Perfect Rigour

by Masha Gessen

  So Morgan tried his luck with other questions. He had several sets of pressing ones. First, he wanted to see Perelman’s preprints in published form—for the historical record if for no other reason. He suggested he would edit them himself and place them in a journal he coedited. He also invited Perelman to Columbia University: “Would you like to come for a week, a month, a semester, a year, for the rest of your life?” Morgan inserted these sorts of questions carefully between his mathematical requests. “And I would get back responses like: ‘The answer to question one is that; here’s the answer to question two. I have no answers to your other questions.’ So he did acknowledge them, which is more than he did with most people.” But he certainly did not answer them. After a while, Morgan ran out of mathematical questions.

  When Morgan and Tian completed their manuscript in 2006, they mailed it to Perelman. The package came back stamped SERVICE REFUSED.

  PERFECT RIGOR

  THE MADNESS

  10

  The Madness

  PERELMAN RETURNED TO St. Petersburg in May of 2004. Late spring is the only time Petersburg becomes not just livable but attractive; its usual grayness gives way to a soft, cool light that refuses to dim well into the night. The city’s residents pour out onto its sidewalks and embankments and start taking all the strolls they did not take in the cold, wet winter months. Perelman, who always walked, and Rukshin, who made a point of doing all things beautiful in St. Petersburg, went for a walk. The weather must have been much the same as it had been in Boston weeks earlier, when Perelman had walked along the Charles with Tian. He said many of the same things too, but more emphatically this time—or Rukshin heard them more clearly and louder than Tian had heard them. Perelman said he was disappointed in the world of mathematics.

  “It took him eight or nine years to solve the Poincaré,” Rukshin told me, recalling that conversation. “Now imagine that for eight years you did not know whether your child, who was born ill, would survive. You have spent eight years caring for him day and night. And now he has grown strong. From an ugly duckling, he has turned to a fine swan. And now someone says to you, ‘Why don’t you sell your baby to me? Here is some grant money, for half a year, or perhaps a year, we could publish the work together, we’d make this a joint result.’”

  Normally, when you’re having a conversation with a mathematician, pointing out logical errors will enrich the exchange. This was clearly not the case here. First, no one sends a child into the world at the tender age of eight, nor would one perceive it as offensive if, say, one’s eighteen-year-old child were offered a spot at university. The thing was, even if Rukshin twisted the logic of what Perelman had said to him, he was likely still conveying the emotions correctly. In a sense, the point was precisely that this was a bad comparison: Perelman’s proof of the Poincaré Conjecture was not as vulnerable or as valuable as a human child, but Perelman’s experience of the incongruity of his accomplishment and the rewards he had been offered was like that of a doting parent who had been offered money for his baby. Rukshin, who was highly suspicious of the world in general and given to feeling slighted, surely added his own interpretations to Perelman’s emotional charge. This was how, in the retelling, offers of professorial appointments turned into not-so-veiled attempts to buy the right of coauthorship of the proof; and how in Rukshin’s and perhaps Perelman’s imaginations, Kleiner and Lott’s, and later Tian and Morgan’s, work on interpreting the proof turned into attempts to usurp credit for it.

  Concluded Rukshin: “The world of science—the science that Perelman had considered the most honest of the sciences—had turned its other side to him. It had been soiled and turned into market goods.”

  Perelman presented similarly charged recollections of his lecture tour to several other St. Petersburg colleagues. They too embellished his narrative with details that served to justify his anger and pain. For example, one person told me that Perelman had been hurt when Hamilton “walked out of the lecture, stomping his feet.” When I asked for clarification, my interlocutor conceded, “I added the part about his stomping his feet. But from what I have been told, he did leave demonstratively.”

  When Perelman spoke to two New Yorker writers1 in the summer of 2006, he told them that Hamilton had shown up late for his lecture and asked no questions during the discussion session or the lunch—a recollection that is at odds with Morgan’s. In all likelihood, Hamilton had asked no questions that indicated to Perelman that the older mathematician had made a serious effort to understand his work. “I’m a disciple of Hamilton’s, though I haven’t received his authorization,” Perelman told the New Yorker and added, “I had the impression he had read only the first part of my paper.”

