# Birth of a Theorem: A Mathematical Adventure

In his book *The Grapes of Math* Alex Bellos describes Cédric Villani as follows: "...no ordinary looking university professor. Handsome and slender, with a boyish face and a wavy, neck-length bob, he looks more like a dandy from the belle epoque, or a member of a pretentious student rock band. He always wears a three-piece suit, lavaliere cravat—the kind folded extravagantly in a giant bow—and a sparkling, tarantula-sized spider brooch". And Villany is a remarkable appearance indeed. The media love him, and this is probably mutual since Villani seems to be happy to collaborate. His flamboyant picture is on the front cover of the hardback version of this book. In this paperback edition, it is on the second cover page and you only see his left eye through a peephole in the cover. But pictures of him are easily found on the Web. He also featured in a documentary by Olivier Peyon *Comment j'ai détesté les Maths* (How I despised math) that came out in France in 2013. The film wants to show (young) people that mathematics is really important in our society and several mathematicians are staged, telling the spectator what it is like to be a mathematician and why mathematics is improtant. Villany is one of them and you see him at work, for example in Oberwolfach, and we see him at the IMU congress in 2010 where he received the Fields Medal.

The present book is a translation of the original French version *Théorème vivant* (Grasset, 2012). It is a literary kind of diary with first entry on 23 March 2008 in Lyon when he starts up a project on proving the regularity of the inhomogeneous Boltzmann equation with his former student Clément Mouhot. The last entry is on 17 November 2010 in Saint-Rémy-lès-Chevreuse near Paris just after the turmoil caused by him winning a Fields Medal and the paper they started at the beginning of the book being accepted for *Acta Mathematica*. It is published about a year later as *On Landau damping*, Acta Mathematica, Volume 207, Issue 1, pp 29-201. September 2011. Throughout the book we learn how they worked on the problem, tried dead ends, reworked the paper, started all over, and even when they finished and submitted a first version, it was rejected. So they had to work some more, could solve the problem, and even generalized the result, until it was finally accepted. Although the mathematics are not explained, and they do not really matter to evoke the gestation period that finally leads to the birth of a theorem. This is really the genesis of a (nontrivial!) theorem, so in my opinion the English title is even better than the original one *Théorème vivant* (Living theorem).

The literary form that Villany has chosen is a set of short chapters that are like diary fragments: where he is at that moment and why, what he is doing, how he is feeling, his hopes, his joys and fears. In short, the private life and work of a mathematician, or rather what it is like to be Cédric Villany, because Villany can hardly be called an average mathematician. Most mathematicians do not win the EMS Prize, the Fermat Prize, the Henri Poincaré Prize, the Fields Medal, and many others as he did. Most mathematicians do not have the choice between being appointed director of the *Institut Henri Poincaré* (IHP) or getting a position at the *Institute of Advanced Studies* (IAS) in Princeton. Being Villani he can negotiate very favorable conditions for accepting the directorship and sees them all accepted. So he eventually chose for the IHS (because in the US, as he mentions, tongue in cheek, they do not have the crunchy French baguette bread and they lack all the good cheese that is abundantly available in France). Most mathematicians do not contemplate whether or not they are still in the running for the Fields Medal. And most mathematicians (I should say most people) do not have the dress code and the looks of Cédric Villani, but that is a different story.

Throughout the diary we follow him while he is still in Lyon, when he attends several conferences, then to Princeton where he stayed with his family (January-June 2009), and back to the IHP. That we learn from the diary fragments, but in every chapter there is always a second part following this personal section. That can be about anything. For example e-mail exchanges between Mouhot and Villani. These are about mathematics, but you do not get any wiser form them. They just give an idea how mathematicians communicate and how these two were probing different possibilities with hopes and disillusions. Or there can even be several pages from a draft of the paper they are writing (but again being fragments without defining the notation, they do not help to understand the problem). Or sometimes you can find a bio of one of the mathematicians that were mentioned (Boltzmann, Fourier, Poincaré, Mittag-Leffler, ...). There is even some explanation about TeX and Donald Knuth. All the big-shots that he meets in IAS deserve a discussion (P. Sarnak, J. Lebowitz, A. Chang, E. Lieb, J. Nash,...). For most of these persons, a pencil drawn portrait is added. There is also some history of mathematical institutions (IAS, IHP, Oberwolfach, CIRM near Marseilles, the Princeton Plasma Physics Lab (PPPL), etc.) These sections also include some mathematical problems like the Collatz 3n+1 conjecture, and other stories like the one of Perelman who solved the Poincaré conjecture and then, disappointed completely retired in oblivion, refusing all Prizes bestowed upon him.

But Villani is also very open with very personal accounts: his worries when he has to give a lecture and he had not finished the proof the night before, or the critique when he has given a lecture that he thought went very smooth. He also tells that he needs a pot of tea to start a long night of calculations, and that he loves to listen to music while he works. His music taste is very diverse, including Tom Waits and Francis Poulenc, and everything in between, but he has a special interest in Catherine Ribeiro, a French experimental folk singer. When he is installing his office at the IHP, he has a big photo of her on the wall. There is a whole chapter just listing all the music he seems to love. Sometimes a poem or song lyrics are added (in English translation). We are informed about some of his curious habits like lying down on the floor with a rug under his head to relax, or listening to a lecture standing up in stocking feet to stay alert. Some of it could be image building, but that's how he is.

Near the end he commemorates some friends that he lost already (e.g. M. Schatzman in Lyon who died moments after she heard he won the Fields Medal). The book ends with an epilogue, a diary entry of 24 February 2011, about a visit to the home of Gábor Domokos in Budapest. Domokos, together with Péter Vákonyi designed the Gömböc, an homogeneous object with only two stationary points of which one is unstable and the other superstable so that it will always wobble to its equilibrium state no matter how you put it down. The object was predicted by V. Arnold and Villani wants to have one of these on display in the library of the IHP.

It should also be mentioned that the translation has been done very carefully. It is not trivial since there are poems to be translated and there are many references to typical French culture and French situations. For those not so familiar with these aspects, the notes added at the end can be helpful. It is after all an interesting story of an interesting personality. It gives an interesting peek behind the scenes of what the life of a mathematician (or more precisely Cédric Villani) is like. For the non-mathematician, it should be repeated that it is not an average case. The majority do not have their papers of over 150 pages published in *Acta Mathematica* on a regular basis. However mathematicians, whatever topic they are working on, will love the book. And for the ambitious ones, if they are interested, they can watch and learn how to become a mathematical rock-star.

**Submitted by Adhemar Bultheel |

**7 / May / 2016