Perfect Rigour

by Masha Gessen

  Best regards,

  Grisha

  About a dozen U.S. mathematicians received this message. It said that the day before, Perelman had posted a paper on arXiv.org, a website hosted by the Cornell University Library and created for the express purpose of facilitating electronic communication among mathematicians and scientists. The preprint was the first of three papers that contained the results of Perelman’s seven-year attack on the Poincaré and Geometrization conjectures.

  “So I start looking at the paper,” Michael Anderson told me. “I’m not an expert on Ricci flow—nevertheless, looking through it, it became clear that he had made huge advances, that the solution to the Geometrization Conjecture and therefore the Poincaré Conjecture was within sight.” Every recipient of the e-mail had been on his own crusade against one of the problems for many years. Every one of them had a conflicted reaction to the news: if Perelman had indeed proved the conjectures, this was a mathematical accomplishment of monumental proportions, and it had to inspire a sense of triumph—but it was someone else’s triumph, and it dashed many mathematicians’ hopes for their own breakthroughs. Anderson had been working on Geometrization for almost ten years and was, as he told me, “getting bogged down in technical issues. I was still hoping that I’d have some insight or some breakthrough but really came to the conclusion that it wouldn’t happen. But if anybody was going to do it, good that it was Grisha. I liked him. So the next day I invited him to come here, and a day later I was really surprised that he said yes.”

  Meanwhile, a flurry of e-mails began to travel among American and European topologists. Mike Anderson sent out a few that read as follows:

  Hi [Name],

  Hope all is well with you. I don’t know if you’ve noticed yet but Grisha Perelman has put up a paper on the Ricci flow at mathDG/0211159 that you and your friends working in the area may want to look at. Grisha is a very unusual and also very bright guy—I first met him about 9 years ago, and we used to talk about Ricci flow and geometrization of 3-manifolds a fair amount in the early 90’s. Out of the blue he sent me an e-mail yesterday informing me of his paper.

  Basically, I know very little about the Ricci flow, but it seems to me he has answered, in this paper, many of the fundamental problems that people have been trying to solve. It may be that he is even very close to the solution of Hamilton’s goal, i.e. proving Thurston’s conjecture. The ideas in the paper appear to me to be completely new and original—typical Grisha. (He solved a number of other outstanding problems in other areas in the early 90’s and then “disappeared” from the scene. It seems he has now resurfaced.)

  Anyway, I wanted to inform you of this, and also ask if you could keep me “in the loop” on discussions/rumors regarding this work . . . Of course, what I’d really like to know is how close one now is to solving Thurston’s conjecture—since this affects some of my work a lot. I’m assuming here that his paper is correct—which to me is a reasonable bet, knowing Grisha.

  I’m sending a similar message to a few other friends I know working on Ricci flow.

  Best regards, Mike

  Someone who had never heard of Perelman might be forgiven for not taking the paper seriously: work claiming to prove the Poincaré Conjecture appeared regularly, yet in almost a hundred years, no one had solved the problem. Everyone, including mathematicians of great repute—indeed, including Poincaré himself—had made mistakes. Purported proofs appeared every few years, and all of them had been debunked—some sooner, some later. One had to know Perelman—be aware that he never produced lemons, as his math-club mates used to say, and have a sense of his propensity for the well-prepared gesture—to know just how seriously this particular attempt at the Poincaré should be taken.

  But how was one to determine whether it was in fact correct? The paper pulled together techniques and even problems from several distinct specialties inside mathematics; they were not even all limited to topology. In addition, Perelman’s presentation was so condensed that a judgment on his proof would first require, in essence, deciphering his paper. Nor did he help by stating up front what he proposed to do and how. He did not even claim he had proved the Poincaré and Geometrization conjectures until he was asked the question directly. Anderson’s e-mails were some of the first steps in starting this process of verification. This guy should be taken seriously, he was saying, and please let me know whether he has done what I think he has done. Anderson wrote this e-mail message at 5:38 in the morning the day after he’d received Perelman’s e-mail alerting him to the preprint.

