Perfect Rigour

by Masha Gessen

  This was how Perelman had been taught mathematics should be practiced. He went to the lecture hall every day to fulfill his destiny, and this explained both his clarity and his patience. But in the world outside the Stony Brook classrooms, things increasingly diverged from his expectations. On the day he arrived at Stony Brook, the New York Times published another article.13 This one too started out by stating, inaccurately, that Perelman claimed to have proved the Poincaré Conjecture and linked that solution to the million-dollar prize, and it then went on to quote a single source: Michael Freedman, who had received the Fields Medal after solving the Poincaré for dimension four, and who was now working at Microsoft. Freedman, incredibly, called Perelman’s achievement “a small sorrow” for topology: Perelman had solved the biggest problems in the field, which made it less attractive, he reasoned, and so “you won’t have the brilliant young people you have now.”

  This was probably a fairly serious insult. After Perelman’s falling-out with Burago, his reference group, which was small to begin with, had shrunk to include just a few people who were in a position to understand his proof. Back at MIT, he had told Tian he thought it would take a year and a half or two years for his proof to be understood. But someone like Freedman might have been expected to have an immediate overall grasp of the elegance—and the correctness—of Perelman’s solution. For Freedman to frame Perelman’s proof as a setback for their once-shared area, and do so in an interview with a newspaper whose readership would never understand the problem or the solution, had to be hurtful—all the more so because Freedman’s reaction seemed so illogical.

  If anyone could speak authoritatively on what Perelman had done—particularly on what he had laid out in his first paper—it was Richard Hamilton. After all, Perelman had followed Hamilton’s program. One of the oddest and most tragic aspects of this story is the extent to which Perelman’s and Hamilton’s orbits missed each other. Perelman did not belong to what Anderson and others called the “Ricci flow community,” which had grown up around Hamilton in the two decades he had been trying to force the matrix to conform to the conjecture. Perelman had apparently approached Hamilton twice—once following a lecture of his, and once in writing, after Perelman had returned to St. Petersburg. Both times, Perelman was asking for a clarification of something Hamilton had said or written. On the second occasion, Hamilton failed to respond—something Perelman might have understood perfectly if he held others to the same standards of behavior to which he held himself. Indeed, for reasons that were probably entirely different from Perelman’s—Hamilton, by all accounts, was an atypically sociable mathematician—Hamilton tended to be elusive, occasionally reclusive, and usually very slow to respond to letters and calls. But rather than recognize familiar patterns, Perelman probably felt significantly frustrated by Hamilton’s silence; he generally expected his own needs, few as they were, to be met.

  Now, too, Hamilton was keeping his silence. That he had not attended Perelman’s lectures at MIT might have been disappointing but was understandable. But when Perelman began his stint at Stony Brook, just an hour and a half from New York City, where Hamilton was teaching at Columbia University, Hamilton’s absence became conspicuous. Other mathematicians from New York attended. One of them, John Morgan, asked Perelman to lecture at Columbia over the weekend. Perelman agreed, and then agreed to give another lecture that weekend at Princeton.

  On Friday, April 25, Perelman lectured at Princeton. The university again made him an offer. Perelman turned it down. On Saturday, he lectured at Columbia. Hamilton came and stayed for the discussion after the lunch break—until the only people in the room were he, Perelman, Morgan, and Gromov, who was then at Courant. “Everybody was waiting for Richard to say either he got it or he doesn’t get it,” Morgan told me. “It’s his theory, his idea. This is the way to do it. He’s the obvious person to pass judgment.”

