Perfect Rigour

by Masha Gessen

  “It was so much fun,”33 one mathematician told me. “It came out right during the congress, and the copy machines immediately started working at full capacity. I might have been bored there otherwise, but as it was, it was really fun.”

  On August 29, the day after the New Yorker’s cover date, the daily ICM newsletter published back-to-back interviews34 with Cao and Jim Carlson, head of the Clay Institute. Cao extolled Hamilton and Perelman, saying that they “have done the most important fundamental works,” and adding, “They are the giants and our heroes!” But he stopped conspicuously short of saying that it was Perelman who had proved the Poincaré and Geometrization—indeed, he made both Hamilton and Perelman sound like giants from the past on whose shoulders modern-day mathematicians had stood to construct the ultimate proof. Carlson, on the other hand, was decisive: “Perelman fulfills all the requirements of the Millennium Prize,” he said, naming the work of Kleiner and Lott, Morgan and Tian, and Cao and Zhu as the papers that completed the Clay Institute’s refereed-publication requirement.

  Mathematicians are not accustomed to controversies this heated and publicity this broad. There had been arguments over authorship and credit before—including one involving a Russian topologist, Alexander Givental, and Yau and one of his students, who claimed to have completed a proof Givental had begun—but they had never spilled over into the mainstream media. Unlike social scientists or even doctors, mathematicians whom Nasar and Gruber interviewed had no experience talking to the press. When they saw their words in print—and copiously reproduced for their colleagues’ entertainment—they were aghast. Yau engaged a lawyer, who wrote a letter to the New Yorker35 demanding a correction and an apology, because, Yau now claimed, he had never tried to wrest credit away from Perelman. Three mathematicians quoted in the article wrote what amounted to letters of apology to Yau and allowed them to be posted on various websites. Anderson was among those who claimed to have been quoted out of context. And when I spoke to him a year later, he was extremely reluctant to go on the record. He also tried to convince me that the Yau controversy had been unnecessarily exaggerated by nonmathematicians.

  Perelman likely did not follow this story. He had positioned himself outside the mathematics community, and he had never been much of a Web surfer. But Rukshin, who was expert at scouring blogs and tracing links, was in his element following this unprecedented mathematical scandal. It would have given him satisfaction to report back to Perelman what both had long suspected: the mathematics community did not stand up for its own, not even for one who had given mathematics its biggest gift in a hundred years.

  The mathematics community in the United States, and even in the world, is very small and very peaceful. “And that’s one of the great joys of being a mathematician,” John Morgan told me about a year after the controversy. “It’s not like sociology or history, where it does become quite political. And maybe that’s another reason why people shy away from these controversies, hoping they’ll go away. You know, you start having war in camps and then suddenly the department explodes. The X supporters are separated from the Y supporters and the anti-Y supporters and, you know, that doesn’t do anybody any good. Keep it a pleasant place to do the work. So few people understand what we do, appreciate what we do, it’s nice. This community is actually a community of people who respect each other and treat each other decently.” Most of the people, most of the time, that is. In a community this small, one cannot afford to burn bridges. Yau, with his academic positions and his army of professor-students on two continents, is not only extremely powerful institutionally but also central to a large and vibrant intellectual community, being shut out of which would amount to a tragic loss for most mathematicians.

  The contemporary Western mathematics community acts like a corporation, albeit a very small one: it protects its own from the outside world, and it depends on peace, cooperation, and communication to function. But being a very small corporation, it also sometimes acts like a family, sacrificing ideals and principles for shared history and interdependence. Perelman had almost as little use for family, outside of his mother, as he did for corporations. He simply did not understand either. And he did not like to deal with things he did not understand. In fact, he refused to deal with them.

