Perfect Rigour

by Masha Gessen

  His trouble with relating his solutions may also be interpreted in this light. If Perelman has Asperger’s, the lack of an ability to see the big picture may be one of his curious shortcomings. British psychologists Uta Frith and Francesca Happé have written on what they call “weak central coherence,”19 a quality that characterizes the thinking of people with autism-spectrum disorders, who focus on detail to the detriment of the big picture. When they are able to arrive at a big picture, it is usually because they have arranged elements—say, the elements in the periodic table—in a pattern, which systemizers find extremely satisfying. “The most interesting facts are those which can be used several times,20 those which have a chance of recurring,” Henri Poincaré, one of the great systemizers of all time, wrote more than a hundred years ago. “We have been fortunate enough to be born in a world where there are such facts. Suppose that instead of eighty chemical elements we had eighty millions, and that they were not some common and others rare, but uniformly distributed. Then each time we picked up a new pebble there would be a strong probability that it was composed of some unknown substance . . . In such a world there would be no science . . . Providentially it is not so.”

  Aspergians learn the world pebble by pebble, ever grateful for the periodic table that allows them to recognize patterns of pebbles. Discussing the existence of Aspergians in the social world, Attwood used the metaphor of a “jigsaw puzzle of 5000 pieces,”21 where “typical people have the picture on the box of the completed puzzle,” which accounts for their social intuition. Aspergians do not have that picture and have to painfully assemble the puzzle by trying to fit the pieces together. Perhaps rules such as “never untie your fur hat” and “read the books on the school’s reading list” were Grisha Perelman’s attempts to envision the missing picture on the box, elements of his periodic table of the world. Only by sticking to them could he live his life.

  The amount of human interaction in which Perelman engaged had been dwindling for eight years. Whatever social skills he had once had—he had exercised them in graduate school and as a postdoc, and they had been adequate, though minimally nuanced—had grown rusty with disuse. So had his tolerance for the behavior of others. Aspergians, it appears, are by and large capable of adjusting to social relationships, though this does not come naturally to them, as it does to neuronormals. John Elder Robison, the author of a memoir of life with Asperger’s, described the process as a tradeoff: socialization seemed to rob the person22 of some of his extraordinary powers of systemizing concentration. Conversely, intense concentration over the course of several years seemed to have robbed Perelman of any social skills he had had. One can imagine how grating he had found the heated political argument between Cheeger and Anderson at Anderson’s party, how disinclined he was to engage in anything superfluous and how completely unwilling to entertain any ironies, real or imagined, connected with his work—such as the idea that his proof might drive people away from topology. And he had had such high expectations. He had given mathematics something great, something truly valuable. Mathematics had responded feebly, trying to convince him to accept substitutes for true recognition. No wonder he was disappointed in mathematics.

  For the moment, though, Perelman’s disappointment was limited to the international mathematics establishment. The Steklov Institute was exempt, or rather his laboratory, his safe harbor after the falling-out with Burago, was exempt. Perelman resumed his activities, such as they were, at the institute: he attended seminars, sometimes several times a week, and he occasionally went by to check his e-mail. In the months before he had left for his lecture tour, he had maintained an even relationship with Ladyzhenskaya, the head of his new lab. She had died in January 2004, at the age of eighty-two, and after that Perelman rarely talked to anyone. As soon as Perelman returned, he wrote up the final installment of his proof, which he posted on the arXiv in June, and then he seemed to be exploring other problems. He was reticent, as usual, to talk about them, but he had apparently moved closer to Ladyzhenskaya’s research interests.

