Perfect Rigour

by Masha Gessen

  Zalgaller called Gromov “the best thing Leningrad University ever produced.” Gromov defended his PhD dissertation there in 1968, at the age of twenty-five; his adviser was Vladimir Rokhlin,4 the topologist whom Alexander Danilovich Alexandrov had saved from persecution. Gromov, whose mother was Jewish, despaired of getting a research position5 at the Steklov or even a less desirable, to him, professorial appointment at Leningrad State, and in the late 1970s he emigrated to the United States, where he worked at the Courant Institute at New York University. Later, having established himself as one of the world’s leading geometers, he started dividing his time between the Courant and the extraordinarily prestigious Institut des Hautes Études Scientifiques, outside of Paris.

  I interviewed Gromov in Paris at the Institut Henri Poincaré, the part of the Université Pierre et Marie Curie reserved for conferences and seminars in mathematics and theoretical physics. So said the university’s website6 and so said laminated signs placed on the large round wooden tables in the institute’s cafeteria: RESERVED FOR THE USE OF MATHEMATICIANS AND THEORETICAL PHYSICISTS. As I arrived in the cafeteria, I saw Gromov7 engaged in an animated discussion with the American topologist Bruce Kleiner, whom I had interviewed in New York a couple of months earlier. Kleiner rose to leave when I approached the table but seemed too agitated by the discussion to say hello to me. Instead, he turned back to face Gromov and said that a science in which nothing had to be proved was no science at all. Gromov responded that an alternative system could still be consistent. “Have you ever talked to a street person?” Kleiner demanded, apparently infuriated. “They have some great ideas.” I think he meant to say something about every crazy person having an internally consistent system to offer, but Kleiner had become too upset to articulate the idea. Gromov too became enraged, waving his arms and saying, “No, no!” He looked very much like a street person himself: his clothes hung loosely and awkwardly on his very thin frame; his black belted jeans were stained; his light green button-down shirt had thinned on the chest and frayed at the cuffs; and both his gray beard and his gray hair stuck out haphazardly in every direction.

  Kleiner stomped off, and Gromov turned to me, apparently still irritated. First, he bristled at my questions about his reasons for leaving the Soviet Union. “Why not?” he asked, speaking Russian that was distinctly affected by the three decades he had spent away from the mother country. “Everyone was leaving, and I left too. I was offered a job in America and went there, and then I was offered a job here and came here.” I already had enough information to know he was not exactly telling me the truth, but I also knew not to push: he was obviously in no mood to talk about the long-ago hardships of Jewish emigration from the Soviet Union.

  “I understand you are the person who brought Perelman to the West,” I attempted.

  “I took part in it,” Gromov responded, still annoyed. “But it was Burago’s initiative.”

  “A lot of people have told me that you were the one who came and said there was this great new mathematician.”

  “Burago told me that. I may have mentioned it to other people.”

  “And what had Burago told you?”

  “He said he had this good young mathematician.”

  “Who needed to be brought here?”

  “Yes, it had to be arranged for him to come here.”

  Gromov arranged for Perelman to spend a few months at IHES as soon as he defended his dissertation at the Steklov in 1990. At IHES Perelman began working on Alexandrov spaces, a topological phenomenon named for Alexander Danilovich Alexandrov. The old man had abandoned this topic in the 1950s, but now three of his mathematical descendants—Burago, Gromov, and Perelman—had come together to work on it.

  In 1991, Gromov helped bring Perelman to the Geometry Festival,8 an annual event held on the East Coast of the United States at a different location each year. That year it was at Duke University. Perelman gave a talk on Alexandrov spaces that the following year became his first major published work,9 coauthored with Gromov and Burago. Gromov mentioned Perelman to all the right people10 to ensure that he would be invited to do postdoctoral research in the United States.

  As Gromov and I talked, I began to understand Gromov’s motivation—or, rather, the depth of his commitment to the Perelman project. “When he entered geometry,” Gromov said, “he was, at the time, the strongest geometer. Before he went underground, he was certainly the best in the world.”

