Perfect Rigour

by Masha Gessen

  You could certainly say that. Cheeger encountered this aspect of Perelman’s personality when he tried to convince the younger mathematician to expand one of his papers to allow more exposition of his ideas. “One of the papers he wrote while he was here was very short; it was a mixture of power and arrogance. It was very striking. I read it and admired it a lot. But I felt it was a little bit too terse, a little bit not making the insights as manifest as they could be. So I said this to him and he said he would consider it. But I couldn’t really get him to change. I don’t know. Have you seen the film Amadeus?” The scene Cheeger was recalling was the one in which Mozart presents an opera he has written and the emperor suggests the piece is wonderful but not perfect: it has too many notes. “Just cut a few and it will be perfect,” he says. “Which few did you have in mind, Majesty?” responds Mozart. By 1992, Perelman was apparently quite certain he was the Mozart of contemporary mathematics. No one, not even an outstanding mathematician twenty-three years his senior, was going to tell him what to do or how to present his ideas to the world.

  For the spring semester of 1993, Perelman went to the State University of New York’s Stony Brook campus—one of the best American graduate programs in mathematics.6 Located just sixty-five miles from New York City, Stony Brook was probably as different from St. Petersburg and New York as any place Perelman had ever visited. Its architecture was square, and its landscape consisted of parking lots, low buildings, and large fields. Its railroad station was a tiny two-room structure across the tracks from campus. To an outsider—and Perelman would always be an outsider, wherever he went—it must have felt utterly desolate.

  Mike Anderson,7 a geometer Perelman had met earlier—currently the director of SUNY Stony Brook’s graduate math program — helped Perelman find an apartment. Perelman’s criteria were “quiet and small,” and he found a studio apartment that cost roughly three hundred dollars a month. He slept on a futon he borrowed from the Andersons. The pay for a postdoc at the time was about thirty-five to forty thousand dollars a year, and Perelman, who lived on bread and yogurt, put most of that money away in his bank account. His mother stayed in Brooklyn but came to visit frequently.

  Perelman continued to wear the same brown corduroy jacket. People continued to notice his long hair and fingernails. His personal hygiene may have deteriorated slightly; he gave the impression of someone who bathed regularly, but the futon on which he slept took on a smell so strong that the Andersons had to throw it out when he returned it. His extraordinary long nails, however, remained clean.

  Perelman taught a course on Alexandrov geometry. The following summer, he traveled to Zurich to speak on Alexandrov spaces at the International Congress of Mathematicians.8 It was a prestigious opportunity; the congress took place just once every four years, and that year only fifty-five of the world’s top mathematicians,9 most of them significantly older than Perelman, had been invited to speak—four Fields Medalists, past and future,10 among them. With his proof of the Soul Conjecture, Perelman had become an undisputed young star. In Zurich, he spoke on the paper he had coauthored with Gromov and Burago. His first talk at the congress probably attracted people who wanted to see the twenty-eight-year-old who, if Gromov was to be believed, was doing the best work in the world in his field. But Perelman apparently exhibited the worst of his public-speaking habits during the talk. He started by sketching something on the board and then began pacing back and forth as he talked. His speech seemed vague and disconnected and essentially incomprehensible.11

  If Perelman was true to his habit of describing his personal relationship with the problem rather than the problem itself, it might explain why his Zurich talk was a disaster. He had lectured on this paper before—at the Geometry Festival at Duke in 1991, and at a couple of American universities immediately following the festival. He had been clear at the time—as the geometer Bruce Kleiner,12 who heard him speak at both Duke and the University of Pennsylvania that year, recalled, it was obvious that “the mathematics was very, very good.” But by 1994, his relationship with Alexandrov spaces had grown complicated.

