Book review: Perfect Rigor

by Diana Davis

Perfect Rigor cover image

Perfect Rigor, by Masha Gessen.

Before reading this book, I had only vague ideas about Grigori Perelman. I thought that, after being a postdoc in the US, he disappeared back to Russia, lived with his mother for a number of years, then suddenly uploaded his proof of the Poincaré Conjecture to the arXiv and then disappeared back into silence. However, this is not true! The story, as you might expect, is far richer than that. Masha Gessen’s book explains it all.

This book traces Perelman’s life from the age of 10, when he joined a math club, through high school and math competitions, through the university, his postdoc positions and proof of the Soul Conjecture, through his celebrated proof of the Poincaré and up to the present. Whatever you may have thought about math education under the USSR, it is probably far from the fertile reality that Perelman experienced: one-on-one attention with teachers and graduate assistants who listened to every student’s solution to every problem, and taught students to verbally explain their solutions to their classmates. Gessen describes how Perelman (universally known as “Grisha”) was not the outspoken star of the class, but he could solve every problem, and never made a mistake, never producing a false solution. If Grisha said he had solved the problem, then he had done it correctly. Many years later, this caused his colleagues to take his arXiv papers much more seriously than they had taken the other supposed “proofs” of the Poincaré that others had published over the years: they knew that he never had any false solutions, so if Grisha said it was true, then it was not a matter of verifying the proof, just of understanding it.

Though Gessen was not able to interview Perelman, she interviewed just about every important person in his life, and paints a complete portrait of him and his mathematics. Though it is true that Perelman eventually stopped communicating with the media, immediately following his posting preprints on the arXiv he embarked on a month-long lecture tour in the United States, speaking daily on his proof and answering mathematicians’ questions for hours. Gessen explains how Perelman was initially eager to give the mathematical community this rare gift, his proof of the Poincaré Conjecture. However, she argues that he experienced repeated slights (such as the Fields Medal committee refraining from claiming that he had proved the conjecture, instead just praising his “contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow”) from the mathematical community. Over the years, this caused Perelman to retract himself first from mathematics, and then from the rest of the world.

At this point, Gessen uses a wonderful mathematical metaphor (p. 172):

The more Perelman talked about his disappointment with the mathematical establishment, and the more his acquaintances decorated his stories with demonizing details, the more Perelman’s sense of betrayal deepened. His world, which had begun narrowing in his first university year and then broadened slightly both times he had traveled to the United States, was now headed for its final, disastrous narrowing. Like a rubber band slipping inexorably off a sphere, his world was about to shrink to a point.

My only critique of this book is that Perelman rarely speaks — Wikipedia has more Perelman quotes than this entire biography. In particular, Perelman is silent for his entire childhood; in my mind he is a silent character who walks around with his shoelaces untied and his hat flaps tied firmly around his chin, rubbing his hands on his pant legs and humming, then writing down perfect proofs. The first time Perelman speaks, it is when he is a university student, summarizing his notes for classmates who skipped the class, and even then we do not hear him speak (no quotations); we only know that he spoke. Later, we hear that he gives math talks at conferences, but these are only mentioned, never quoted. Of course, Gessen couldn’t interview Perelman, but she could have quoted secondary sources, such as the New Yorker article published about the Cao/Zhu controversy, which quotes Perelman several times, in which he speaks about how he feels about math and the mathematical community. These would have enriched the reader’s understanding of Perelman’s character, as well as his current situation.

I was especially impressed with Gessen’s treatment of the hypothesis that Perelman has Asperger’s syndrome. I happen to have an interest in Asperger’s, and have read widely on it, mostly because I get annoyed at people who casually diagnose others with the syndrome despite no psychological training whatsoever. Gessen alleges that he has Asperger’s, but presents it in a very compelling way: Perelman’s math club is at a summer camp with many other, non-mathematically-inclined children (pp. 174-175):

… at the beginning of the camp season [the math club boys] attended a lecture on foreign affairs. “The international situation,” said the speaker, a young Komsomol worker, “is particularly tense today.” The entire mathematics contingent broke out laughing. It is particularly tense today! Get it? It is like it was not at all tense yesterday but is particularly tense today.

If you do not find that especially funny, then chances are you do not have Asperger’s syndrome. …

Gessen goes on to explain that, on the autism-spectrum quotient test, university students studying humanities score about the same as random controls; scientists score higher, and mathematicians score the highest. While Gessen could not give the test to Perelman, she took it herself and scored quite high, which she said would be expected because she attended a Russian math high school. There is, in fact, an entire chapter (titled “The Madness”) exploring the connections between Asperger’s-type behavior and Russian mathematics, exploring other eminent Russian mathematicians’ social awkwardness, and explaining that the Russian math system is very accepting of mathematicians’ weirdness.

As much as this is a biography of Perelman, it is also a biography of Russian mathematics. Let me end with Gessen’s description of the mathematical community under communism, which I quote in full because it paints such an idyllic scene (pp. 13-14):

On paper, the jobs that members of the mathematical counter-culture held were generally undemanding and unrewarding, in keeping with the best-known formula of Soviet labor: “We pretend to work, and they pretend to pay us.” The mathematicians received modest salaries that grew little over a lifetime but that were enough to cover basic needs and allow them to spend their time on real research. “There was no such thing as thinking that you had to focus your work in some one narrow area because you have to write faster because you had to get tenure,” said [Israel] Gelfand. “Mathematics was almost a hobby. So you could spend your time doing things that would not be useful to anyone for the nearest decade.” Mathematicians called it “math for math’s sake,” intentionally drawing a parallel between themselves and artists who toiled for art’s sake. There was no material reward in this — no tenure, no money, no apartments, no foreign travel; all they stood to gain by doing brilliant work was the respect of their peers. Conversely, if they competed unfairly, they stood to lose the respect of their colleagues by gaining nothing. In other words, the alternative mathematics establishment in the Soviet Union was very much unlike anything else anywhere in the real world: it was a pure meritocracy where intellectual achievement was its own reward.

Let us endeavor to practice mathematics such that even if this is not the mathematical culture we experience, we can have this reality within ourselves.

P.S. After reading this book, I can recommend this song about Perelman’s proof — read the clever lyrics, but skip listening to the song.

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