Cédric Villani looks like a time traveler. With his long hair parted in the middle, three-piece suit, watch chain, beautiful silk ascot tie, and a spider brooch on his left shoulder, he wouldn’t look out of place in a 19th century European city. He’s also a brilliant mathematician, the director of the Institut Henri Poincaré in Paris, a winner of the Fields Medal and several other important prizes. And yes, a talented writer. His Théorème Vivant is an impressive book, a memoir of the period during which he and his student Clément Mouhot proved an important theorem, one of the results that earned Villani the Fields Medal.
Most books on mathematics for the “general reader” have as their goal to explain some bit of mathematics. Villani, instead, aims to communicate the experience of being a (top-level) mathematician. The mathematics is there, and there are explanations of the parts that are easily explained. But the book has more to say about how mathematicians interact, about late-night doubts and moments of exhilaration, about how cryptic emails with TeX code in them can make you laugh or cry.
The problem Villani and Mouhot are working on is “l’amortissement Landau” (Landau damping). Villani describes, early in the book, a conversation in Oberwolfach with two experts in statistical physics. They tell him that Landau damping is a “weird, strange” phenomenon, perhaps just “a chimera of physicists’ fertile imagination, with no hope of a mathematical formulation.” But Villani is recalling the conversation because Mouhot has suggested that they try to understand the phenomenon.
The book is full of accounts of mathematical places (Oberwolfach, the Institute for Advanced Study, IHP, many universities) and mathematical people. Villani is a talented describer with a knack for capturing details that make places and conversations come to life. He often gives quick (and generous) accounts of various mathematicians’ work. Some of them have something to contribute to the project. Others he admires despite not really understanding any of their work, and they talk about other things. There are also conversations with ordinary people, including a few where he explains something about himself and his mathematics.
Villani’s family appears in the book, but he is very careful to protect their privacy. The story is not about them, but about the drama in Villani’s mind as he tries to solve what turns out to be a difficult problem. Bits of mathematics are displayed, but clearly they are there to be noted, perhaps admired, but the reader is not expected to understand. Villani includes many emails, extracts from papers, all of which serve the purpose of making a bit more concrete the story being told.
It’s a victory at the end, of course, despite occasional obstacles. The paper is initially rejected, and the rejection proves to be correct: under pressure, Villani and Mouhot improve the result significantly, and the new version is accepted. And, of course, there’s eventually the phone call from the president of the International Mathematical Union.
I understand that Théorème Vivant was a bestseller in France. This must be one of those rare cases in which something really good turns out to be really popular. Those who can read French should go for the original just to enjoy Villani’s writing and for the pleasure of getting to know him. (Those who want more should check out his web page, which includes occasional bits of writing in the same spirit as the book. One of the posts, in English, is almost like an epilogue for the book.) I understand there’s an English translation in the works; don’t miss it.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.