  After a semester at Stony Brook, in the fall of 1993, Perelman moved to the West Coast to take up a two-year Miller Fellowship—an enviable position at the University of California at Berkeley that offered generous funding for research in one of the basic sciences without any teaching responsibilities. In fact, the conditions of the fellowship stated explicitly13 that fellows were “granted more independence than other postdocs on campus” and could participate in the lives of their host departments as much or as little as they desired. This was the kind of setting for which Perelman had been raised by his early mathematical mentors—the kind of setting he had praised in his conversations with Russian colleagues—but it did not work. Or something didn’t work. Perelman had been trying to press on with Alexandrov spaces, and he had gotten stuck.

  “That’s normal,” Gromov told me. “Out of everything you try, most things don’t work out. That’s just the way life is.” Gromov might have been talking about life in mathematics or life in general, but in either case, he was speaking from experience, which Perelman, even in his late twenties, simply did not have. Improbably, with the possible exception of his second-place showing at the All-Soviet Math Olympiad at the age of fourteen, he had never failed to accomplish what he had set out to accomplish, or to receive what he was due, or to solve a problem he had taken on. Moreover, all the hours of practice and all the behind-the-scenes anxiety and intrigue notwithstanding, in the eyes of observers, he had accomplished everything with ease. At this point, following the Soul Conjecture proof and the international congress, he had more mathematical eyes trained on him than ever before—and he was facing the unfamiliar experience of failure.

  Kleiner spent the 1993–1994 academic year at Berkeley too, and he and Perelman “had several math conversations during that year,” he recalled. Perelman occasionally ventured into areas adjacent to Alexandrov spaces. He talked about the Geometrization Conjecture, a long-unsolved problem that included the Poincaré Conjecture; that is, if someone proved Geometrization, Poincaré would also be proved along the way. He talked about the possibility of applying Alexandrov spaces to Geometrization, and “there was no obvious way or scheme,” said Kleiner. Perelman also considered dipping into Ricci flow, an approach invented by another mathematician to prove the Poincaré Conjecture—but that mathematician had himself gotten stuck years before. Perelman wondered out loud whether Ricci flow might be applied usefully to Alexandrov spaces. Had there been any indication that Perelman might actually take up the Poincaré and Geometrization conjectures? No — but, recalled Kleiner, “he was not very open about what exactly he was working on or thinking about. He was no more reticent than many people would be in a similar situation. It’s not necessarily a good idea to share your ideas openly because, unless you really know the person and trust them, they could start working on it themselves or they could pass information on to a third party who might start working on it. You’ll find someone competing against you using your same ideas, which is not a very comfortable situation.” Kleiner’s own area of research lay quite near Perelman’s, so Perelman’s reticence seemed reasonable to him.

  But there was probably another reason for the reticence, one that Perelman articulated in a conversation with Cheeger in 1995. As Cheeger recalled, Perelman stopped by his office while he was briefly in New York City and they discussed some issues related to Alexandrov spaces but not to the specific aspects Perelman had studied in the past. This time, however, Perelman was very interested and even referred to one of the questions as the “holy grail” of the subject. “And I ask him, ‘Didn’t you say you had no interest in it?’” Cheeger recalled. “And he said, ‘Well, whether a problem is interesting depends on whether there’s any chance of solving it.’” As pompous as that statement sounds, Perelman was probably telling an important emotional truth about himself: he could become engaged with a problem
only if he could fully grasp it—and if he grasped a problem fully, down to the nature of every minute technical complication, he could certainly solve it. What had happened between Perelman and Alexandrov spaces was that he had come up against technical difficulties he could not penetrate, and so he had grown emotionally disengaged. Hence the nebulous, rambling talk at the congress.

  Perelman’s term as a Miller Fellow ended in the spring of 1995. His paper on the Soul Conjecture had come out the previous year, and he had spoken at the International Congress of Mathematicians, so it is not surprising that even though he put no effort into securing an academic position after Berkeley, he was courted by several leading institutions. He turned all of them down, and the way he did it—specifically, the way he rejected Princeton—has become part of American and Russian mathematical lore. I had heard about it on both sides of the Atlantic before I asked one of the immediate participants what had happened, and his account differed little from what I had been told.

