Go Forth and Multiply

by Andrew Crumey

Review of Birth Of A Theorem by Cédric Villani. Literary Review, April 2015.

There is no Nobel Prize for mathematics, but there is the Fields Medal. Cédric Villani won it in 2010, and his curious memoir tells how. Rather than being a straightforward account of the mathematician's life, or a step-by-step explanation of the problem he worked on, the book is a collage of emails, equations, quotations and observations, painting a dizzying picture of frankly unfathomable genius. Put it this way: I'm a PhD-level mathematician, and after reading this book I still can't figure out exactly what Villani did. But it was a fun ride.

Although comparable to the Nobel Prize in prestige, the Fields Medals are crucially different, being awarded only every four years to a maximum of four mathematicians under 40. Any contender has to be something of a wunderkind, and Villani (born in France in 1973) fully fits the bill, both in terms of technical brilliance and flamboyant eccentricity. As he explains in the book - and as we see from his cover photograph - Villani favours bright cravats, a fob watch and a spider brooch. His prose style - which one guesses to be an accurate reflection of his normal speaking manner - is frenetic, leaping between arcane jargon and childlike whimsy. This may not fit the traditional stereotype of the dry and distant mathematician, but it does match perfectly with a more postmodern stereotype. Villani could walk straight into the role of Doctor Who whenever Peter Capaldi calls it a day.

Given the four-year cycle of the Fields Medal, and its age restriction, Villani knows he has only one shot at it. He gives no impression of being a careerist, despite all the professional networking and jetting around to research institutes that he describes, but is admirably frank about his sense of personal ambition, and has a realistic sense of his place in the pecking order of mathematical greatness. In 2006 the medal was offered to Grigori Perelman for his proof of the Poincare Conjecture, but the reclusive mathematician turned it down. When Villani gets the call next time round, he detects hesitancy about whether he will accept. "But I'm hardly on Perelman's level," Villani writes, "and I have no qualms about saying yes."

So what did he actually do? The medal was awarded "For his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation." Right at the start of the book we are told that this has to with plasma physics, and we are shown the relevant equations. The implicit and perfectly reasonable assumption is that nearly everyone who reads this book will have no idea what the equations mean; they might as well be hieroglyphics. Communicating highly technical ideas in an accessible way is an art in itself, and requires a willingness to simplify concepts almost to the point of caricature, omitting all the details that actually constitute most of the work. Villani has little interest in that art; he cares about the details, which is why he got the prize. As he says, "Appreciating a theorem in mathematics is rather like watching an episode of Columbo: the line of reasoning by which the detective solves the mystery is more important than the identity of the murderer." What makes maths hard is the necessity to keep hold of each stage of logic: miss a step and you're lost. And if, from the very first step, you can't understand the basic concepts, then it's like watching an episode of Columbo in a foreign language. You can see people moving around with expressions on their faces, but you don't really know why they're doing it.

This state of captivated bewilderment is what Villani achieves in his book. I can't reproduce the lecture notes and draft articles that fill a significant number of pages, since it would require the use of TeX (the typesetting language used by mathematicians worldwide for putting their symbolic formulae into print). Here, though, is a typical extract from the emails that Villani exchanges throughout the book with his collaborator:

"* the first one has to do with the need to stratify the estimates on < \nabla h^k \circ \Om^n > (subsection 9.4). This is tricky, as I explain in the file, we can't rely on recurrence, and we can't rely on regularity since \Om^n is highly irregular. The only solution I see is to use the additional Sobolev regularity of the characteristics, which is propagated uniformly over n."

Last year I published a research paper in an astronomy journal, and a novelist friend asked to see it. I knew she wouldn't understand it, but she told me she liked the evocative vocabulary. This is the sort of pleasure to be had from Villani's frequent emails, which most readers will be content to skim quickly. The remainder is generally more accessible: meetings with fellow mathematicians, potted biographies of figures past and present, and some explanations of mathematical problems of a more readily comprehensible kind. Landau damping and the Boltzmann equation feel almost like MacGuffins by comparison.

Villani's personal reflections include a list of favourite music tracks (a couple of pages' worth) and, more interestingly, occasional vignettes of family life. Watching his young offspring in a school music performance, his mind drifts back to equations. At home, pacing circles in a darkened room, his wife declares, "This is getting weird!" His obsession is total, and it's left to the reader to ponder what that might mean in human terms.

Few books make so little concession to their reader as this one does. Yet at the same time, it is presented in a manifestly market-savvy way, with Villani's Byronic good looks lavishly adorning the cover, along with a sticker promoting its serialisation on Radio 4. It is a commodity that expects to be purchased rather than understood. I'm glad my review copy was free.