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Conceived in 1904, the Poincaré conjecture, a puzzle that speaks to the possible shape of the universe and lies at the heart of modern topology and geometry, has resisted attempts by generations of mathematicians to prove or to disprove it. Despite a million-dollar prize for a solution, Russian mathematician Grigory Perelman, posted his solution on the Internet instead of publishing it in a peer-reviewed journal. This book "tells the story of the fascinating personalities, institutions, and scholarship behind the centuries of mathematics that have led to Perelman's dramatic proof." The author also chronicles dramatic events at the 2006 International Congress of Mathematicians in Madrid, where Perelman was awarded a Fields Medal for his solution, which he declined.… (more)
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In this book, O’Shea takes us through the history of the conjecture and the attempts at solving it, and also takes some time to give us the historical context along the way by describing the social and political climate surrounding each mathematician that has sought to prove the conjecture. He does a good job of providing relatively clear and simple explanations of the complex ideas in topology and non-Euclidean geometry involved, but the book does move at a fairly brisk pace (minus the notes at the end the main text is only 200 pages long) so some work is still required to follow along, but I never felt completely lost. This book contains a nice mix of mathematical ideas and history for a general audience, and it managed to keep my interest throughout.
Despite the subtitle
O'Shea does a terrible job at making topology look useful: the idea that with everything we knew about maps at the time of Columbus, we could have concluded (but we did not) that the earth is a doughnut is irritating and ludicrous. It does not help. You can skip the first chapters and start at page 21 and get to the meaty part of the thing.
What I like about the book:
1. It is a neat problem
2. I understand what the author says
3. It has a compassionate look on the fate of mathematicians, and the history part is very well done.
Firstly, the book introduces many concepts by name, with some short descriptions, and then goes on to discuss them in some qualitative detail; how one concept leads to another; how concepts fail to connect. For me, at least, this was difficult to
Secondly, there are too many sections where names and dates and attempted proofs of such-and-such a conjecture/theory/etc. are listed; in these sections it very much feels like the only people who would be able to pull much meaning would be already quite familiar with the topics. There is much more of this in the last third or quarter of the book.
The middle 85% of the book isn't about the Poincare Conjecture per se. In this, I would describe the book as the history of mathematicians and mathematics, from ancient times to today, as told from the point of view of the Poincare Conjecture. An analogy might be something like a book that details the life of some famous figure by telling the history of their family/ancestry and the times and events their family lived through.