  “What does that mean?”

  “He did the best work,” Gromov responded with perfect precision. I immediately remembered a joke told to me by a mathematician: A group of people taking a ride in a hot-air balloon get carried away by the wind. After drifting some distance, they spot a man below and shout to him, “Where are we?” The man, who happens to be a mathematician, responds, “You are in a hot-air balloon.”

  But as we talked more, I realized Gromov thought Perelman was actually the best man in the world—not just the best geometer, but the best human being involved in mathematics. Gromov compared Perelman to Isaac Newton, then immediately amended the comparison to say, “Newton was a rather bad person. Perelman is much better. He has some faults, but very few.” His faults, Gromov explained, sometimes led him to attack his friends, but these conflicts were minor compared to Perelman’s overwhelming natural goodness. “He has moral principles to which he holds. And this surprises people. They often say that he acts strange because he acts honest, in a nonconformist manner, which is unpopular in this community—even though it should be the norm. His main peculiarity is that he acts decently. He follows ideals that are tacitly accepted in science.”

  In other words, Perelman was what a mathematician—and a man—ought to be. Later that day I walked around Paris with a French mathematician and historian of science11 who bemoaned the state of French mathematics, the commercialization of science, and the unprincipled participation of people like Gromov who, this man claimed, stood by while IHES printed up vapid fundraising brochures. I realized that Gromov probably wished he could be as principled as Perelman, as resolutely removed from the institutionalization of mathematics, and as sincerely disdainful of empty recognition. Which was clearly why Gromov had adopted Perelman as a cause—and also why he resisted taking credit for having helped him.

  So continued the line of Perelman’s guardian angels: Rukshin shepherded him into competitive mathematics, Ryzhik coddled him through high school, Zalgaller nurtured his problem-solving skills at the university and handed him off to Alexandrov and Burago to ensure that he practiced mathematics uninterrupted and unimpeded. Burago passed him on to Gromov, who led him out into the world.

  PERFECT RIGOR

  ROUND TRIP

  7

  Round Trip

  HAD GRIGORY PERELMAN been born ten or even five years earlier than he was, his career may well have ground to a halt around the time he finished writing his dissertation: it would have been difficult, if not outright impossible, for a Jew to defend his dissertation at the Steklov and stay on in a research position there; even the intervention of someone as influential as Alexander Danilovich Alexandrov might not have guaranteed success. Had Perelman been born ten or even five years later than he was, he might never have entered graduate school at all: State anti-Semitism would no longer have been an issue, but his family would likely have been unable to afford to keep him in school at a time when a graduate student’s stipend barely bought three loaves of black bread. But Grisha Perelman was born at just the right time, and when he completed his dissertation, he was in exactly the right place: a country that was collapsing, which freed its citizens to travel abroad for the first time in seven decades. He belonged to the luckiest generation of Russian mathematicians. Like millions of other Soviet citizens, Perelman began a new life sometime around 1990, a life out in the world. So fortuitous was the timing of this change that Perelman might be forgiven for believing the world w
orked exactly as it should. Just when Perelman needed to broaden his circle of mathematical communication, the opportunities to do so presented themselves.

  In this new part of Perelman’s life, a new cast of characters appeared. Whether they knew it or not—and most likely they did not, for Perelman was as reserved with them as he was with most people—and whether he cared or not, they would play important roles in the development of his career. In addition to Gromov, these included Jeff Cheeger, Michael Anderson, Gang Tian,1 John Morgan, and Bruce Kleiner.

  Cheeger is an important American mathematician, a generation older than Perelman. He works at the Courant, in a large sparse office in a high-rise building on the NYU campus. Like other American acquaintances of Perelman, Cheeger seemed to find him both sympathetic and inscrutable, if occasionally slightly infuriating, and he spoke carefully, hoping to avoid offending him. Cheeger recalled that he first heard about Perelman from Gromov: “He came back and mentioned that he had met this young fellow who was extraordinarily impressive.” In 1991, Cheeger saw Perelman at the Geometry Festival at Duke. And then Perelman came to the Courant as a postdoctoral fellow in the fall of 1992. He was still working on Alexandrov spaces.

