Birth of a Theorem: A Mathematical Adventure, by Cédric Villani

In recent years, a number of eminent mathematicians have written popular books about their practice, notably Marcus du Sautoy (Finding Moonshine: A Mathematician’s Journey through Symmetry) and, more recently, Edward Frenkel (Love and Math: The Heart of Hidden Reality). Birth of a Theorem, similarly aimed at a general readership, offers an account of the breakthrough that led in 2010 to its author, Cédric Villani, being awarded a Fields Medal – arguably the greatest accolade for a mathematician. But this book is more than a description of the work – “proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation”, to give it its formal name – that won him this honour. Villani’s compellingly readable account is peppered with literary references and quotations, insights into his family life and his eclectic taste in music and, in italicised chunks of text, background on the famous mathematicians, fascinating mathematics, elite institutions and disciplinary journals that play key roles in recent developments in the discipline.

The story is told in part via emails exchanged by Villani with his collaborator Clément Mouhot – including, in testament to the obsessive hold such research can have on its devotees, on Christmas Day. These exchanges offer a captivating, blow-by-blow account of their dedication, their delight at breakthroughs and, not infrequently, the despair over setbacks that follow. For anyone particularly interested in how mathematicians think and solve problems, as I am, these first-person records of their collaboration are fascinating.

This account very lucidly describes the desolation felt by mathematicians when a solution simply refuses to be found

Our tale begins in 2008: Villani is working as a professor of mathematics at the École Normale Supérieure in Lyons while wrestling with the Boltzmann equation, which grew out of physicist Ludwig Boltzmann’s work on the theory of gases in the 1870s. Boltzmann provided proofs for much – but not all – of his work, and the lacunae have exercised mathematicians, physicists and engineers for years. But with input from others, including Mouhot, Villani has spotted a connection between the Boltzmann equation and Landau damping, a concept he was previously unfamiliar with.

In 2009, Villani heads to the Institute for Advanced Study in Princeton for five months, intending to put his questions about Landau damping to rest in a couple of weeks. The IAS has a long history of supporting great thinkers, from Albert Einstein to Kurt Gödel and André Weil. But despite the wonderful surroundings and the opportunity to interact with some of the world’s most inspired minds, Villani finds that the proof concerning Landau damping does not come as quickly as he had hoped – even though the time difference between France and the US allows him and Mouhot, between them, to work around the clock as they painstakingly go back over their work, hunting out mistakes and wrong assumptions.

Even among mathematicians, Villani is a strikingly unusual figure. Dubbed “the Lady Gaga of mathematicians”, he dresses more like a 19th-century poet, with shoulder-length hair, a three-piece suit and silk cravat, a fob watch and a giant spider brooch. In an anecdote he uses to illustrate how he communicates his work to non-specialists, Villani recalls hitching home from a concert by the punk-folk band Têtes Raides and being given a lift by a family who assume he is a musician or artist. He reveals that he is a mathematician, and explains that he has “developed a synthetic notion of Ricci curvature lower-boundlessness in complete, locally compact metric measure spaces”. For the next 30 miles, he enlightens them about what this involves. The journey concludes with the family asking him to write down a mathematical equation, aware that they have met someone rather special even if they do not fully appreciate the depth of his research.

Villani is particularly good at detailing the practice and culture of mathematics. He offers a stomach-churning account of facing up to a long-standing engagement, around the end of his first month in Princeton, to go to Rutgers, the State University of New Jersey to deliver a talk about his results. Alarmingly, when the time comes, he does not have any definite results. Although the talk goes well on the surface, on the way home Villani realises that what he has presented is not going to work. The talk has marked a turning point, he says: “To announce results you have not yet proved is a grave sin. The violation of the bond of trust that unites a speaker and his audience. My back was to the wall: if I was ever to make up for my transgressions, I had to prove the result I had just announced.”

As they redouble their efforts, Villani and Mouhot’s frustration mounts. Reading this, it’s hard not to think of the old problem about a snail climbing out of a drain, moving up x units each day but falling back y units each night. The mathematicians’ progress is painfully slow, and for each breakthrough, there are errors to correct and several new problems to solve. At one point, the dream is nearly shattered: Mouhot drafts an email to Villani saying that the technical hurdles are insurmountable and he is giving up. Mercifully, before hitting “send”, he goes through the details of their work one last time and spots another direction that they have yet to explore. I am not aware of any other account that so lucidly describes the desolation felt by mathematicians when a solution simply refuses to be found. Indeed, the very notion of such a distinguished mathematician so often making mistakes and being stuck might come as a surprise to many.

I suspect that many readers will wonder what possesses someone to take on work that is frequently so frustrating and provides so few victories. But as Birth of a Theorem shows, the exhilaration when a breakthrough occurs is beyond compare. After one major advance, Villani quotes Weil: “Every mathematician worthy of the name has experienced, if only rarely, the state of lucid exaltation in which one thought succeeds another as if miraculously…Unlike sexual pleasure, this feeling may last for hours at a time, even for days.”

Finally the work is over; Villani and Mouhot’s proof is complete and has been recognised by the mathematical community. In February 2010, a phone call changes Villani’s life and catapults him into the spotlight: the Fields is his. His description of the awards ceremony in India makes clear that it is a glorious occasion, and yet three months later there comes a time of darkness and depression when he wonders if his achievement really is quite so great after all. It seems extraordinary that he could feel such doubt after all the accolades he has won.

Villani’s account of his journey through the darkness of mathematical research also touches on important themes of academic life, including the balance to be struck between work and family life, between research and teaching, and between meeting the expectations of one’s own institution while pursuing other related opportunities. There is much in Birth of a Theorem to interest non-mathematicians, and they are urged not to be put off by the mathematics – which, incidentally, is likely to be incomprehensible to most mathematicians as well. This book’s great strength is that it remains a compelling and revealing tale nevertheless.