I just finished reading author Masha Gessen’s new book about Grigori Perelman, Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century. It’s a short but very well done account of the life of Grigori Perelman, how he came to prove the Poincare Conjecture, and what has transpired since.
The book is really not about mathematics, but about mathematicians and their culture, especially that of Russian mathematicians. Only one chapter deals with the mathematical content of the Poincare Conjecture, with the bulk of the book about Perelman and his career. Perelman’s talent’s were recognized early, and were nurtured in Leningrad by a system designed to train students for mathematical competitions. He won a gold medal at the International Mathematical Olympiad in 1982. The institutionalized anti-Semitism of the Soviet mathematics establishment of this period is described in detail in the book, together with the intense efforts made by Perelman’s supporters (including Alexandrov) to overcome this. He did his graduate work at the most prestigious institution in Leningrad, and then went on to a research position there at the Steklov Institute.
Gessen never managed to interview Perelman himself, but did talk to many if not most of the mathematicians he interacted with. He was brought to Courant by the intervention of Gromov, and for a few years worked there, at Stony Brook and at Berkeley. By the end of this time, he had started to develop a significant reputation in the math community, but he chose to return to Steklov and pretty much dropped out of sight, communicating with very few people for several years. It was during this period that he developed his proof, finally posting what could be described as a detailed outline in a series of three papers submitted to the arXiv.
The story of what happened then is rather remarkable, but it’s a story I’m pretty familiar with since I got to watch much of it from up close (Perelman’s preprints and the question of whether he really had a proof were discussed intensively here at Columbia, where Richard Hamilton and John Morgan are among my colleagues, and quite a few other people work in this area). Gessen does a good job of telling this story, adding some details I was unaware of.
Perelman turned down the Fields medal awarded him for this work, and sadly, he seems in recent years to have cut himself off from even his closest friends in the math community. Indications are that he is no longer actively working on research mathematics. The book contains speculation from several mathematicians who know Perelman about his thought processes and the reasons for his behavior, but they remain somewhat of a mystery. Some amount of paranoia seems to be at work, together with an intense distaste for any sort of politics, even the most innocuous workings of the mathematical community and its institutions.
The last chapter of the book has some news I hadn’t heard. Last year, Jim Carlson, who runs the Clay Mathematics Institute and is responsible for the process that will determine the award of the million-dollar Millennium prize for the proof of Poincare, traveled to St. Petersburg. He talked to Perelman on the phone, but Perelman refused to meet with him. According to the book, Clay was planning on convening a committee to decide on the prize this past May, with a report planned for August. Presumably this all has already happened by now, and perhaps Carlson has already made another trip to St. Petersburg in a last attempt to see if Perelman can be convinced to accept the prize. Perhaps we will be finding out the results soon…
Update: Today’s Wall Street Journal has an article by Gessen about Russian mathematics that summarizes part of her book.