A hermitian operator in self-imposed exile

Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century
Masha Gessen (Houghton Mifflin Harcourt, 2009)

Pure mathematicians have the reputation of being otherworldly and divorced from practical matters. Grisha or Grigory Perelman, the Russian mathematician who at the turn of this century solved one of the great unsolved problems in mathematics, the Poincare Conjecture, is sadly or perhaps appropriately an almost perfect specimen of this belief. For Perelman, even the rudiments of any kind of monetary, professional or material rewards resulting from his theorem were not just unnecessary but downright abhorrent. He has turned down professorships at the best universities in the world, declined the Fields Medal, and will probably not accept the 1 million dollar prize awarded by the Clay Mathematics Institute for the solution of the some of the most daunting mathematical problems of all time. He has cut himself off from the world after seeing the publicity that his work received and has become a recluse, living with his mother in St. Petersburg. For Perelman, mathematics should purely and strictly be done for its own sake, and could never be tainted with any kind of worldly stigma. Perelman is truly a mathematical hermit, or what a professor of mine would call using mathematical jargon, a "hermitian operator".

In "Perfect Rigor", Masha Gessen tells us the story of this remarkable individual, but even more importantly tells us the story of the Russian mathematical system that produced this genius. The inside details of Russian mathematics were cut off from the world until the fall of the Soviet Union. Russian mathematics was nurtured by a small group of extraordinary mathematicians including Andrey Kolmogorov, the greatest Russian mathematician of the twentieth century. Kolmogorov and others who followed him believed in taking latent, outstanding talent in the form of young children and single-mindedly transforming them into great problem solvers and thinkers. Interestingly in the early Soviet Union under Stalin's brutal rule, mathematics flourished where other sciences languished partly because Stalin and others simply could not understand abstract mathematical concepts and thus did not think they posed any danger to communist ideology. Soviet mathematics also got a boost when its great value was recognized during the Great Patriotic War in building aircraft and later in work on the atomic bomb. Mathematicians and physicists thus became unusually valued assets to the Soviet system.

Kolmogorov and a select band of others took advantage of the state's appreciation of math and created small, elite schools for students to train them for the mathematical olympiads. Foremost among the teachers was a man named Sergei Rukshin who Gessen talks about at length. Rukshin believed in completely enveloping his students in his world. In his schools the students entered a different universe, forged by intense thought and mathematical camaraderie. They were largely shielded from outside influences and coddled. The exceptions were women and Jews. Gessen tells us about the rampant anti-Semitism in the Soviet Union which lasted until its end and prevented many bright Jewish students from showcasing their talents. Perelman was one of the very few Jews who made it, and only because he achieved a perfect score in the International Mathematical Olympiad.

Perelman's extreme qualities were partly a result of this system, which had kept him from knowing about politics and the vagaries of human existence and insulated him from a capricious world where compromise is necessary. For him, everything had to be logical and utterly honest. There was no room for things such as diplomacy, white lies, nationalism and manipulation to achieve one's personal ends. If a mathematical theorem was proven to be true, then any further acknowledgment of its existence in the form of monetary or practical benefits was almost vulgar. This was manifested in his peculiar behavior in the United States. For instance, when he visited the US in the 90s as a postdoctoral researcher he had already made a name for himself. Princeton offered the twenty nine year old an assistant professorship, a rare and privileged opportunity. However Perelman would settle for nothing less than a full professorship and was repulsed even by the request that he officially interview for the position (which would have been simply a formality) and submit his CV. Rudimentary formalities which would be normal for almost everyone were abhorrent for Perelman.

After being disillusioned with what he saw as an excessively materialistic academic food chain in the US, Perelman returned to Russia. For five years after that he virtually cut himself off from his colleagues. But it was then that he worked on the Poincare Conjecture and created his lasting achievement. Sadly, his time spent intensely working alone in Russia seemed to have made him even more sensitive to real and perceived slights. However, he did publicly put up his proofs on the internet in 2002 and then visited the US. For a brief period he even seemed to enjoy the reception he received in the country, with mathematicians everywhere vying to secure his services for their universities. He was unusually patient in giving several talks and patiently explaining his proof to mathematicians. Yet it was clear he was indulging in this exercise only for the sake of clarifying the mathematical concepts, and not to be socially acceptable.

However, after this brief period of normalcy, a series of events made Perelman reject the world of human beings and even that of his beloved mathematics. He was appalled by the publicity he received in newspapers like the New York Times which could not understand his work. He found the rat race to recruit him, with universities climbing over each other and making him fantastic offers of salary and opportunity, utterly repulsive. After rejecting all these offers and even accusing some of his colleagues of being traitors who gave him undue publicity, he withdrew to Russia and definitively severed himself from the world. The final straw may have been two events; the awarding of the Fields Medal which, since his work was still being verified, could not explicitly state that he had proven the Poincare conjecture, and the publication of a paper by Chinese mathematicians which in hindsight clearly seems to have been written for stealing the limelight and the honors from Perelman. For Perelman, all this (including the sharing of the Fields with three other mathematicians) was a grave insult and unbecoming of the pursuit of pure mathematics.

Since then Perelman has been almost completely inaccessible. He does not answer emails, letters and phone calls. In an unprecedented move, the president of the International Mathematical Congress which awards the Fields Medals personally went to St. Petersburg to talk him out of declining the prize. Perelman was polite, but the conversation was to no avail. Neither is there any indication that he would accept the 1 million dollar Clay prize. Gessen himself could never interview him, and because of this the essence of Perelman remains vague and we don't really get to know him in the book. Since Gessen is trying to somewhat psychoanalyze her subject and depends on second-hand information to draw her own conclusions, her narrative sometimes lacks coherence and meanders off. As some other reviewers have noted, the discussion of the actual math is sparse and disappointing, but this book is not really about the math but about the man and his social milieu. The content remains intriguing and novel.

Of course, Perelman's behavior is bizarre and impenetrable only to us mere mortals. For Perelman it forms a subset of what has in his mind always been a perfectly internally consistent and logical set of postulates and conclusions. Mathematics has to be done for its own sake. Academic appointments, prizes, publicity and professional rivalries should have no place in the acknowledgement of a beautiful mathematical proof. While things like applying for interviews and negotiating job offers may seem to us to be perfectly reasonable components of the real world and may even seem to be necessary evils, for Perelman they are simply evils interfering with a system of pure thought and should be completely rejected. He is the epitome of the Platonic ideal; where pure ideas are concerned, any human association could only be a deeply unsettling imposition.