Birth of a Theorem by Cédric Villani

Review by Annabel

birth of a theoremIt is a well-known fact that Stephen Hawking was persuaded to remove all the equations bar the single famous one, E = mc2, from the draft of his bestseller, A Brief History of Time. In that book’s acknowledgements, Hawking himself notes that he was told that for every equation, his readership would be halved. Of course, A Brief History of Time was primarily about cosmology, and diagrams and models could more easily be used instead of equations to explain the science.

However, if you want to talk about the world of pure mathematics and the furthest reaches of theoretical physics, it’s all about the equations – they can’t really be avoided. Cédric Villani is no ordinary mathematician though, and his memoir of his work leading to him being awarded a Fields Medal in 2010 for ‘Outstanding Discoveries in Mathematics’ succeeds in making equations sexy!  More on that later.

Villani was born in 1973, so he’s only 41 now. He’s a flamboyant Frenchman, with long hair, a penchant for cravats and fob watches. He’s now director of a top Parisian institute and a professor at Lyon university but, he is also addicted to manga comics, loves all kinds of music, and he’s a family man through and through. He has been described as a rock star mathematician – which doesn’t appear to be far from the truth.

The story of how he and his doctoral student, Clément Mouhot, spent several years developing their paper on non-linear Landau damping (believe me, you don’t need to understand what it actually is) is a real roller-coaster. It begins with Villani finally working out that he wants to work on the Landau damping proof, and he wins us over with this comparison:

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.

Cédric and Clément are rarely in the same country at the same time – Villani often has residences at Princeton or other centres of mathematical excellence around the world. As in many a collaboration, the two men work separately on bits then swap – time and time again, each time improving the maths, and emailing each other…

But the composition will certainly never be continuous in an L^1 space, so that can’t be right, probably we’ll have to be fairly devious and begin by “integrating” the \etas. That would leave an L^2 analytic norm in the variable k.

Conclusion: We’ll have to go on being devious.

More later


That may seem rather dry, but the two men are quite chatty in their emails, acknowledging towards the end that their proof is ending up as ‘a bit of a monster’.

In between this however, Villani takes time to introduce us to the key people working in the field – both historical and living.  We get character portraits with accompanying pencil illustrations (by Claude Gondard). These are inserted into the text whenever they come up, or wherever Villani meets them.

You may have even heard of one of them. One day Villani finally gets to meet his hero – John Nash – the mathematician who won the Nobel Prize for Economics for his work on Game Theory forty five years previously. (Nash was the subject of Sylvia Nasar’s biography A Beautiful Mind – later adapted into the movie with Russell Crowe as the mathematician who was then suffering from paranoid schizophrenia).

But back to the equations… Looking at a page of them, with a few instructions here and there reminds me of studying a musical score – flicking through to find the musical directions, where the horns come in – that kind of thing.  If you’ve done A-level maths, you’ll recognise that the tall italic f-like symbol is an integral sign – if you know music, you’ll think of it as the shape of the sound-holes on a violin or cello. The terms in an equation often flow from each other as if they were music, you can appreciate their form without understanding a single note. It proves to me how close maths and music really are. See, I told you that in the hands of a good storyteller, equations can be sexy!

What was fascinating though, was exploring Villani’s mind and where the inspiration comes from. Sometimes it comes at night from the deep recesses of a tired mind that just can’t sleep. Other times a throwaway remark from a colleague will trigger a new train of thought or a new approach to a sticky area. There is also the rigour of peer review to go through before their ‘monster’ of a paper can be published – all 180 pages of it. Their theorem has a difficult birth.

I used to think that globe-trotting professors giving lectures around the world or having sabbaticals at other celebrated establishments was a bit of a gravy train. Maybe in some disciplines it is, but it’s clear that getting out and meeting fellow mathematicians and theoretical physicists and the consequent cross-fertilisation of ideas that grow from these talks and collaborations is a real enabling factor that permits the leaps of imagination, resulting in real progress.

Malcolm De Bevoise’s translation from the French appears seamless and Villani is a very engaging companion in this ‘mathematical adventure’. I’d highly recommend this to anyone looking for a different style of popular science book.

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Annabel is one of the editors of Shiny New Books and barely scraped through the maths module in her science degree and is now having to relearn algebra to help with her daughter’s homework.

Cedric Villani, Birth of a theorem (Théorème Vivant) translated by Malcolm De Bevoise. (Bodley Head, London, 2015) 978-1847922526, 260 pp., hardback.


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    1. Of course you’d say that! 🙂 But equations in a text are a hard sell to those who don’t read science much, hence the music analogy which I think works very well…

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