Birth of a Theorem: A Mathematical Adventure (36 page)

This identity plays a key role in the representation formula

where F
(
v
,
v
_*
)
=
f
(
v
)
f
(
v
_*
)
and G
(
v
,
v
_*
)
is the average of all the products f
(
v
′)
f
(
v
′
_*
)
when
(
v
′
,
v
′
_*
)
describes all pairs of postcollision velocities compatible with the precollision velocities
(
v
,
v
_*
).
This formula lies at the heart of the solution of Cercignani’s conjecture.

Carlo Cercignani

Theorem
(Villani, 2003). Let S
(
f
)
=
−
∫
f
log
f denote the Boltzmann entropy associated with a velocity distribution f
=
f
(
v
)
. Let B be a Boltzmann collision kernel satisfying
for some constant K
_B
>
0,
and denote by
the associated entropy production functional,

Let f
=
f
(
v
)
be a probability distribution on
with zero mean and unit temperature. Then

where

FORTY

Paris

February 16, 2010

Late afternoon in my spacious office at the Institut Henri Poincaré. I had the handsome blackboard enlarged and got rid of a few pieces of furniture to make more room. I’ve given a lot of thought to how I want to redecorate.

First, the bulky air conditioning unit has got to go—it’s normal to be hot in the summer!

Against the wall, a large display cabinet will hold some personal items and a few of the jewels from the Institute’s collection of geometric models.

To the left of my desk I plan to install the bust of a rather stern-looking Henri Poincaré that his grandson, François Poincaré, generously donated to the IHP.

And behind me a large space is reserved for a portrait of Catherine Ribeiro! I’ve already selected the image, found it on the Internet: Catherine with her arms spread wide in a vast gesture signifying struggle, peace, strength, and hope. Arms spread wide like the rebel in Goya’s
El tres de Mayo
,
kneeling before Napoleon’s firing squad, or like Miyazaki’s Nausicaä before the soldiers of the royal house of Pejite. An image of strength, but also of abandonment and vulnerability. This idea appeals to me: you can’t expect to go forward if you’re not prepared to expose yourself to chance, risk, even danger. The iconic image of the
pasionaria
,
the hopelessly passionate, vulnerable artist, which occurs also in Baudoin’s magnificent
Salade Niçoise
—I need it to watch over me, I need to confront it myself every day, face to face, in the person of Catherine.

Today, a day like every other, filled with appointments, discussions, meetings. This morning, a long telephone conversation with the chairman of my board of directors, the CEO of an insurance company who is deeply committed to enlisting the private sector in the service of scientific research. And this afternoon, a photo session to illustrate an interview I’ve given to a popular science magazine. Nothing very disagreeable about any of this. It’s been more than six months since my life took a fascinating turn: new people to meet, new things to learn, new ideas to talk about.

The photographer was unpacking his equipment in my office, setting up the tripod and reflector, when the telephone rang. Absent-mindedly, I picked up the receiver.

“Allô, oui.”

“Hello, is this Cédric Villani?”

“Yes.”

“This is Lázló Lovász calling from Budapest.”

For a moment my heart stopped. Lovász is president of the International Mathematical Union—and, by virtue of his office, chair of the Fields Medal Committee. This, by the way, is all I know about the committee; apart from him, I haven’t the faintest idea who’s on it.