The story of Grigory Perelman and the Poincaré Conjecture is certainly one of the more compelling and, to most people, strange stories in modern mathematics. You start with a long-standing open conjecture and combine this with a genius who has a conflicted relationship with the mathematical community and whether he wants to be a part of it. The genius may or may not have solved the problem and the story has all kinds of narrative tension as the rest of the mathematical community sorts out the details. There are other mathematicians who jump in and try to claim credit and still others who are filled with professional jealousy. And the whole story climaxes with an award ceremony that Perelman refuses to attend for reasons that most people find incomprehensible.There are so many interesting characters and pieces of the story that if I were a Hollywood executive, I would be trying to get the rights to make a movie of this story as quickly as possible. (Of course, many things would be different if I were a Hollywood executive…)
There may not yet be a movie of the story of the Poincaré Conjecture, but Masha Gessen has written a book about the story, Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century, and it does not disappoint. While Gessen did not get to interview Perelman himself, she interviewed a large number of other people who knew Perelman or were involved in the story. Gessen does a good job of articulating the looming questions about Perelman — why would he refuse to submit his papers for publication? Why does he turn down the Fields medal and refuse to accept honors and positions at prestigious institutions? — and trying to give answers to them. And while she never really “solves” these questions, she gives a compelling portrait of Perelman, starting with his childhood and his early involvement with math competitions in Russia through the current day.
I should be clear that Gessen has not written a math book, choosing to focus on the people and cultures in the story rather than on the mathematics itself. She keeps the description of the Poincaré Conjecture and the various details of the proof at a very superficial level, and when she does describe mathematics, she often gets some of the small details wrong in a way that will cause many mathematicians to bristle. On the other hand, she seems to “get” mathematicians, and does a good job of portraying our community and what motivates us, even in the opening lines of the book:
As anyone who has attended grade school knows, mathematics is unlike anything else in the universe. Virtually every human being has experienced that sense of epiphany when an abstraction suddenly makes sense. And while grade-school arithmetic is to mathematics roughly what a spelling bee is to the art of novel writing, the desire to understand patterns — and the childlike thrill of making an inscrutable or disobedient pattern conform to a set of logical rules — is the driving force of all mathematics.
In addition to being a biography of Perelman and of the mathematical community’s response to the solution of the Poincaré Conjecture, Perfect Rigor is a portrait of mathematics in Russia during the second half of the twentieth century. Knowing very little about this subject, I found her portrayal of a community filled with intense competition, strict rules, complicated bureaucracy, and not a small amount of anti-semitism to be fascinating.
Perfect Rigor is not a perfect book. The book would have benefitted from direct quotes from Perelman, and while I understand Gessen’s choices I personally would have preferred a little more math. I also could have done without a section late in the book about Asperger’s Syndrome, which tried to diagnose Perelman from a medical point of view, and which I found a bit too condescending towards mathematicians and buying into stereotypes. But given the choices that Gessen made and the parameters she had to work in, she wrote a very interesting and readable book about one of the more compelling stories of our time.
Darren Glass is an Associate Professor of Mathematics at Gettysburg College, whose research interests include Galois theory, number theory, and cryptography. He can be reached at firstname.lastname@example.org.