  The more Perelman talked about his disappointment with the mathematical establishment, and the more his acquaintances decorated his stories with demonizing details, the more Perelman’s sense of betrayal deepened. His world, which had begun narrowing in his first university year and then broadened slightly both times he had traveled to the United States, was now headed for its final, disastrous narrowing. Like a rubber band slipping inexorably off a sphere, his world was about to shrink to a point.

  From the moment Perelman entered Rukshin’s math club at the age of ten—or perhaps from a much earlier point, when his mother told her professor she was leaving mathematics to have a baby—Perelman had been a human math project. He was raised by his mother, reared by Rukshin, coddled by Ryzhik, coached by Abramov, directed by Zalgaller, protected by Alexandrov, tended by Burago, and promoted by Gromov so that he could do pure mathematics in a world of pure mathematics. Perelman repaid his teachers and benefactors by doing just that: solving the hardest problem he could find—and by devoting himself to this process fully. And when he was done, he expected certain things. Just as he had been convinced that he should not untie his hat and had always, against all evidence, believed in meritocracy, so now he had in his head a perfect picture of how things should go. He had, in essence, a script. This script apparently indicated that Hamilton would attend all of Perelman’s lectures at Stony Brook, possibly even his first lecture at MIT, and Hamilton and the whole Ricci flow community would delve deeply into Perelman’s proof, making every effort to understand it. Other mathematicians would do this too; this would be their natural way of responding to his contribution and of showing mathematical appreciation.

  Perelman’s disappointment in Hamilton was all the more painful because he had apparently perceived Hamilton as a member of the pure-mathematician caste. In his conversation with the New Yorker journalists, Perelman recalled his first encounter with Hamilton, at Princeton, in a way that made this clear: “‘I really wanted to ask him something,’ Perelman recalled. ‘He was smiling, and he was quite patient. He actually told me a couple of things that he published a few years later. He did not hesitate to tell me. Hamilton’s openness and generosity—it really attracted me. I can’t say that most mathematicians act like that.’” So striking and stable was this image of Hamilton in Perelman’s memory that he seemed to have ignored Hamilton’s nonresponse to his initial letter regarding Ricci flow and Hamilton’s nonreaction to the first preprint—and so he kept expecting that Hamilton would stick to the script during the lecture tour.

  The script also contained rules, obvious ones. People should not talk about things they do not understand; if it was going to take a year and a half for anyone to understand the proof, no one should talk about the proof until then. Great mathematical achievement should be rewarded with professional recognition, which can take only one form: the form of studying and understanding the work that the person has done. Money is no substitute for work. In fact, money is insulting. If you think it is natural for a university to offer money to someone who has solved a huge problem even though no one at this university understands the solution, imagine the following parallel: a publisher approaches a writer, saying, “I have not read any of your books; in fact,
no one has gotten to the end of one, but they say you are a genius, so we want to sign you to a contract.” This is a caricature. There was no place for caricatures in Perelman’s script.

  Back in the summer of 1981, the first year Sergei Rukshin managed to organize a summer mathematics training camp, Grisha Perelman lived away from home for the first time. Rukshin transported a score of his club members, age thirteen to sixteen, to a pioneer camp outside of Leningrad, a grouping of low stone buildings situated scenically in a mixed wood with easy access to a cold lake. Rukshin’s agenda called for roughly four hours of problem-crunching per day, diluted with some swimming, hiking, walking through the woods listening to Rukshin recite poetry, and resting indoors listening to classical music. The arrangement with camp officials2 stipulated that the mathematicians would be a unit within the camp; they would have their own sleeping quarters and their own schedule, but they would have to wear Young Pioneer uniforms—white or blue button-down shirts and red neckerchiefs—and participate in some campwide activities, like politics lessons.