  Within a few hours, Anderson started getting responses from geometers who apparently had also stayed up all night reading the paper. They reported that what the mathematicians called “the Ricci flow community” was in a frenzy—and noted that none of them had heard of Perelman before.

  None of the topologists with whom Perelman had been acquainted in the United States belonged to the Ricci flow community, which centered around Richard Hamilton—the most important addressee of Perelman’s e-mail announcement and, in a sense, of his entire paper. As the e-mails flew back and forth among geometers, Hamilton remained conspicuously silent. “Has there emerged yet any impressions of Perelman’s work?” Anderson wrote to another Ricci flow-er a few days later. “Are some of you in Hamilton’s group going over the paper? Does Hamilton know about it? Any ideas how close [Perelman] may be to finishing the program?”

  Hamilton knew about the paper, the correspondents reported. The paper appeared very important indeed.

  In fact, it took Perelman less than half of his first paper to get past the point at which Hamilton had been stuck for two decades. No wonder Hamilton was silent. One can only imagine what it must have felt like to see one’s life’s ambition hijacked and then fulfilled by some upstart with unkempt hair and long fingernails. One can imagine, that is, if one understands that ambition, competitiveness, and a sense of professional self-worth are what likely motivates human behavior—not, say, the best interests of mathematics. Grisha Perelman did not have that understanding.

  Indeed, one of the most remarkable aspects of the story of Perelman’s proof is the number of mathematicians who temporarily set aside their own professional ambitions to devote themselves to the deciphering and interpretation of his preprints. In November 2002, Bruce Kleiner was traveling in Europe. Just as he was about to begin a lecture at the University of Bonn, Ursula Hamenstaedt, a local professor who was in the audience, asked him: “Oh, by the way, did you see the preprint that Perelman just posted with proof of the geometrization of the Poincaré Conjecture?” At least, this was what he remembered her saying. She might in fact have been more cautious in her assessment—but Kleiner knew just how seriously Perelman was to be taken.

  “Nobody who knew his papers or had listened to his lectures had ever suggested he made claims that would later collapse, or would say things that he hadn’t thought through carefully,” Kleiner told me. “And here he was posting something on the arXiv, which is a very public forum. So, unless there was some personality change that had taken place since the early nineties, I thought there was a very good chance there was something there or maybe he had solved it completely.” And this meant that Kleiner’s professional life was taking a sudden turn. Like Anderson, Kleiner had for years been working on an aspect of the Geometrization Conjecture, though using an entirely different approach. Unlike Anderson, he did not yet suspect that his pursuit would prove fruitless. He did know that, as he put it, “it was a high-risk project,” a famous conjecture with which someone else might succeed sooner, but he was hardly prepared to hear, just before his own lecture, that his project was effectively over. For the next year and a half, Kleiner would be working on Project Perelman.

  Perelman, meanwhile, was preparing for his trip to the United States. He had received invitations from Anderson at Stony Brook and Tian, now at MIT, and he decided to spend two weeks at each place. He had told
Anderson at the outset1 that he would be in the United States no more than a month because he could not leave his mother alone for longer. The plan later changed to include his mother on the journey, but Perelman stuck to the original length of the trip.

  Perelman now seemed fully re-engaged with the world. He handled the U.S. visa formalities2—burdensome even for people seasoned in dealing with bureaucracies—on his own, securing visas for himself and his mother. He bought his tickets himself, apparently using money still left in his American bank account. He had been living frugally the past seven years, using his postdoc savings—he even added a footnote to that effect to his first preprint,3 obsessively true to his ideal of giving credit where credit was due, however irrelevant it was to the matter at hand. He corresponded with Anderson and Tian regarding the scheduling and logistics of his travels, including medical insurance, an issue that apparently concerned him a great deal.

