The Poincaré ConjectureReview by Andrew Crumey

Published in Scotland On Sunday

The Poincaré Conjecture

Donal O’Shea

Allen Lane, £17.99This book is subtitled “In search of the shape of the universe”, and if that is not tempting enough, the dustjacket promises “An unsolved mystery, a reclusive genius and a race to win a million dollars.” All true enough, though somewhat misleading to potential readers.

The Poincaré Conjecture is a mathematical problem first formulated in the late nineteenth century and still unsolved a hundred years later when the Clay Foundation named it one of their “millennium problems”, offering a million dollars to anyone who could crack it. Russian mathematician Grigory Perelman attempted what so many others had failed, and in 2006 was declared to have succeeded. He had been beavering away at the conjecture for years, and the cash was unimportant to him. He was also awarded the Fields Medal – the mathematicians’ Nobel Prize – but turned it down.

Those are the facts alluded to in the dustjacket synopsis; but the trouble with reclusive geniuses is that unless someone is prepared to intrude upon their privacy, very little can be said about them. In a couple of pages we learn that Perelman went to a top Russian school, was a brilliant student, and has never shown much interest in anything except mathematics – all of which we could have guessed from the start. Working largely in seclusion, his only race was against his own demons, whatever they were.

And what of the Poincaré Conjecture itself? Here the trouble is that it is very hard to explain in simple language; but mathematicians these days are adept at promoting the relevance of their work to an uncomprehending public, hence the attention-grabbing line about the shape of the universe. If you had lots of maps showing little bits of space, the question is whether you would be able to stitch them all together in the right way to make a complete cosmic atlas. You might quibble that we don’t actually have maps of every bit of the universe; but remember this is pure mathematics, not astronomy. So Perelman’s epoch-making achievement hasn’t really told us anything about whether space goes on forever or curves round on itself, and nor does this book.

What all this means is that readers lured in by the promises on the cover are apt to be disappointed, which is a shame because this is an interesting and well written book by a professional mathematician who knows his subject inside out. But it is not a book about space, money or tortured geniuses: it is a potted history of geometry and topology written for the sort of people who can get excited about differentiable structures on three-spheres - in other words mathematicians. Lovers of the subject will welcome the small chunks of biography – the coldly aloof Gauss, tragic Riemann and suave Poincaré – and even the occasional historical excursions that are dubiously relevant but offer relief to over-worked brains, while rigorous endnotes make up for the tacky illustrations. As for whoever thought up the jacket copy – they’d make a great estate agent.