I wasn’t sure what to expect from Cédric Villani’s memoir Birth of a Theorem (first published in French as Théorème Vivant). I had read glowing reviews from Noel Ann Bradshaw and Hannah Fry and decidedly less glowing reviews by Alexander Masters and Jacob Aron. A math book for a general audience isn’t supposed to have pages of emails between researchers or portions of technical papers. I had misgivings, but I’m so glad I decided to read it anyway.
If you want to learn about Villani’s research in nonlinear Landau damping and the Boltzmann equation, Birth of a Theorem is not the way to do it. Mathematically, we arrive in media res. The first we hear of math is a conversation he has with his collaborator Clément Mouhot:
“My old demon’s back again—regularity for the inhomogeneous Boltzmann.”
“Conditional regularity? You mean, modulo minimal regularity bounds?”
“Completely? Not even in a perturbative framework? You really think it’s possible?”
“Yes, I do. I’ve been working on it again for a while now, and I’ve made pretty good progress. I have some ideas. But now I’m stuck. I broke the problem down using several reduced models, but even the simplest one eludes me. I thought I’d gotten a handle on it with a maximum principle argument, but the whole thing collapsed. I need to talk.”
If I were writing about Villani’s journey to the Fields Medal, I would pause and define what mathematicians mean by regularity and the maximum principle. I could probably do a good job of explaining some of these ideas and why they are important not just for Villani’s work but broadly in math. That’s kind of the point, though; I could do a good job writing about Villani’s work, and so could other math expositors. What we can’t do is show you the journey, the sleepless nights and agonizing moments of doubt. We would write about math, he writes about feelings.
Interspersed with the story of his and Mouhot’s work are some of his favorite songs and poems, sketches of the august mathematicians he interacts with, and vignettes about Villani’s day-to-day life as a working mathematician, granted one with a cushier job than most.
It seems as if every profile of women in STEM tells us about their children and domestic life, sometimes before mentioning their scientific work. Some argue that it’s a good thing; we are all people before we are scientists, after all. I agree to some extent; I enjoy learning about world-class researchers as people. The problem with this argument is that only women get this treatment. Men struggle with work-life balance as well, and I’m glad Villani lets us see how he manages to take care of a family while devoting himself to his work.
One email to Mouhot reads, “This week is going to be a little hard for me because I’ll be taking care of the kids myself…” Later, he reports staying up until after 2 am working on one of his biggest breakthroughs only to be awakened at 4 by a middle-of-the-night kid emergency. “That’s life,” he writes.
I think many people will be able to enjoy getting inside Villani's head, but I would recommend this book particularly to young mathematicians and students. It was comforting to see many scenes and feelings that were familiar to me: getting obsessed with a problem, watching a solution disintegrate in front of you, taking the kids to brunch at Alice Chang's house (OK, not that one). I know next to nothing about the Boltzmann equation, but the Boltzman equation itself is not the point.
It is not a book about mathematics; it is an art project. It is the brief story of an astonishing adventure, told with many pictures. It could also be read as a 15-minute play script about a man’s hunt for a medal, quirkily interleaved between 200 pages of instructions to the set-designer. The one thing it has almost nothing to do with is mathematics. Looked at in this light, Birth of a Theorem is a remarkable book and I advise everyone to buy it.
I'm not sure whether Masters is being sarcastic here, but he’s right: this is art—literature, or perhap theater—not mathematics. Birth of a Theorem succeeds in giving us a glimpse, even a small one, of what it feels like to be Cedric Villani.