"The compelling and uncompromising diary of a prizewinning French enthusiast and eccentric."
"To really appreciate mathematics, you have to see it evolve, to work through the twists and turns yourself; it’s almost never enough for someone to just tell you about it. These wise words from my secondary school maths teacher have stuck with me ever since. Cédric Villani knows this, too. To put it in his words: “Appreciating a theorem in mathematics is 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.” Villani should know. He is widely regarded as one of the most talented mathematicians of his generation. His work has won him almost every prize going: the Fermat prize (a big deal) the Poincaré prize (a very big deal) and even the Fields medal (off-the-chart big deal). For those who aren’t familiar, the Fields medal is often referred to as the Nobel prize of mathematics but, as it is only awarded once every four years, and even then only to mathematicians under 40, it is much, much harder to win. Villani’s medal was awarded in 2010 for his work in mathematical physics describing the behaviour of particles in gases and plasmas.
The Birth of a Theorem is written as a diary, taking us through the early evolution of the idea in Lyon in 2008, through six months of frustration trying to wrestle the beast of a theorem to the ground at Princeton (his words, not mine) and culminating in the news that he has won the most coveted prize.
This twisting mathematical path is there for you to experience, not just to spectate. At the end of each entry, Villani includes email correspondence with colleagues about the progress of the theorem in its full technical glory. He partners the sections with a historical and mathematical overview of the methods applied, along with pencil sketches of each inventor. There are even extracts from the prizewinning theorem in proofs that go on for pages and pages with nothing but the most severe mathematical notation to rescue the faint-hearted reader.
I like these extra sections. The change in typeface that signposts their presence makes it clear that they are not compulsory. Instead – like a “pick your own adventure” book – they are there to add to the story if you happen to choose that route. But more detail isn’t always a good thing. There are times when the density bleeds into the story of Villani’s discovery and risks obscuring what makes this book so interesting. For example, on the second page, he recalls a light and playful conversation with one of his colleagues: “My old demon is back again – regularity for the inhomogeneous Boltzman.” “Conditional regularity? You mean, modulo minimal regularity bounds?” “No, unconditional.” “Completely? Not even in a perturbative framework?” And so it goes on. For pages.
Perhaps Villani believes that the Columbo analogy extends to the story as well as the mathematics: that it is important to fully experience and understand the back-and-forth of his exchanges to appreciate his achievement. But if so, the level of technical knowledge he expects from his readers is pretty astonishing. I have a PhD in a relevant subject and have worked as a professional mathematician with the Boltzmann equation for the last four years, and I still found large sections of the book impenetrable.
Thankfully, just before I wept with sorrow at my own mathematical shortcomings, I spotted a note at the end of the book (presumably from the editor): “No attempt has been made to expand upon, much less explain fine points of mathematical detail many of which will be unfamiliar even to professional mathematicians.” No kidding.
As well as being a brilliant mathematician, Villani is also an eccentric. On the cover of the book, he wears a black suit, a flamboyant red cravat, a pocket watch and a large spider-shaped brooch on his left lapel. On screen and in life he speaks with a thick French accent and oozes enthusiasm and unconventional charm. To me, the book is at its best when Villani’s character, and not the maths, takes centre stage.
He tells us how scandalously poor the cheese is in Princeton – “My entire family has been suffering since we got here!” – and how he broke in to the university in the middle of the night to replenish his dwindling tea supplies: “Without the stimulating leaves of Camellia sinensis, I couldn’t possibly face the hours of calculation that lay in store.”
But most of all, he manages to vividly convey how much mathematics – and the medal – mean to him. Ultimately, this is a story about the limits of what can be achieved. And in that respect it has everything: partnership, courage, doubt and anxiety, elation and despair. Villani’s path to success was not always easy, and he writes vividly of his setbacks and obstacles, detailing the inner monologue of self-doubt that we all experience, regardless of our ability. In one memorable moment, he reflects on how close he came to disaster. Working on the theorem on the other side of the Atlantic, his colleague and friend Clément Mouhot had decided that their goal was impossible. He had written Villani an email that began with the line: “All hope is lost.” But just as he was about to send it he hesitated. He wanted to find the right words to convince (and console) Villani that their dream was over.
Working until the small hours, Clément went back through the theorem taking notes of all the paths they had explored and why they were dead ends. But as he worked, he finally found – to his amazement – the right path opening up before him. This key moment ended up sending them down the right path. Villani awoke to find an excited email from his friend, unaware of how close they had come to failure."