutely not.”
“That’s very kind of you.”
“Before you go, would you write down a mathematical formula for me?”
* * *
Fig 14.4 – The meaning of distortion coefficients. Because of positive curvature effects, the observer overestimates the surface area of the light source; in a negatively curved world, the observer underestimates it.
Fig 16.2 – The lazy gas experiment. To go from state 0 to state 1, the lazy gas uses a path of least action. In a nonnegatively curved world, the trajectories of the particles first diverge, then converge, so that at intermediate times the gas can afford to have a lower density (higher entropy).
[From Cédric Villani, Optimal Transport: Old and New (Berlin: Springer, 2008), pp. 408, 460, with slight modifications]
EIGHT
A village in the Drôme
December 25, 2008
Back home with family for the holidays. I’ve made a lot of progress.
Four computer files, simultaneously updated as we go along, contain everything we have learned about Landau damping. Four files that we have exchanged, added to, corrected, reworked, and sprinkled with notes—marked “NdCM” in boldface for Clément’s comments, “NdCV” for mine. Composed in the TEX language developed by the universal master, Donald Knuth, these files are ideally suited for the preliminary maneuvers we’re engaged in at the moment.
How I could say such a thing was completely beyond him. He had a point, I must admit. I had spoken too soon, without thinking it through. At the time this statement seemed to me to be self-evident. Later, after I’d thought about it some more, not only couldn’t I say what justification I had for writing the inequality, I no longer understood why it had seemed obvious to me in the first place.
I still don’t know why I believed it went without saying. But I’ve come to realize that it is in fact true! It’s true because of Faà di Bruno’s formula.
My differential geometry professor at the École Normale Supérieure in Paris long ago introduced me to this formula. It’s used for obtaining the successive derivatives of composite functions, and it’s unbelievably complicated. I can still picture the scene: lots of murmuring, nervous laughter in the lecture hall; the professor looked at us rather sheepishly, as though he felt he had to apologize, and said, “Don’t laugh, it’s very useful!”
He was right, Faà di Bruno’s formula really is very useful: thanks to this formula, my mysterious inequality is true!
But I had to be patient. As Boltzmann, Knuth, and Landau are my witnesses, I swear that for sixteen long years it was of absolutely no use to me whatever. Even the name of the formula’s author I had forgotten, unusual though it is.
Somewhere in the back of my mind, however, lurked the thought: there is a formula for derivatives of composite functions. Thanks to Google and Wikipedia, it took only a few moments to find the author’s name and the formula itself.
The appearance of Faà di Bruno’s formula is a sign of the unexpected combinatorial turn that our work has suddenly taken. Ordinarily my drafts are covered with what looks like the sound hole on a cello—the integral sign ∫ (I’ve written this symbol so many times that I see it in my mind the moment I start thinking about a problem). But this time they’re infested with exponents between parentheses (multiple derivatives: f (4) = f ′′′′) and exclamation points (factorials: 16! = 1 × 2 × 3 × … × 16).
Birth of a Theorem: A Mathematical Adventure Page 4