EPS Review #166 - The Poincaré Conjecture

Is the universe doughnut shaped? The Poincare Conjecture: In Search of the Shape of the Universe, by Donal O'Shea, Penguin, 2007, 292pp.

You have to wonder who is the intended audience for popular math books. If it is for math groupies like me, then why put in basic descriptions of Euclidean, spherical and hyperbolic geometry? Perhaps authors live in hope that somebody else will read it too. Anyway, I am used to that. I felt this book was going wrong when I was tempted to skim, but what I was reading was not rehashed basics, but pure human history about mathematicians. Poincaré's life story is new to me and should have been interesting. But in this book, it was not. There is a lot of saying how exciting math is, instead of showing. Perhaps it works better in O'Shea's lectures at Mt Holyoke. And perhaps it is more interesting to career academics, since it covers a lot of history of academic institutions.

This is a book about topology in which the diagrams are poor! Though the portraits are not. This is inexcusable in the age of computers, where even the Grateful Dead show Not Knot at concerts. I will console myself by poring over my copy of A Topological Picturebook.

Cool factoid: there are 28 inequivalent ways of defining a calculus on the seven-sphere.

Most popular math books save the really great stuff for the last chapter. I expect this book was commissioned because of the amazing story of Perelman solving the conjecture and declining the million-dollar prize, and the exciting piece in the New Yorker about it, which among other things accuses the famous mathematician Shing-Tung Yau of trying to steal credit. O'Shea calls the article sensational, but then does not say much else about the controversy. The book ends a few pages later.

My favorite recent book in this genre remains the one on the four color map theorem. In the copious notes, O'Shea recommends The Shape of Space, which might be good. Symmetry and the Monster looks tempting too.