Cédric Villani’s *Birth of a Theorem* tells the story of a mathematical theorem, from its initial conception as a vague, throwaway observation, to its development and formal statement, right through to its proof and eventual publication, a journey lasting several years. The theorem in question is the important result of Villani and his former PhD student Clément Mouhot on the phenomena of ‘Landau damping’, a result that ultimately won Villani the 2010 Fields Medal and widespread acclaim.

* Birth of a Theorem* is like no other popular maths book I have read. Indeed, it is not really a book about maths at all, or even about mathematicians. Instead, it aims to tackle a much more ambitious topic: the mathematical process, the actual way in which mathematical results are obtained and proved. In lesser hands this might be pretty dry, but Villani’s flair for storytelling, drawing on fables, metaphors and anecdotes, ensures that it is never boring.

What makes *Birth of a Theorem* so interesting is that Villani does not just describe the mathematical process, but takes great care to actually show it taking place. Through a series of journal entries and lengthy email correspondence between Villani and Mouhot, we see in great detail the ups and downs of doing maths, the long periods of frustration punctuated by moments of inspiration. We feel the despair, the temptation to give up and the fear of failure. But we also experience the shared delight in doing maths together, the friendship and camaraderie. To be sure, this book banishes the cliché of the lone mathematical genius, for this is maths as a genuinely collaborative effort, not so much conjured out of thin air but created though passion, hard work and luck, and after many, many false starts.

All the same, if *Birth of a Theorem* was aiming to popularise maths to a wider audience, it takes huge risks with much of its content. Pages and pages of impregnable mathematics, often lifted verbatim out of journal articles, are supplemented by the briefest of explanations. Even the expository sections, intended for a general audience, are extremely short, leaving more questions than answers. Given Villani’s well-known talent for the popularisation of maths, this is perhaps surprising. The only explanation I can offer is that the effect is deliberate, considered the best way to show the mathematical process in its unadulterated form.

For me, what was most admirable about *Birth of a Theorem* was its honesty. This is a very personal account – more diary or scrapbook than autobiography – and Villani makes no attempt to sugar-coat or gloss over the banality of doing mathematics. On full show are the drudgery of administrative work, the interminable obligations, and the petty disputes and politics of workplaces everywhere. Also on display are Villani’s vulnerabilities, his fear of ridicule, of disgrace, or even worse, of complete irrelevance. In this light, his obsession with winning the Fields Medal reveals an intense craving for respect and recognition. It is perhaps no accident that the end of the book, the moment marking the ‘birth’ of the theorem, is not when the problem is cracked open, or even when the proof is complete, but a full year later when *Acta Mathematica* finally accepts the paper as worthy of publication.

*Birth of a Theorem* is at its best when it conveys not only the joy and passion that can be found in mathematical exploration (‘*that wonderful feeling of not knowing what continent you are on*’), but also the frustration, pain and even depression that Villani suggests is the inevitable result of such passion. For a mathematician, this is fascinating stuff, a rare glimpse into the mind of someone at the very top of their game. It is also a book that, impregnable maths aside, might be shared with non-mathematical family and friends, for it gives a real insight into what it feels like, what it is, to be a mathematician. If, as Villani puts it, ‘there is nothing more precious than an unlit path’, this book does a good job of explaining the joy and fulfilment to be had in stumbling around in the dark.

*You can watch the talk Cédric Villani gave at TEDxOrangeCoast on YouTube:
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https://www.youtube.com/watch?v=h4jVZatICAo

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