  Perelman got a promotion at the Steklov: he now held the title of lead researcher. Russian academic institutions assign their researchers to one of four levels, lead researcher being the top. Simple PhDs rarely hold this title; Russia maintains a two-tier dissertation system, in which the first dissertation—the one Perelman had written at the end of his graduate studies and that qualified him as a doctor of philosophy in the United States—ranks one as a candidate, while a second dissertation entitles one to be called Doctor. Steklov well-wishers kept telling Perelman to write his second, doctoral dissertation. The process required a traditional publication, and a defense. Perelman, naturally, scoffed at the idea. “He didn’t think he needed it,” Steklov director Sergei Kislyakov23 told me in a slightly puzzled tone of voice. Kislyakov seemed to personify the attitude that grated on Perelman the most: he liked Perelman and wished him well, but Kislyakov sincerely thought rules were the same for everyone, and this meant that a lead researcher should really get his act together and write and defend a second dissertation. Perelman, of course, also thought that rules were rules—but by now this applied only to rules of his choosing and, increasingly, of his own invention. He considered other rules to be sorts of impostors, all the more offensive for pretending to be real rules.

  Meanwhile, the Russian Academy of Sciences was putting its house in order, trying to restore itself, after the chaos of the 1990s, to its former buttoned-down glory. On the one hand, Academy property was gradually being repaired—the Steklov got a decent paint job and new plumbing—and salaries were going up; a lead researcher’s pay had gone from what literally amounted to pennies in the early 1990s to about four hundred dollars a month in 2004 (though Perelman would have made more had he secured his doctorate). On the other hand, the Academy was now demanding paperwork, reports on research and publishing activity. Perelman, predictably, bristled at the very idea of filling out paperwork to justify his mathematical existence. Ladyzhenskaya’s successor, Grigori Seregin, shielded Perelman, ensuring his continued peaceful existence at the Steklov.

  In late 2004 Perelman even traveled to Moscow to represent the St. Petersburg branch of the Steklov at a year-end Academy meeting. He gave a talk on the Poincaré. When he returned to St. Petersburg, he was unable to file his expense report.24 Russian law required that a person dispatched by an institution on official business have his documents stamped at his final destination in order to qualify for reimbursement. Surely someone who just a few months earlier had navigated the U.S. visa maze could easily have managed the Russian business-trip maze. In fact, Perelman had not had his documents stamped on principle: “I cannot go robbing the institute,” he told the staff at the accounting office back in St. Petersburg. The accountant had to mail Perelman’s documents to the Academy in Moscow so they could be stamped and returned. Still, Perelman would not accept the reimbursement money until the accountant had shown him the books proving that the reimbursement would come out of a special travel fund that was entirely separate from the Steklov’s salary budget. Clearly, Perelman’s rules on handling money had grown as exacting and as convoluted as his rules on footnoting. And as with footnotes, while the standards were known only to Perelman himself, he believed they were universal—and if he caught anyone violating them, he was merciless.

  Merciless he was in the summer of 2005, when he showed up at the Steklov accounting office to ask why he had been paid more than his usual monthly salary. By this time the Steklov was depositing its researchers’ pay directly into their accounts, so Perelman had made his discovery at a bank machine. The accountant, a short, overweight woman in her fifties who had seen a lot of mathematician weirdness in her nearly thirty years at the Steklov, confirmed that Perelman had been paid eight thousand rubles—a bit less than three hundred dollars—over his usual monthly amount, thereby receiving almost double his normal monthly pay. The reason was no mystery: his lab had completed a project and had some
grant money left over. In keeping with the usual practice, Seregin, the head of the lab, had instructed the accounting office to divide the leftover funds among the staff of his lab. He had made one mistake. Perelman’s previous bosses had known he did not approve of the practice—much as he had not approved of exam-time cooperation at the Mathmech, another generally accepted activity that could probably be seen as violating the letter of the law—and so they had always left him off the list of beneficiaries. Seregin did not know of Perelman’s position and so placed him on the list.