  About a year before all hell broke loose in the summer of 2006, the ICM program committee sent Perelman a letter inviting him to give a lecture at the Madrid congress. The program committee and the medal committee worked independently; the members of both were kept secret until the congress, and only the names of the chairs were released. Perelman did not respond to this letter or to subsequent others. A committee representative then called Kislyakov—Perelman was still on staff at the Steklov then—and Kislyakov called Perelman at home. Perelman explained to Kislyakov that he had not responded to the letters precisely because the names of committee members were kept secret. He would not, he said, deal with conspiracies.

  Kislyakov conveyed Perelman’s reasoning back to the committee, which followed with another letter, this time disclosing the names of its members. Perelman again did not respond; the committee again requested Kislyakov’s intervention; and the Steklov director again called Perelman at home. Perelman explained that the committee’s disclosure was too little, too late—and he would not entertain further discussion.

  Perelman’s refusal to deal with the program committee, which amounted to his refusal to speak at the congress, was an almost debilitating blow to the ICM organizers. It was obvious that the topic of the Poincaré Conjecture would dominate the congress. At the same time, the Fields Medal committee had decided that Perelman should be one of the recipients. The Fields Medal, often called the Nobel Prize of mathematics (there is, in fact, no Nobel Prize for mathematics), is awarded every four years to two to four mathematicians age forty or younger. Perelman would turn forty just before the congress, making it the last year he would be eligible. And although by the summer of 2005 a consensus had formed among topologists that Perelman had indeed proved the Poincaré—and the committee was aware of this consensus, because Jeff Cheeger was one of its members—final certainty was lacking. Kleiner and Lott and Morgan and Tian were not yet done with their explorations of the proof, so no one could guarantee that a major flaw—or even a fatal one, as Yau had implied—would not emerge. The committee drafted a carefully worded invitation36 to Perelman to accept the Fields Medal—an invitation that, much like the press release a year later, did not state that he had proved the Poincaré Conjecture.

  Normally the names of Fields Medalists are not released to anyone, including the laureates themselves, until they are announced at the ICM. Naturally, though, the medal recipients are usually present at the congress and already scheduled to give speeches. But Perelman had refused to speak, and this was what necessitated the special invitation. Imagine Perelman’s reaction. Was this all the mathematics community had to offer him, after all he had contributed? Recognition along with three other mathematicians, none of whom had accomplished anything as momentous as the proof of the Poincaré? And recognition that was carefully worded so as to avoid giving Perelman true credit for what he had done! If ever Perelman had seen mathematics taking on the worst traits of politics, it was then.

  To ensure that Perelman would agree to attend the congress and accept the medal, the Fields Medal committee dispatched its chairman—president of the International Mathematical Union, Oxford professor Sir John Ball—to St. Petersburg. This was an unprecedented mission, but then, there had never been as difficult a problem as the Poincaré Conjecture or as difficult a medal recipient as Grisha Perelman. The week before Perelman was due to be awarded the medal, he and Ball spent hours speaking at a conference center in St. Petersburg. Perelman would not accept the medal. Ball offered him a number of alternatives, including the delivery of the medal to St. Petersburg—as had been done decades before when Soviet mathematicians were not allowed to travel to the ICM and the medal had been aw
arded whenever it could physically meet its recipient—but Perelman refused.

  On August 22 in Madrid, during the ICM opening ceremony, John Ball announced the names of the four Fields Medal recipients. They were Andrei Okounkov, a Russian mathematician working at Princeton; Perelman; Terence Tao, a onetime Australian wunderkind now at the University of California at Los Angeles; and the French mathematician Wendelin Werner. Perelman came second on Ball’s list, as the list was arranged alphabetically. “A Fields Medal is awarded to Grigory Perelman,37 of St. Petersburg, for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow,” said Ball. “I regret that Dr. Perelman has declined to accept the medal.”