  And did he? This is where it gets tricky. “Richard from the beginning was willing to and did acknowledge that what was in the first paper was correct and it was a huge advance,” said Morgan, now trying to tread carefully so as not to offend a colleague. The first paper dealt solely with Ricci flow, which was Hamilton’s invention and his area of total confidence. The second paper dealt with Ricci flow with surgery, which was also Hamilton’s invention but which in Perelman’s treatment intermingled with Alexandrov spaces and the work Perelman had done with Gromov and Burago. Hamilton was less of an expert here, and that may have made him both less confident and, perhaps, more hopeful that Perelman had failed. “I think maybe he thought, Well, this is a mistake,” said Morgan, “and if it’s a mistake, that would leave room for me to produce something more that I wanted to contribute. So I think he was sort of withholding judgment, waiting to see.” If there was a chance that Perelman had taken things in the wrong direction with his second paper, then someone else—most logically, Hamilton himself—could build on the breakthroughs of Perelman’s first paper. All of this, however, is conjecture: when Hamilton spoke of Perelman’s work publicly, he always did so graciously; he just did it far less frequently than many—including Perelman—might have expected.

  That day at Columbia, as Morgan remembered it, “It was proper but distant. There didn’t seem to be any overt tension. Grisha was not going to aggressively approach anybody. If you looked at it from the outside, it looked like any other math conversation: ideas coming in and going out. In other words, whatever Richard’s private feelings were about his distance, in this conversation at least, he was sort of normal about it.”

  Morgan invited Perelman to come to his house for brunch the following morning. “And he said, ‘Well, who would be there?’ I said, ‘Oh, my wife, my daughter, I may invite a couple other people.’ He said, ‘Oh, no. I don’t think so.’ So my take on that is, had it been a mathematical gathering, maybe he would have come. But a social gathering he was not at all interested.” That day, Perelman walked around New York with Gromov and talked to him about the Poincaré Conjecture and about his conflict with Burago. Then he went back to Brighton Beach, where his mother was staying, planning to return to Stony Brook the following night for another week of lectures and discussions.

  Perelman went back to Stony Brook discouraged. He told Anderson he was disappointed at the level of questions Hamilton had asked him: it seemed the inventor of Ricci flow had not taken the time to delve deeply into Perelman’s proof. In all likelihood, the reasons for this were complex: Hamilton was conflicted about engaging Perelman’s work, and, in addition, it may have been both psychologically and mathematically difficult to absorb a sudden break in the wall against which he had been beating his head for twenty years. But, just like twenty years earlier, while Perelman could be endlessly patient in reiterating his explanations to interested listeners, he could not imagine that anyone might have a difficult time with something that seemed, to Perelman, transparent and nearly self-evident.

  Perelman was annoyed too with Princeton’s insistent courtship attempts. Someone from that university called Anderson following Perelman’s lecture to ask for help in recruiting Perelman. At Perelman’s request, Anderson declined to help, but Princeton got a formal offer in the mail to Perelman anyway—and this he found upsetting. “They are being pushy,” he told Anderson. Among Perelman’s many rules of behavior, articulated and perhaps even formulated a couple of years after the Princeton offer, was the rule that “one should never force oneself on anyone.”14 Princeton, which had offended Perelman by asking him to apply for a job, now offended him by being too persistent in its affections.

  Anderson, who in addition to his genuine admiration for Perelman also seemed to have a keen sense of Perelman’s boundaries, apparently managed not to offend Perelman while pursuing the same agenda as all of Perelman’s other American hosts: to convince him to stay at his university and to draw him out socially. Anderson daily took great pains to convince Perelman to go out to dinner, and occasionally he succeeded. He also h
eld a party for Perelman at his house, which, in retrospect, seemed a bit of a disaster: Anderson and his friend Cheeger got into a loud argument over the U.S. invasion of Iraq, which Cheeger supported and Anderson did not. Anderson remembered getting very angry. “Grisha just listened,” he recalled. “He didn’t seem to have an opinion.” Except, of course, for the firmly held opinion that the discussion of politics was beneath the dignity of a mathematician.

  Anderson took Perelman to meet Jim Simons, the extraordinary man who had transformed the Stony Brook math department into one of the top such departments in the country and then become a hedge-fund manager and amassed impressive wealth, which he shared with many charities as well as with the university at Stony Brook. “So Simons made it clear that he’d like Grisha to come here—any terms he wants, any salary, or even one month a year,” said Anderson, “because Simons has the influence and money to make this possible. Grisha says, ‘Thank you, that’s very nice, but I don’t want to talk about this now. I have to go back to St. Petersburg to teach high-school students.’ He had a commitment in fall 2003.”