  By the time Perelman arrived in the United States, he was twenty-six, no longer pudgy but tall and apparently fit. His beard had passed out of its extended awkward-tuft stage and was thick, black, and bushy. His hair was long. He did not believe in cutting hair or fingernails—some people thought they remembered his saying something about the unnaturalness of such trimming, but no one can vouch for this recollection and chances are at least as good that Perelman found the conventions of personal hygiene and appearance both taxing and unreasonable. “He was very, you know, known as eccentric,” said Cheeger, citing the nails, the hair, the habit of wearing the same clothes every day—most notably a brown corduroy jacket—and his holding forth on the virtues of a particular kind of black bread that could be procured only from a Russian store in Brooklyn Beach, where Perelman walked from Manhattan.

  Structurally, the life of a postdoc in the United States differed little from the life of a graduate student in Russia. Perelman was left largely to his own devices, but he apparently saw no reason not to spend most of his time at the Courant. The institute was conveniently located in a concrete-block tower as square and impersonal as anything that had been built in Russia in the previous thirty years. It looked out on Washington Square Park, a place as flat, geometrical, and ceremoniously architectural as any park in St. Petersburg or Paris, where Perelman had just spent several months. To complete his sense of familiarity, Perelman had to travel to the outer reaches of Brooklyn to get his bread and fermented milk—and by making the journey on foot, he ensured that he had both solitude and the usual measure of physical hardship. After a while, he had his mother waiting for him at the other end of the journey to Brooklyn; she had followed him to the United States and was staying with relatives in Brighton Beach. Within Courant Institute itself, Perelman did not find the social demands taxing; the typical regimen of mathematics seminars was accompanied by a familiar array of faces, since Gromov, Burago, and other St. Petersburg mathematicians were occasional residents there.

  Perelman made a friend at Courant. I am not sure Gang Tian knew he was Perelman’s friend, but Viktor Zalgaller, Perelman’s old teacher in Israel, was certain he was. “He made a friend there, a young Chinese mathematician,” he told me. “They suited each other.” This they certainly did. I went to see Tian at the Institute for Advanced Study at Princeton, one of the world’s most prestigious mathematical institutions, where Tian now occupied another cold concrete box. He spoke very softly and sadly, if not as reluctantly as Cheeger. He had already made the mistake of speaking to the media, and he believed this was why Grisha had not responded to his letters in several years. Tian did not think he and Perelman had been friends. “We talked quite often,” he acknowledged, but it was all about math. “I don’t think we talked much other than that. There are probably other people with whom he was friendly and talked more about other things. He did talk about bread. He somehow cared a lot about bread. He found a place to buy good bread in Brooklyn and near the Brooklyn Bridge.” What kind of bread was it? I asked. “I’m not so sure,” responded Tian, “because I’m not that fond of bread. I eat bread but I don’t really care which one.” Aside from the bread issue, Tian and Perelman really were perfect for each other: both were interested in little outside of mathematics, and their mathematical interests were shared.

  It was with Tian that Perelman started going to lectures at the Institute for Advanced Study at Princeton. Cheeger came along too. During one of these visits Perelman surprised Cheeger by joining a game of volleyball after a lecture. “You look at him and think this is something he’d have no interest in or couldn’t do,” recalled Cheeger. “But I remember one time watching it, the game, and he said, you know, ‘Well, I think I could do that.’ And you know, he was pretty good.” I nodded. My lack of surprise surprised Cheeger. I explained that Perelman had had to take part in numerous games of volleyball while he was training for the International Mathematical Olympiad as well as when he was at the math camps. Then Cheeger looked slightly annoyed. Even on this small score, he had been misled by Perelman’s habit of underplaying both his abilities and his interest. This was, of course, the same man who later told no one he was working on the Poincaré Conjecture and who posted his solution on the Internet without claiming it was in fact the solution. It was only after someone asked him if he had proved the conjecture that he said he had. Most likely, if Cheeger had asked him directly whether he had had much volleyball practice, Perelman would have said yes. He still believed in telling the entire truth—but only when asked. He just didn’t see the utility of volunteering information, especially information about himself. I suspect he also took some pleasure in demonstrating that he could solve any problem he picked—even a game of volleyball.