  Peter Sarnak,14 a Princeton professor who became chair of the mathematics department in 1996, first heard of Perelman from Gromov, who, Sarnak recalled in an e-mail message,15 had said Perelman was “exceptionally good.” In the winter of 1994–1995, Perelman came to Princeton to give a talk on his proof of the Soul Conjecture. Few people showed up, but the math department’s brass was there: distinguished professor John Mather, then–department chair Simon Kochen, and Sarnak all attended. Perelman gave a great lecture: clear, precise, and engaging—probably because his personal relationship with the Soul Conjecture had been brief and satisfying and was resolved. “After the lecture the three of us approached Perelman saying we would like to arrange for him to come to Princeton as an assistant professor,” recalled Sarnak. Legend has it—though Sarnak did not remember it—that at this point Perelman asked why they would want to bring him to Princeton when no one there was interested in his areas of research—an impression perhaps intensified by the nearly empty auditorium and which, Sarnak acknowledged, was an accurate reflection of the situation, “which we were eager to change.” Sarnak remembered Perelman making clear “that he wanted a tenured position, to which we responded that we would have to look into that and in any case we need some information from him such as a CV. He was surprised by the latter, saying something like ‘you have heard my lecture, why would you need any more information?’ Given that he wasn’t interested in a tenure track position we didn’t pursue this any further. History has proven that we made a mistake in not being more aggressive in recruiting him.”

  Perelman told several people at the time16 that he would settle for nothing less than immediate tenure—an audacious position for a twenty-nine-year-old mathematician with few publications and only a semester’s worth of teaching experience. But Perelman’s own logic was impeccable. He was not out looking for work, so the job offers were coming from institutions—or, rather, people—who, as Cheeger put it, “knew how terrific he was.” In other words, they knew what Perelman and Gromov knew: that he was the best in the world. Why, then, would they want to put him through the conventional paces of earning his full professorship? Why even make him submit his CV before they offered him his well-deserved job? It would not have occurred to Perelman that his well-intentioned interlocutors did not perceive his place in the mathematical hierarchy quite the same way he did and simply did not realize that his would be a star presence in any university mathematics department. Or his insistence on immediate tenure might merely have been a way of setting the bar so high as to cut off any further discussion of his staying in the United States. The University of Tel Aviv, where Perelman’s sister was by then a student, actually offered him a full professorship, and Perelman, as Cheeger recalled, “ended up turning them down or not responding at all.” So Sarnak might take some consolation in the knowledge that even if Princeton had been more aggressive, it probably would not have succeeded in drawing Perelman.

  Getting ready to return to Russia, Perelman told his American colleagues he could work better back home—the exact opposite of what he had told his family in Russia three years earlier but in all likelihood the exact same sort of solipsism. Back when breakthroughs came easily to him, his American environment had seemed to be on his side; now that he was stuck, a return to Russia held the promise of rejuvenation, a renewed ability to work. What it was he was working on, no one knew. The questions he asked Cheeger when he was passing through New York on his way to St. Petersburg in 1995 seemed to indicate he was broadening his focus on Alexandrov spaces—in a way that, in retrospect, may have meant he was edging closer to tackling the Poincaré Conjecture.

  Back in St. Petersburg, Perelman took up residence in Kupchino with his mother and reclaimed his spot in Burago’s laboratory at the Steklov Institute. He would not have any teaching responsibilities—or, for that matter, any obligations at all. By the mid-1990s, institutions of the Russian Academy of Sciences had fallen into physical disrepair and organizational chaos. Researchers no longer had to submit regular reports on their work or account for their time in any way; institute rolls gradually filled up with dead souls — or, in any case, long-absent émigré souls. Buildings, which had been maintained at architectural subsistence levels in the Soviet era, literally began to crumble after about five years of neglect. The Steklov building in St. Petersburg, a once-lovely low-rise structure on the Fontanka River in the very center of town, grew increasingly cold and drafty. Researchers’ salaries were so far out of sync with inflation as to be laughable; many people did not even bother showing up at their institutes to pick up the wads of worthless cash that was their pay. They sought sources of income elsewhere — mostly in the West, where many stayed all the time while others created complicated schedules of semester-on/semester-off teaching. But none of this bothered Perelman. At the institute, there was heat, and there was electricity, and the phone lines worked—most days, anyway. At home, his mother catered to his ascetic needs. The subway continued to run from the center of town to Kupchino. And Perelman had saved tens of thousands of dollars while he was in the United States; in 1995, a family of two in St. Petersburg could live well enough on less than a hundred dollars a month. It seemed he would never again have to worry about anything but mathematics. With the distraction of exams, competitions, dissertation, and teaching behind him, he would lead the life he had been raised to live: the life of the pure mathematician.