  Perelman’s answer might have been fully understood only by Perelman himself. A popular Russian joke tells of an actor courted by a major Hollywood studio. The actor is going to star in a film, and he is very excited, until he finds out that the filming is scheduled for December. “I can’t do it,” he says. “I have New Year’s parties,” meaning that he is scheduled to play Grandfather Frost (Santa’s Russian cousin) at children’s parties—and since he values this gig, he will have to pass up the opportunity of a lifetime. Perelman’s excuse sounded equally absurd and touching—but apparently it was just an excuse. From what I could tell, his sole commitment in the fall of 2003 was to attend a daylong math competition at a physics-and-math school in St. Petersburg,15 which he did, but which in no way would have precluded his accepting any of the many offers American institutions were making. The real reason he didn’t was simple: he abhorred the idea of being some department’s prized possession.

  Perelman went back to Russia at the end of April. He submitted the third and last in his Poincaré series of preprints16 on July 17; this time, it was just seven pages. The discussions went on without him. In June, Kleiner and his University of Michigan colleague John Lott started a Web page17 where they posted their notes on Perelman’s first paper. Toward the end of the year, the American Institute of Mathematics in Palo Alto and the Mathematical Sciences Research Institute in Berkeley held a joint workshop on the first preprint;18 Kleiner, Lott, Tian, and Morgan were its most active participants. In the summer of 2004, all four attended a workshop at Princeton sponsored by the Clay Institute, which, as the administrator of the million-dollar prize, had a stake in encouraging the appraisal of Perelman’s proof. Around the time of the Clay workshop, the four mathematicians most involved in closely reading the papers seemed to dispense with any residual doubts that the proof was correct. There were some mistakes, it seemed, and there were many gaps in the narrative Perelman presented, but none of this any longer seemed to challenge the assertion that Perelman had proved the Poincaré Conjecture and, probably, the Geometrization Conjecture (consensus on geometrization would come a bit later). Just as Perelman had predicted, this understanding came about a year and a half after his colleagues started studying his proof.

  Following the summer 2004 workshop, Tian and Morgan decided to collaborate on a book about Perelman’s proof; it was eventually published by the Clay Institute, which also funded Kleiner and Lott’s work. In the summer of 2005, the institute sponsored a month-long workshop on the proof. The study of Perelman’s preprints was turning into a mathematical cottage industry, which was just as it should be; many of the mathematicians involved had spent significant portions of their professional lives attacking these conjectures, and now each sacrificed the hope of a starring role for the opportunity to play a supporting part in the greatest mathematical production of the age.

  Had Perelman followed the more traditional route—had he written a conventional paper or papers and submitted them to a mathematical journal—his work could hardly have been subjected to any more scrutiny. A journal would have sent his papers to be reviewed by his peers—who, the world of topology being so small, would have been some of the same people who pored over his preprints now. The difference is that, as reviewers, they would have read the papers in private, not in a seminar, workshop, or summer-school setting, and they would have revealed the results of their examination in a letter to the journal rather than in notes posted on the Web for all interested parties to see. The process Perelman set in motion by posting his proof, in highly concentrated form, on the Web probably involved as many people as journal publication would have, but it turned out to be far more collaborative and public than the traditional procedure. It was also faster: before going public, Perelman did not take the typical months or years to frame his results in a traditional mathematical narrative. Perelman’s revolt against the conventions of scientific publishing was not based on an ideology; he simply had no use and therefore no regard for them.

