Birth of a Theorem: A Mathematical Adventure is the recently released English translation of Cédric Villani’s Théorème Vivant. Director of the Institut Henri Poincaré in Paris, Villani cuts a unique figure, frequently wearing large spider broaches on his jacket lapel. He is an expert in partial differential equations and their application to problems in statistical physics. His book recounts his frustrating but exciting journey to winning the 2010 Fields Medal for work on Landau damping.
Villani describes Landau damping as a phenomenon in physics that explains a “return to equilibrium without any increase in entropy.” In the abstract of the final proof, Villani and his collaborator, Clément Mouhot, assert that they have shown the damping phenomenon to be caused by “phase mixing” instead of the exchange of energy between phases. Their work illustrates that particles within a plasma are accelerated or decelerated to match the velocity of the electromagnetic waves acting on them.
While this phenomenon seems complicated, the primary goal of the text is not to explain the math behind Villani’s proof but to describe the years leading to his breakthrough. When beginning to pursue the Fields Medal — or ‘FM’ as he says — Villani did not know which problem to focus on. He only knew that he had to prove something amazing and that he had do it in a short period of time — at 38 years old, he was two years away from the age cutoff for the quadrennial award. One could imagine that this timeline would put a great deal of stress on a person, and Villani only made it worse by giving lectures about his proof before he knew it existed. In narrative form, Villani’s book illustrates the author’s fears and passions in a way that makes him seem more human and less of a surreal genius.
The book dives into Villani’s life as a father and scientist, as well as into the history of physical mathematics. Some of the most fascinating pieces are the short italicized anecdotes that Villani provides about the world’s most famous mathematicians. The most interesting excerpt is about Henri Poincaré. The aside divulges that Poincaré published a paper on the stability of the solar system in a prestigious magazine, won an award for his work, and then realized that he was completely wrong! Like any rational man with a hint of hubris, he spent all of the money he earned from the prize and more on ensuring all the printed copies returned to the publisher (there was no internet in 1889). Furthermore, Poincaré went on to publish a paper proving the exact opposite of what he initially claimed, and became the founder of chaos theory and the sponsor of one of the world’s premier mathematics institutes.
The story also includes a sizable number of emails exchanged between Villani and Mouhot in which they converse about their damping theorem. This correspondence can be tedious to read; however, the emails are interesting at times and shouldn’t be a deterrent from exploring the heart of the work. Ultimately, Villani’s purpose is to take readers inside of his psyche and illustrate the fact that even the most respected professionals doubt themselves, have a family, get nervous, are afraid to speak to icons, and start with a dream.