The image of the eccentric genius runs deep in the public perception of mathematicians. Mostly, it's nonsense, but occasionally not ...
What about the shape of the universe, the book's subtitle? Poincaré after something far more important: how to tell what shape anything is. The universe, or a doughnut, were just examples.
Mathematics creates general tools, which scientists and others use to solve specific problems, and tool-making itself is motivation enough for doing mathematics. And, in fact, mathematical physicists are currently using topological methods to understand the shape of the universe, as the book explains in its proper place.
It is always unfair to review a book by comparing it with another book which exists only in the reviewer's imagination, with the title What He Should Have Written Instead. In this ideal book, page lengths are so malleable that every conceivable side issue can be pursued, and expository difficulties miraculously vanish. But I would like to have seen a bit more about the beautiful geometrical ideas in Perelman's proof, and I would have been willing to forgo some of the earlier history to make room. Be that as it may, The Poincaré Conjecture makes one of the most important developments in today's mathematics accessible to a wide audience, and it deserves to be widely read.