  But outside of a traditional publishing framework, what were the roles of people like Kleiner, Lott, Tian, and Morgan, who had set out not only to understand but also to explain Perelman’s proof? In a sense, they became his coauthors. Perelman had coauthored one of his most important early papers in a similar manner. When I asked Gromov what it had been like to write an article with Perelman, he said, “It wasn’t like anything. I didn’t actually interact with him. Burago came here and we talked, and then Burago went back and they talked, and I guess Perelman wrote it up.”

  “So you didn’t look at the manuscript?” I asked, incredulous.

  “No.”

  “But wasn’t there the risk that someone would have gotten something wrong along the way?”

  “There is, there always is. It often happens that somebody writes part of the work and someone else writes another and it actually doesn’t come together. Some very well-known mathematicians have had bad articles like that.”

  “But that’s not a reason to read the manuscript?”

  “The manuscript? Of course not. It’s not interesting to read about work that you’ve already done. You do it—and you forget about it.”

  This was Perelman’s school. While Perelman was lecturing at Stony Brook, Kleiner and Lott found him as approachable and willing to engage in conversation about his proof as any mathematician could be. But when, toward the end of Perelman’s stay, Kleiner and Lott asked him whether he would look over their notes once they were done, Perelman said he would not. “He could have spent a half-hour sort of looking through and making some comment,” said Kleiner, who five years later still seemed perplexed by Perelman’s reaction. “That would be sort of the typical thing one might expect at the minimum. But, you know, he’s not a typical guy.” As Kleiner recalled, Perelman had explained that looking at their notes would make him in some way responsible for the work Kleiner and Lott had done. This was a perfect combination of Perelman’s exaggerated sense of personal responsibility and his equally solipsistic perception of the importance of any given mathematical problem. At the center of the universe in which Perelman stood, the Poincaré Conjecture was fading into the past. As Gromov said, “You do it—and you forget about it.” Perelman knew that months later, once Kleiner and Lott had finished their notes, he would no longer be interested in discussing the Poincaré.

  Kleiner and Lott went on to work on Perelman’s papers without Perelman. They found some problems along the way—at one point, in fact, Kleiner was convinced that they had found a serious, possibly fatal flaw, but Lott disabused him of this idea—and they found that even in the highly condensed preprints, Perelman had stayed true to his manner of relaying not so much the solution to the problem as the history of his own relationship with the problem. As Kleiner and Lott’s exploration moved toward the end of the first preprint, they realized that some of the earlier sec
tions of the paper were self-contained pieces that had no bearing on the eventual trajectory of the proof.

  In September 2004, following the Clay workshop, Tian sent Perelman an e-mail note “saying that we now understood the proof.” He pointed out that a year and a half had passed since their walk along the Charles River. Tian asked him if he would be publishing his preprints, for he and Morgan were now thinking of a book. Perelman did not respond. “He may think that he had done enough for publication by posting his preprints on the arXiv,” Tian suggested when he talked to me. “Or he could be already uncomfortable with me by that time. I tried to avoid talking to reporters, because first, I didn’t really enjoy talking to reporters; secondly, it takes time.” But in the spring of 2004 Tian had, at a friend’s request, broken his silence and talked to a freelance reporter for Science magazine—and now he suspected that Perelman was aware of this breach and had not written back for this reason. Most likely, though, Perelman simply had nothing to say. His prediction about the proof had come true, and he had never planned to publish his preprints—why would any further comment be necessary?

  Morgan had better luck with Perelman. In the Tian-Morgan tandem, it was Morgan who wrote to Perelman to ask mathematical questions. He was consistently amazed with the precision of the responses he received. “I would ask him a mathematical question and I would almost immediately get back the answer I was looking for,” Morgan told me. “Now a much more typical mathematical interchange is: You ask a question, the person you ask it to either doesn’t quite understand what you’re asking or because he’s coming at it from inside a different point of view answers it slightly obliquely from what you’re looking for. So then you ask it again. You reformulate it; refine it. And then maybe you get back an answer that is really what you’re looking for. That never happened with Perelman, I’d ask him a question and it was like he knew exactly what point I was confused about or didn’t understand, and exactly what I needed in order to clarify the situation.”