  When the New Yorker writers visited Perelman earlier that summer, he told them it was the prospect of being awarded the Fields Medal that had forced him to make a complete break with the mathematics community: he was becoming too conspicuous, getting roped into the limelight. He might have been engaging in a bit of justification postdating: when he had quit the Steklov in early December 2005, declaring on his way out that he was abandoning mathematics altogether, the Fields Medal, while certainly a predictable possibility, was not yet a subject of discussion. “At a certain level you could say he lives absolutely by his principles,” Jeff Cheeger said to me almost two years later. “But he is certainly not entirely open about his motivations, and in particular I believe he’s quite an emotional person. And he uses his powerful mind to sort of explain his emotions after the fact.”

  The Fields Medal debacle seems to have tried Cheeger’s patience with his brilliant younger colleague. “It’s sort of like he is above it and maybe there is something wrong with practitioners in general,” Cheeger told me, trying as hard as he could to choose words that would not offend Perelman, on the infinitesimal chance that he ever reads this book. “His behavior was supposed to be purer than pure, but it wound up having the effect of essentially focusing all the attention on him—not just because of the extraordinary importance of what he had done, but seemingly paradoxically. To the relative exclusion of all the other Fields Medalists.”

  If part of what insulted Perelman about the Fields Medal was the suggestion of his sharing with three other mathematicians what he felt should have been a singular honor, then by rejecting the medal, he set himself firmly apart. The same way that Perelman’s refusal to accept the European honor in 1996 had hurt Vershik, now a number of his colleagues felt slighted, insulted, or at least misunderstood and puzzled by Perelman’s behavior. Only Gromov claimed to understand Perelman’s reasoning perfectly and to support it fully.

  “When he got the letter from the committee inviting him to give a talk, he said he wouldn’t talk to committees,” Gromov recounted for me. “And that is absolutely the right thing to do! There are all sorts of things that we accept that we shouldn’t accept. And he looks extreme only against the backdrop of conformism that is characteristic of mathematicians in general.”

  “But why shouldn’t one talk to a committee?” I asked.

  “One doesn’t talk to committees!” Gromov exclaimed, exasperated. “One talks to people! How is it possible to talk to a committee? Who is on that committee? It might be Yasir Arafat is on it.”

  “But they sent him the list of committee members and he still refused to talk,” I objected.

  “After the way it started, he was right not to talk to them,” Gromov persisted. “The moment the community begins to act like a machine, you have to stop dealing with it—that is all! The only strange thing is that more mathematicians don’t act that way. That is the strange thing! Most people are perfectly content to talk to committees. They are satisfied to travel to Beijing and accept a prize from the hands of Chairman Mao. Or the king of Spain, which is the same thing.”

  Why, I pleaded, was the king of Spain undeserving of the honor of hanging a medal around Perelman’s neck?

  “Who the hell are kings?” Gromov was really cranked up now. “Kings are the same kind of crap as communists. Why should a king give a mathematician his prize? Who is he? He is nothing. From a mathematician’s point of view, he is nothing. Same as Chairman Mao. So one of them seized power like a robber while the other got it from his father. That’s no difference.” In contrast to these people, Gromov explained, Perelman had actually made a real contribution.

  Following my interview with Gromov, I walked around Paris with a French mathematician who had refashioned himself as a historian of science. I had met Jean-Michel Kantor at a conference on mathematics and philosophy. Here was a classic French intellectual, a short, disheveled man who had to rush off to an editorial meeting of a highbrow book-review journal following our walk. As we walked, he criticized Gromov. The geometer, he said, had stood idly by as French mathematics sank into the abyss: mathematical institutions now issued fundraising brochures, blatant appeals for money that contributed nothing to the mathematical discourse. And professors shamelessly entered into salary negotiations, sometimes even making their plans contingent on the remuneration. Where was their love of the science and their will to sacrifice material comforts for the common cause of mathematics?

  What this man was describing was the Americanization of French mathematics. And what I found invaluable about his perspective was that he still managed to see the money-centric, marketing-driven messages of the mathematical establishment as outrageous rather than obvious and expected, as they are in the United States. To someone like this—and to someone like Gromov, who seemed sensitive to criticism that he was becoming a capitalist conformist—Perelman, with his disregard for money and aversion to institutions, appeared very much like the Platonic ideal of a mathematician.

