Théorème vivant

When I ordered this book, Théorème Vivant (Alive Theorem), by Cédric Villani, I had misgivings about it being yet another illustration of the, pardon my French!, universal “pipolisation” process that turns values upside down and sets mundane aspects of major contemporary figures above their true achievements like, say, winning a Fields medal! However, as soon as I started reading Théorème Vivant, I realised it was a fascinating delve into the way mathematicians operate and how they build theorems. Of course, as an “insider”, I can find many entry points to relate to, some quite mundane and unrelated like entering the common room of a conference centre in the middle of the night to “steal” some life-saving tea bags or an aversion to taxi rides, not mentioning an addiction to French cheeses… And I have the advantage of being able to read the math formulas given in the book (even though this is not at all my area of expertise and I find the wording of the theorems and proofs rather unusual at times). But I think Théorème Vivant can be read by non-mathematicians as well, provided they take those formulas and paper extracts as pictures, just like the drawings of mathematicians interspeded throughout the book and do not get annoyed at not understanding the meaning of them (I do not get the deepest levels either!). Nothing to be afraid of: Théorème Vivant is another impressive illustration of the ability of Cédric Villani to explain mathematics to the general public and to surf upon his popularity with the medias. (The book is currently available in French only, but should soon be translated into English. Possibly polishing the least politically correct statements…)

Théorème Vivant can be read as a mathematical thriller that tells the story of a search for the general version of the existence of the Landau damping, the main villain (an easy acronym of Villani and an unintended pun!) being the proof of the theorem which escaped several times its pursuers, either on its own or with the help of mathematicians and physicists at seminars or coffee breaks! The result was established by Clément Mouhot (then at Paris-Dauphine as a CNRS researcher) and Cédric Villani in a collaboration that took many months of intense work, an invited position at Princeton Institute of Advanced Study, at least 60 versions, and hundreds of emails to reach its conclusion. The Fields medal and the resulting fame (beyond mathematical circles) comes as an epilogue to the book in that, obviously, the medal was not awarded in connection with the theorem at the centre of the book, but happened in the same time period.

“Dans les années 1950, une révolution scientifique s’est produite quand on a compris que, pour explorer un système trop riche en possibles, il est souvent préférable de s’y déplacer au hasard, plutôt que le quadriller méthodiquement ou d’y choisir des échantillons successifs de manière parfaitement aléatoire. C’était l’algorithme de Métropolis-Hastings, c’est aujourd’hui tout le domaine des MCMC, les Monte Carlo Markov Chains, dont l’efficacité déraisonnable en physique, en chimie, en biologie, n’a toujours pas été expliquée. Ce n’est pas une exploration déterministe, ce n’est pas non plus une exploration complètement aléatoire, c’est une exploration par marche au hasard.” C. Villani, page 201

The book is written as an intimate journal (or a blog) and sometimes feels too exhibitionist or even indecent (!) in its description of the working of a Beautiful Mind, to borrow unsubtly from the title of this movie on Villani’s hero John Nash, hero that he had the opportunity to meet in Princeton. But it also operates at another level by describing how two persons can collaborate on breaking a hard problem from both sides of the Pond, how chance encounters and coffee break discussions can produce advances in one’s research, what makes a mathematical result an important result, how those results are discussed and dissected and rewritten before being submitted to a journal, along the side occupations of a top mathematician (like taking over the Institut Henri Poincaré directorship). Théorème Vivant also contains inserts about mathematical characters that impacted the field and are mentioned in the current chapter, contributing further to the vulgarisation effort. Overall, the style is very fresh and engaging, if not particularly literary!, uses different typographies to separate historical inserts from the main stream, reproduces some (apocryphal?) emails between the authors, as well as excerpts from the paper published in Acta Mathematica. Théorème Vivant reads very well and quickly, even when paying attention to the mathematical details (I ingested it within two days, when travelling between Paris and Des Moines). A note to statisticians: the book mentions MCMC algorithms (while inverting Markov chains and Monte Carlo) as illustrated in the above quote!