  So it was at the beginning of the camp season that Rukshin’s boys attended a lecture on foreign affairs. “The international situation,” said the speaker, a young Komsomol worker, “is particularly tense today.” The entire mathematics contingent broke out laughing.3 It is particularly tense today! Get it? It is like it was not at all tense yesterday but is particularly tense today.

  If you do not find that especially funny, then chances are you do not have Asperger’s syndrome. The condition got its name from the Austrian pediatrician Hans Asperger, who was long believed to have been the first to define it, in the 1940s. In fact, it seems it was the Soviet child psychiatrist Grunya Sukhareva4 who first grouped the symptoms, in the 1920s; she, however, called the syndrome a schizoid personality disorder, which may partly account for why it has not become a popular diagnosis in Russia. Asperger’s is a disorder that’s part of the autism spectrum. Unlike most autistics, Aspergians tend to have normal or high IQs, but their mental development still proceeds in ways that are markedly different from neuronormals’, as people in Aspergian circles call them. Hans Asperger observed that these children’s social maturity5 and social reasoning were delayed, and some of their social abilities remained, as he kindly put it, “quite unusual” for life. They had difficulty making friends; they had trouble communicating—the tone, rhythm, and pitch of their speech were often odd and off-putting to others; they had trouble understanding and controlling their emotions; and many of them needed profound assistance in organizing their lives, so they were often dependent on their mothers for their day-to-day functioning.

  More than forty years after Hans Asperger, a British psychologist named Simon Baron-Cohen6 came to study autism and Asperger’s syndrome and figured out several things that seem to me to be very useful in understanding Grigory Perelman. First, Baron-Cohen suggested that the autistic brain was lopsided in a particular way. Where a neuronormal brain has the ability to both systemize and empathize, the autistic brain might be excellent at the former but is always lousy at the latter—causing Baron-Cohen to dub the autistic brain “the extreme male brain.”7 Baron-Cohen defined systemizing as “the drive to analyze and/or build a system (of any kind) based on identifying input-operation-output rules” and theorized that great systemizers might be at increased risk for autism. When he tested this theory on a population of Cambridge University undergraduates,8 it turned out that the mathematicians among them were three to seven times more likely than other students to have a diagnosis of an autistic condition. Baron-Cohen also developed the AQ, or the autism-spectrum quotient, test, which he administered to adults with Asperger’s or high-functioning autism as well as to randomly selected controls and Cambridge students and winners of the British Mathematical Olympiad. The correlation between math and autism and/or Asperger’s was proved again: mathematicians scored higher than other scientists,9 who scored higher than students in the humanities, who scored roughly the same as the random controls. I took the AQ test too when Baron-Cohen e-mailed it to me, and scored as high as Baron-Cohen would probably expect a former math-school student to score, which is very high. Grigory Perelman, as far as I know, never took the AQ test and certainly cannot be diagnosed by someone who has not talked to him, though after I spent an hour on the phone describing Perelman to Baron-Cohen, the famous psychologist volunteered to fly to St. Petersburg to evaluate the famous mathematician—who sounded so very much like many of his clients—thus joining the long list of people who had volunteered help that Perelman did not welcome.

  Had Baron-Cohen chosen Russian rather than British mathematicians as his subjects, the results would probably have been either the same or even more clearly pronounced. After all, Russian mathematical prodigies are often grouped with others of their kind in environments that are especially tolerant of their particular brand of weirdness. The tradition of forgiving mathematicians their autistic rudenesses dates back as far as anyone can remember. Many memoirs of Kolmogorov cite his peculiar manner of walking away in midconversation, demonstrating both his utter disregard for social convention and his pragmatic approach to socializing, which is typical of Aspergians: once he had received the information he sought, he had no further use for communication.10 In one instance, Kolmogorov, then a dean at Moscow University, was accosted in a hallway by a man11 who said repeatedly, “Hello, I am Professor Such-and-Such.” Kolmogorov did not answer. Finally, the professor said, “You do not recognize me, do you?” Responded Kolmogorov: “I do, and I realize that you are Professor Such-and-Such.” In the Aspergian world, conversations are exchanges of information, not exchanges of pleasantries. Most of Kolmogorov’s students cited another of their teacher’s typically Aspergian traits: what they called his “temper”12 and what were actually frightening episodes of apparently uncontrollable rage. That Kolmogorov’s marked social problems did not impair his career is a measure of the degree to which a sort of Aspergian culture was built into the larger Russian culture of mathematics.