  Perelman’s re-emergence from his near hermithood did not seem to impair his ability to continue writing up his proof. He submitted the second of his three preprints4 to the arXiv on March 10, 2003, while he was in the process of obtaining his U.S. visa. At twenty-two pages, this one was eight pages shorter than the first installment. He had apparently formulated the proof so clearly in his mind that distractions, minor and major, did not take away from his ability to devote a couple of weeks at a time to these concentrated write-ups (that spring he would tell Jeff Cheeger that it had taken him three weeks to write the first paper—less time than it had taken Cheeger to read and understand it).

  Perelman arrived at MIT at the beginning of April 2003. To Gang Tian, he looked more or less as Tian had remembered him: lean, long-haired, and with long fingernails, although without the brown corduroy jacket. To those who were seeing him for the first time, Perelman looked striking but entirely within the weirdness bounds of mathematicians. At his lecture, the hall was packed. A number of people in the audience had been reading Perelman’s first paper and writing their own notes on it; several of them were doing this in a seminar started by Tian. But a majority were curious mathematicians who had come to look at the man who might have made the biggest mathematical breakthrough in a century. These mathematicians were qualified to follow the narrative line of his lecture but would certainly have been unable to ask meaningful questions after the lecture—which made them, to Perelman, uninteresting at best and annoying at worst. He had banned videotaping of the lecture and had made it clear he did not want any media publicity, but a couple of journalists made it into the audience that day anyway.

  Almost incredibly, those who had come hoping for a mathematical spectacle got one. In sharp contrast to his speech at the 1994 international congress, Perelman presented an organized, lucid, and at times even playful narrative. He was at the peak of his relationship with the Poincaré Conjecture. If the Poincaré Conjecture were a person, this might have been the moment when Perelman would have chosen to marry it: a time when he could see their entire history together clearly, and when he was most free of doubt and most certain of the future.

  Almost daily for two weeks after his first presentation, Perelman gave talks on his work to smaller audiences. He spent several hours a day answering questions, mostly about the Geometrization Conjecture. In the mornings before his lectures, Perelman made a habit of stopping by Tian’s office to talk, mostly about mathematics. He may have been looking for new problems to tackle; he asked Tian about his own research and even floated some ideas related to Tian’s specialization at the time rather than to geometrization. Tian, unlike Anderson and Morgan—who regularly attempted to draw out Perelman—rarely ventured outside the narrow discussions of mathematical problems. “He was focused and very single-minded,” Tian told me. “I respect that he can ignore many things other people pay attention to and focus on doing mathematics.”

  Perelman seemed so relaxed and friendly during this visit that in one of their morning talks Tian broached the subject of Perelman’s staying at MIT. The university was interested in making an offer, and some colleagues of Tian’s had approached Perelman the previous evening and attempted to convince him that the resources of MIT would allow him to work more productively. Tian asked Perelman for his reaction. Whatever Perelman said to him in response, the polite, exceedingly soft-spoken Tian would not repeat to me. “He made some comments,” Tian allowed. “I don’t want to say them.” The problem was not just that this time Perelman had no interest in staying on in the United States. It was that the idea of being rewarded now with a comfortable university position insulted him. He had expected a full professorship eight years earlier. His brain had been the same then as it was now; he had been just as deserving; and yet they had wanted him to prove that he was good enough to teach mathematics. Now they acted as though he had finally proved it, when in fact he had proved the Poincaré Conjecture, which was its own reward.

  The two returned to their civilized discussions of manifolds, metrics, and estimates. Perelman’s irritation surfaced just once more in their discussions. The first of what Tian called “incidents” must have occurred April 15, toward the end of Perelman’s stay at MIT, when the New York Times published an article5 titled “Russian Reports He Has Solved a Celebrated Math Problem.” Just about every word in the title was an insult to Perelman. He had “reported” nothing; he had been careful to make his claims only in response to direct queries. To call the Poincaré Conjecture “celebrated,” and to do so in a mass-circulation newspaper, was, from Perelman’s standpoint, unconscionably vulgar. And the story itself heaped on the insults. The fourth paragraph of the article began, “If his proof is accepted for publication in a refereed research journal and survives two years of scrutiny, Dr. Perelman could be eligible for a $1 million prize.” This seemed to imply that Perelman had taken on the problem in order to win the million dollars—that he had any interest in the money at all—and that he would actually submit his work for publication in a refereed journal. All of this was demonstrably untrue. Perelman had started working on the conjecture years before the Clay prize was created. While he used money and had some appreciation for it, he felt little need and, certainly, no desire for it. Finally, his decision to post his proof on the arXiv had been an intentional revolt against6 the very idea of scientific journals distributed by paid subscription. And now that he had solved one of the hardest problems in mathematics, Perelman would not be asking anyone to vet his proof for publication.