  Whatever patience he had once had for distractions was now gone. In 1996, the European Mathematical Society held its second quadrennial congress17 in Budapest and awarded prizes to mathematicians under the age of thirty-two. Gromov, Burago, and St. Petersburg Mathematical Society president Anatoly Vershik18 submitted Perelman’s name for his work on Alexandrov spaces. “I was always interested in making sure that our young mathematicians looked good,” Vershik explained to me. “They decided to award it to him, but as soon as he learned about it—I don’t remember whether I was the one who told him or if it was someone else—he said he did not want it and would not accept it. And he said that he would create a scandal if it was announced that he was a recipient of this prize. I was very surprised and very upset. He had in fact known that he was up for the award and had said nothing about this. I had to have some emergency communication with the chairman of the prize committee, who was an acquaintance of mine, to make sure they did not announce the prize.”

  A dozen years after the incident, Vershik, a soft-spoken, bearded man in his early seventies, still seemed to feel betrayed by Perelman’s behavior. He told me he would rather refrain from trying to find the reason for Perelman’s rejection of the prize. If Perelman was opposed to prizes on principle, this was news to Vershik: in the very early 1990s the Mathematical Society had awarded Perelman a prize, which Perelman had accepted; he even gave a talk on the occasion. Later Perelman apparently told someone that the European Mathematical Society had no one who was qualified to judge his work, but Vershik did not recall hearing anything of the sort then—and with G
romov and Burago on board, that would have seemed an odd argument. “He did say one thing to me at the time, and it actually sounded convincing. He said the work was not complete. But I said there were reviewers and the jury had decided he deserved the prize.” Still, the idea that anyone might be better suited than he was to judge whether a paper of his deserved a prize could only have infuriated Perelman.

  Unlike Vershik, Gromov thought Perelman’s behavior entirely acceptable, even though Gromov had been one of the three mathematicians who had submitted Perelman’s name for the prize. “He believes he is the one who decides when he should be getting a prize and when he should not be,” Gromov told me quite simply. “So he decided that he had not fulfilled his program and they can just take their prize and stuff it. And, of course, he also wanted to show off.” Or at least show that he wanted to be left alone.

  He continued to accept invitations to take part in mathematical community events, especially ones involving children. Apparently this was not so much because he had any affection for children as it was that he had respect for the tradition of the clubs and competitions in which he had been reared. But Perelman grew increasingly resistant to entertaining any questions concerning his projects. His American colleagues soon discovered he did not answer e-mail messages. In 1996, Kleiner went to St. Petersburg for a conference on Alexandrov spaces that Perelman also attended. Even though the two men had had a few mathematical conversations at Berkeley a couple of years earlier, Kleiner could not find a way to approach Perelman with questions about his current research. A friend of Kleiner’s, a German mathematician named Bernhard Leeb, who had met Perelman at the International Mathematical Olympiad, did manage to ask a question—but not to get an answer. As Kleiner recalled twelve years later, Perelman said to him, “I don’t want to tell you.” Leeb’s own recollection differed in tone if not substance. “I did ask him what he was working on,”19 he wrote to me. “He told me that he would be working on some topic in geometry but he did not want to become specific. I find this attitude very reasonable. If one is working on a big problem like the Poincaré Conjecture, one is well advised to be extremely reluctant to talk about it.”