  Baron-Cohen’s other key insight is the concept that people with autism do not have a “theory of mind”13—that is, the ability to imagine that other people have ideas, perceptions, and experiences that are different from one’s own. In a striking experiment, Baron-Cohen tested normally developing children, children with autism, and children with Down syndrome. All children watched a brief play involving two dolls and a marble. One of the dolls placed the marble in a basket and left the room. While she was gone, the other doll moved the marble. When the first doll returned, the experimenter asked the children where she would look for the marble. The mentally retarded Down syndrome children and the normal children did equally well on the test: they realized that the doll would look for the marble in the basket, where she had left it. But sixteen out of twenty autistic children were certain she would look for the marble where it really was, not where the doll would have believed it to be. These children were believers in a single truth, utterly incapable of adjusting for human limitations.

  Another world authority on Asperger’s syndrome, an Australian psychologist named Tony Attwood, believes it is the theory-of-mind impairment that causes Aspergians to interpret everything they hear literally. In one of his books he described a child who sketched a picture14 at the end of an essay because the teacher had told students to “draw their own conclusions.” The belief that people mean exactly what they say is what can lead Aspergians to laugh at a political lecture that to them sounds like a weather forecast (“the political situation is tense today”). It also leads them to believe that things work exactly as they are said to work. “I suspect that many ‘whistle-blowers’ have Asperger syndrome,”15 wrote Attwood. “I have certainly met several who have applied a company’s or government department’s code of conduct to their work and reported wrongdoing and corruption. They have subsequently been astounded that the organization culture, line managers and colleagues have been less than supportive.”

 
So it is perhaps no accident that the founders of the dissident movement in the Soviet Union16 were mathematicians and physicists. The Soviet Union was not a good place for people who took things literally and expected the world to function in predictable, logical, and fair ways. But the math clubs, such as the one run by Rukshin, provided a refuge. Rukshin saw it as his mission to shelter the black sheep of Soviet schoolchildren, and he saw a certain posture of social withdrawal as the mark of a gifted mathematician. The first time I interviewed Rukshin, he had a later appointment with an eleven-year-old boy; the child’s mother was bringing him “to be looked at,” which meant that Rukshin would spend an hour or two or three giving the boy math problems in order to decide whether to accept him to the club. At the appointed time, Rukshin opened his office door to see if the boy had shown up yet. He had, and was sitting quietly in the lone armchair in the hallway. “I can tell he is gifted,” Rukshin said, closing the door. “I can spot them.” I knew exactly what he meant: the boy was pale and awkward, and he looked absent. If Attwood and Baron-Cohen had looked at him, they probably would have seen familiar signs as well: physical awkwardness and inappropriate facial expressions are among the outward signs of Asperger’s syndrome.

  Virtually everything people have recounted to me about Perelman’s behavior, starting from the time when he joined the math club, fits the typical picture of a person with Asperger’s syndrome. His apparent disregard for the conventions of personal hygiene is common to Aspergians, who perceive it as a nuisance forced upon them by the incomprehensible world of social mores.17 The trouble he had with articulating his solutions to problems is also classic. “People with Asperger often put in far too much detail,” said Baron-Cohen. “They don’t know what to leave out. They are not taking into account what the listener needs to know.” That is the theory-of-mind problem: the point of telling is not to get a point across but solely to tell. Schoolmates told me Grisha was always willing to answer questions about mathematics; the problems arose if the questioner did not understand the explanation. “He was very patient,”18 a former classmate recalled. “He would just repeat the exact same explanation, again and again. It was as though he could not imagine that somebody found it hard to understand.” She was probably exactly right: he really could not imagine it.