  Perelman asked the accountant to name the exact amount he had been overpaid. He then left the institute and returned a short time later with eight thousand rubles in cash. He wanted to give the money back to the accounting office. The accountant suggested he take it to the lab, where Seregin could decide how to dispose of it. Perelman insisted on returning the money directly to the institute. This is probably the point in the conversation where, as some Steklov staff members later reported, Perelman’s shouting could be heard in the hallways. The accountant, however, denies that there was yelling—though over her years at the Steklov she may have grown accustomed to extreme and unexpected expressions of human emotion. Perelman finally prevailed: he convinced the accountant to write a receipt saying she had accepted the money.

  The grant story, absurd and telling as it is, is famous in St. Petersburg and among mathematicians elsewhere. In fact, I heard it for the first time in the United States. But the first three or four times I heard it, it was purported to be the story of how Perelman left the Steklov. He refused to take the money and walked out, slamming the door behind him, the story went. That would be a very neat narrative, but it was not what happened. Perelman quit his job at the Steklov half a year later, in early December 2005, for no apparent reason. He came to the Steklov and tendered his resignation letter to the secretary. She ran to the director to alert him. Kislyakov asked Perelman to come in. Perelman went into the director’s long rectangular office, with its endless polished-wood conference table, and said calmly, “I have nothing against the people here, but I have no friends, and anyway, I have been disappointed in mathematics and I want to try something else. I quit.”

  Kislyakov suggested it might be a good idea for him to stay until the end of the month, so he would be able to draw the traditional December bonus—four hundred dollars or so. Perelman declined. He canceled his e-mail account at the Steklov and left mathematics by walking out through the heavy oak double doors that led onto the embankment of the Fontanka River and into the oppressing grayness that masqueraded as daylight in St. Petersburg in winter.

  “Something just snapped,” Kislyakov told me, shrugging. He had no idea what had snapped. There was a chance that Perelman encountered a difficulty with a problem he was tackling—but then, he had encountered difficulties before and they had not caused him to reject mathematics. Anyway, he was certainly a marathoner. There was a possibility that his final disappointment had to do with the second anniversary of his posting the first Poincaré preprint. Perhaps he had given the mathematical establishment a grace period. After all, the Clay Institute’s rules said the million-dollar prize could be awarded two years after publication. (In fact, the rules said a committee to administer the prize could be appointed two years following a refereed publication, but Rukshin, for one, willfully ignored the subtleties when he spoke to me about the Clay prize, claiming to represent Perelman’s position.) November 2005 may have been the mathematical establishment’s last chance to redeem itself in Perelman’s eyes. By ignoring the superfluous parts of the rules that made no sense to Perelman and observing only the rules that did make sense, the Clay Institute could have declared Perelman the winner of its million-dollar prize. The money was, as ever, not the issue; the recognition was—and the recognition had to be as singular as Perelman’s achievement. He would have been the first person ever to receive the Clay prize. He would have received it alone. And he would have received it on his own terms.

  This did not happen.

  What happened next was very strange. The June 2006 issue of the Asian Journal of Mathematics came out. The journal’s entire three hundred pages were devoted to an article25 by two Chinese mathematicians, Huai-Dong Cao and Xi-Ping Zhu, titled “A Complete Proof of the Poincaré and Geometrization Conjectures—Application of the Hamilton-Perelman Theory of the Ricci Flow.” At first glance, this might have appeared to be another explication of Perelman’s proof, along the lines of what Kleiner and Lott and Morgan and Tian had been doing—with the important distinction that Cao and Zhu had not been public about their work and had not participated in any of the seminars and workshops sponsored by Clay. They had worked under the tutelage of Shing-Tung Yau, a Harvard professor, Fields Medalist, close friend of Hamilton’s, one of the most powerful mathematicians in both the United States and China, and the editor of the Asian Journal of Mathematics. Yau had been among the recipients of Perelman’s e-mail message drawing attention to his first preprint. He had not responded in any way save for telling Science magazine that he thought26 Perelman’s proof might contain a fatal flaw connected with the number of surgeries required to complete the flow.