  In 2006, the ICM went forward without Perelman. John Lott gave what would ordinarily have been the laudation38 but was instead a presentation devoted to Perelman’s mathematical career trajectory and the trajectory of his proof. Two hours later, Richard Hamilton39 led a discussion of the Poincaré Conjecture. The announcement of this session in the program,40 presumably submitted by Hamilton, adopted a virtuoso approach to apportioning credit: the program for the solution, it said, had been invented by Hamilton and Yau, followed by Perelman, who supplied an important part of the solution and “announced the completion of the program,” crowned by Cao and Zhu’s paper, which Hamilton called “a full exposition.” Such wording did not suggest that Cao and Zhu deserved credit for the proof, but it also did not state that Perelman did—only that Perelman himself believed so. During the actual discussion in Madrid, however, Hamilton was as gracious when speaking of Perelman as he had ever been. One participant recalled that Hamilton said he had not originally believed Perelman’s claims that he had resolved the problems with his Ricci flow program and taken it to its completion but on closer inspection had seen that Perelman was right. “It was an expression of real admiration,” recalled Jeff Cheeger. “Even more so because his initial reaction was ‘this guy has got to be crazy!’”

  By the end of the congress, the international mathematics community had fully accepted the majority topologists’ position: Perelman had completed the proof of the Poincaré Conjecture. The Clay Institute would now use the ICM41 as the starting point for its countdown to the prize.

  Any lingering idea that Cao and Zhu deserved ultimate credit was quietly put to rest the following fall, when a pdf file started circulating42 among mathematicians. Its left column contained excerpts from Kleiner and Lott’s notes on the first Perelman preprint, which had been posted on the Web in 2003; the right column contained excerpts from Cao and Zhu’s later paper. Sizable passages appeared to match verbatim. In an erratum note they submitted to the Asian Journal of Mathematics, Cao and Zhu claimed they had forgotten they had copied the material43 into their notes three years earlier. In early December, Cao and Zhu posted a revised version of their article to the arXiv. Now it was called “Hamilton-Perelman’s Proof of the Poincaré Conjecture and t
he Geometrization Conjecture,” and the abstract no longer claimed to give the complete proof or be the “crowning achievement.” It now read almost contrite: “In this paper, we provide an essentially self-contained44 and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of three-manifolds. In particular, we give a detailed exposition of a complete proof of the Poincaré conjecture due to Hamilton and Perelman.”

  Following the ICM and the New Yorker article, a frenzy broke out where it could hurt Perelman most: the Russian media. Journalists from all sorts of newspapers, including tabloids with press runs of more than a million copies, began calling constantly. Some days, School 239 seemed engaged in a nonstop press conference. Perelman’s old teachers weighed in on the subjects of his sanity and his relationship with the mathematics community. Channel 1, which reached more than 98 percent of Russian households, reported that Perelman45 had turned down the million-dollar prize. Tamara Yefimova, the director of School 239, told a tabloid newspaper that Perelman had not attended the ICM in Spain because he did not have the money to buy a ticket.46 Alexander Abramov, his old coach, contributed an article47 to a highbrow Moscow weekly, arguing that there was “no Perelman mystery,” just the failure of Russian academic institutions to recognize his achievements. Channel 1 called Perelman at home and broadcast the conversation, in which he said he was no longer doing mathematics and had not been since he left the Steklov Institute. “You could say I’m engaged in self-education,”48 he said. “I cannot predict what I am going to be doing.” A camera crew from a Channel 1 tabloid-style talk show burst into his apartment, pushing his mother out of the way on camera in order to film an unmade bed. People began recognizing him in the street and at the opera. He took to saying he was not Grigory Perelman. Strangers snapped pictures of him with their mobile phones and posted them on the Internet.