Birth of a Theorem: A Mathematical Adventure by Cédric Villani – review

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.

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