  Another Perelman incident that surprised Cheeger during the New York period was harder to explain. In 1993 Cheeger and Gromov went to a conference in Israel that had been convened in part to celebrate their fiftieth birthdays. Perelman came, as did his mother—but that was not what surprised Cheeger. What he found startling was that he saw Perelman renting a car at the airport, using a credit card. I have talked to no one else who ever witnessed Perelman drive a car—indeed, some people claimed he rejected cars as “unnatural”—but it is conceivable he could have obtained a driver’s license and a credit card during his first semester in New York. The reason he might have done this was that, for a fleeting moment, Perelman seems to have planned to move permanently to the United States.

  “You see, it often happens that when someone crosses the border with Russia in any direction, he has a very strong reaction,” Golovanov explained to me. “In Grisha’s case, it was the only time he experienced something resembling political enthusiasm. As soon as he ended up there, he started sending letters decreeing that the entire family had to move.” The entirety of the family remaining in St. Petersburg at this time was Grisha’s younger sister, Lena, who had just graduated from high school. Their father had emigrated to Israel, and their mother was in New York hovering over Grisha, so in essence, he was campaigning for his sister to go to college in the United States. Lena decided to move to Israel, where she obtained her PhD in mathematics from the Weizmann Institute2 in 2004.

  To the best of Golovanov’s recollection, Perelman did not try to make a case for the move: he “decreed” it, as Golovanov put it, in accordance with his understanding of his role in the family, which was to “know what is right.” Making an argument to his little sister may also have seemed to him to be beneath his dignity or, in any case, a waste of time. When he talked to colleagues, however, he made the argument that Western mathematicians, while suffering from too narrow a focus3 compared with their Russian counterparts, organized their research more effectively and accomplished more. This may have been a classic
solipsism, for in 1993 Perelman did exactly what postdoctoral researchers who are unencumbered by formal academic obligations and are at the height of their creative and mental abilities are supposed to do at that stage in their lives: he solved an important long-standing problem, and he did it in a way that, to mathematicians, possessed breathtaking beauty.

  Twenty years before Perelman arrived at Courant, Cheeger and his coauthor Detlef Gromoll had published a paper4 outlining a way of deducing the properties of certain mathematical objects from small regions of these objects, which they called the soul of the objects, for, like the imaginary human soul, the imaginary soul of the imaginary mathematical object also possessed all the qualities that made the whole what it was. Cheeger and Gromoll proved part of what they set out to show, and this became known as the Soul Theorem, but they could only suppose the rest, and this became known as the Soul Conjecture. It remained a conjecture—that is, a mathematical supposition without proof—until Perelman showed that it was true. His paper was four pages long.5

  “It seemed to be extraordinarily hard,” Cheeger told me. “At least a couple of people had written very long and technical papers on it. And they only proved part of it. And he realized that everyone had been missing the point, you could say. And he made a very short proof of it. He used something—something nontrivial, but something that had been in the public domain since the late seventies.”

  This was the trick Perelman’s friends at the math club had called his “stick”: absorbing the problem in its entirety and then boiling it down to an essence that proved simpler than everyone had assumed. “Part of it was that the problem was not as difficult as people had thought it was,” Cheeger continued. “Part of it was, you could say, the force of his personality. I mean when you talked to him it was clear you were dealing with an unusually penetrating and powerful mind. A personality that’s very forceful in a certain direction, very believing in his own insights. You could say almost stubborn in a way, not aggressive, but you could almost say a little arrogant.”