  Before coming to the United States, Perelman had made it clear to those who asked—and Mike Anderson, for one, was very careful to ask—that at that point he did not want any publicity outside the mathematics community. Perelman did not say he never wanted publicity; he made it clear he did not think the time was right for it. And as strict as he was on the issue of speaking to journalists, he took a relaxed attitude toward publicizing his lectures and his work among colleagues: he was happy to let the organizers7 of his lectures use their professional mailing lists, or not, as they saw fit. He had an implicit trust in mathematicians of many stripes, and he had just as instinctive a mistrust of journalists. The New York Times article not only reinforced his suspicions of journalists—the author misinterpreted events and motivations in all the ways Perelman had probably feared—but also undermined his trust in his colleagues; one of the reporter’s two quoted sources was a mathematician who had attended Tian’s seminar and Perelman’s talks. Thomas Mrowka was no idle observer, yet he had offered an appraisal that served as the perfect kicker for the article and likely made Perelman cringe: “Either he’s done it or he’s made some really significant progress, and we’re going to learn from it.”

  On the day Perelman left MIT, he and Tian went across the river to Boston’s historic Back Bay to have lunch, which Perelman seemed to enjoy. Perelman even talked of the possibility of returning to the United States; he said he had offers from Stanford, Berkeley, MIT—in fact, by that point he could have had any terms he desired at any mathematics department
in the country. In a perfectly Bostonian flourish, the two mathematicians followed the lunch with a walk along the Charles River. Perelman’s relaxed state must have given way to anxiety, for he confided to Tian that things had gone sour between him and Burago—and, more generally, between him and the Russian mathematical establishment. Tian, again, would not reveal the details to me—saying only that he doubted that his friend was right this time—but the rupture was so much discussed in St. Petersburg that the details were easy to obtain. The conflict involved another researcher at Burago’s laboratory, one whose footnoting practices, Perelman believed, were so sloppy as to border on plagiarism. The man followed a generally accepted footnoting practice of referencing the latest appearance of an item rather than providing all the available truth on its origins. Perelman had demanded that the notoriously tolerant Burago subject his researcher to all but a public scientific whipping. In Perelman’s estimation, Burago’s refusal made him an accomplice to what amounted very nearly to a crime; Perelman’s screaming at his mentor had been heard8 in the halls of the Steklov. Perelman left Burago’s laboratory and found refuge in the lab of Olga Ladyzhenskaya, a remarkable mathematician who was old enough, wise enough, and woman enough to accept Perelman9 just as he was. Everyone else—including Burago and Gromov, who generally saw Perelman as nearly faultless—seemed willing to forgive him, but they were incapable of seeing his approach to footnoting as anything but10 capricious at best and meanly ridiculous at worst.

  Following his lectures at MIT, Perelman went to New York City, where his mother was once again staying with relatives. He stayed the weekend and traveled to Stony Brook by train on Sunday evening. Mike Anderson picked him up at the station and delivered him to the dormitory where he was staying; Perelman had explicitly requested that his accommodations be “as modest as possible.”11 He started lecturing the following day, establishing a consistent schedule for the next two weeks: morning lectures followed by afternoon discussion sessions. To those attending, these sessions seemed nothing short of a miracle. Here was a man some of them had never heard of and others had believed had vanished who had slain the Poincaré and now exhibited fantastic clarity in his lectures and unparalleled patience during the discussions.12