  The abstract of the Cao and Zhu paper read more like a marketing pitch than perhaps any mathematical abstract ever written. In fact, there was nothing obviously mathematical about it. It said, in its entirety: “In this paper, we give a complete proof of the Poincaré and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow.” The authors appeared to be claiming that Hamilton and Perelman had laid the groundwork for the proofs of the Poincaré and Geometrization conjectures but the last mile had been covered by the Chinese mathematicians, hence the breakthrough—and, it would seem to follow, the fame, glory, and the million dollars—rightfully belonged to them. Such is the law of mathematics: the person who takes the final step gets all the credit for the proof. The difference between taking the final step and providing the explication of the proof is substance, and substance can be a difficult thing to measure. Yau held a press conference27 at his mathematics institute in Beijing on June 3, and the acting director of the institute declared, “Hamilton contributed over fifty percent; the Russian, Perelman, about twenty-five percent; and the Chinese, Yau, Zhu, and Cao et al., about thirty percent” (there had apparently been a miracle of arithmetic, among other things, and Yau has disputed this account, which was originally printed in a Chinese paper and later reproduced in the West).

  A week later, Yau held a conference in Beijing that was headlined by Stephen Hawking. Though most of the several hundred people in attendance were physicists, Yau used the occasion to announce Cao and Zhu’s putative breakthrough,28 saying, “Chinese mathematicians should have every reason to be proud of such a big success in completely solving the puzzle.”

  Yau was frantically creating a chronology to support his narrative, in which Cao and Zhu were the mathematical heroes. In an article he published in June 2006, Yau painted the following picture: “In the last three years, many mathematicians have attempted to see whether the ideas29 of Hamilton and Perelman can hold together. Kleiner and Lott (in 2004) posted on their web page some notes on several parts of Perelman’s work. However, these notes were far from complete. After the work of Cao-Zhu was accepted and announced by the journal in April, 2006 (it was distributed on June 1, 2006) [sic]. On May 24, 2006, Kleiner and Lott put up another, more complete, version of their notes. Their approach is different from Cao-Zhu’s. It will take some time to understand their notes which seem to be sketchy at several important points.” In fact, it appears Yau rushed the Cao-Zhu paper through to publication,30 effectively forgoing the review process and preempting previously scheduled content, specifically so the authors could claim not to have read Kleiner and Lott’s notes—which stated clearly, at the outset, that the proof
explicated was Perelman’s.31

  The race was on, because the end of the summer would see the International Congress of Mathematicians—the first such gathering since Perelman started posting his preprints. The Poincaré proof—and the million-dollar prize that went with it—would certainly be the main topics of the congress.

  The ICM in Madrid began on August 22. On the morning of the opening, publications all over the world received a press release—embargoed until noon that day, when the information would be made public—announcing that Perelman would be awarded the Fields Medal “for his contributions to geometry and his revolutionary insights32 into the analytical and geometric structure of the Ricci flow.” The document went on to explain, “As of the summer of 2006, the mathematical community is still in the process of checking his work to ensure that it is entirely correct and that the conjectures have been proved. After more than three years of intense scrutiny, top experts have encountered no serious problems in the work.” In other words, the official press release stopped short of giving Perelman credit for proving the Poincaré. On the same day, the new edition of the New Yorker went on sale; it included an article called “Manifold Destiny,” written by A Beautiful Mind author Sylvia Nasar and science journalist David Gruber. The article traced the story of Perelman’s proof, Cao and Zhu’s paper, and Yau’s promotion of the Chinese scientists’ authorship of the proof, and it even contained excerpts from a conversation with Perelman, whom the authors had convinced to speak with them in St. Petersburg. The article quoted Anderson, who said, “Yau wants to be the king of geometry. He believes that everything should issue from him, that he should have oversight. He doesn’t like people encroaching on his territory.” It also quoted Morgan, who contradicted Cao and Zhu’s claim that Perelman’s proof had contained catastrophic gaps that they had filled in. “Perelman already did it and what he did was complete and correct,” Morgan told the New Yorker writers. “I don’t see that they did anything different.”