Birth of a Theorem: A Mathematical Adventure (6 page)

Therefore

Let us assume

(I’m not sure it’s a good idea to put
4
π
here.…) In certain cases (such as Maxwellian f
_0
)
, p
₀
is positive; but generally there is no reason for this to be true. Note that p
₀
rapidly decays if
;
it exponentially decays if f
₀
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 goes 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