Perfect Rigour

by Masha Gessen

  Two key Perelman traits were displayed here. One was that, as Rukshin put it, “he was deliriously honest even at moments when what was important was that he could have saved time.” Delirious is a wonderful word, conjuring up the drive of someone organically incapable not only of telling an untruth but of telling an incomplete truth. But what if Perelman was wrong? What if the part he had explained was correct and represented the complete solution while the rest was superfluous? In math olympiad slang, a solution—or part of a solution—that looked right to its author but was wrong was called a lipa, a general Russian slang term for fake that literally means “linden” but is probably best translated as lemon. Everyone who spoke to me about Perelman specifically mentioned this trait of his: he had no lemons. None. Ever. Such was the precision of his mind: not only was he incapable of telling a lie—he was even incapable of making an honest mistake.

  Mathematicians make mistakes. This is part of what they do. Unlike humanities scholars, they cannot allow for the possibility of more than one truth. But unlike natural scientists, they cannot check their hypotheses against empirical truths. So they have only the resources of their own minds—and those of their colleagues—with which to subject their imaginary constructions to sets of imaginary rules, to see if they still hold up. This makes the peer-review process in mathematics even more important than it is in any other academic discipline, and it also explains the importance of having the two-year waiting period imposed by the Clay Institute before it awards one of its Millennium Prizes. Even so, mathematicians make mistakes that sometimes take years to catch. Occasionally they catch them themselves—as Poincaré did when he realized that he had not proved his own conjecture. Sometimes the mistakes are discovered by referees, as when Andrew Wiles released his original attempt to prove Fermat’s Last Theorem. The solution turned out to contain a serious flaw, which Wiles fixed himself10 but not until two years later. Young mathematicians, less adept at subjecting their own solutions to scrutiny, make mistakes more frequently than older ones. It is not surprising that Grisha Perelman could not conceive of himself making mistakes; what is surprising is that he actually did not make them.

  So it must have been all the more upsetting to him that when he finally made it to his first national competition that year, he took second place. Both of his coaches—Rukshin and Abramov—claimed that it was after the All-Soviet Olympiad in Saratov that Perelman got mean. He set about ensuring he would never lose to anyone again. “He had now tasted the blood of a freshly killed competitor” was how Rukshin put it. “And his ambitions far exceeded his accomplishments.” Here Rukshin’s ornate language seemed to get the better of his understanding of Perelman. It seems that what bothered Perelman in Saratov in 1980 was what would always bother him about the world: things had not gone logically. If Perelman was so good that he had never had a lemon, if his mind was so powerful that it had never encountered a problem it could not crack, then why had he not taken first place? The only possible answer lay in unforgivable human failure: Grisha Perelman had not practiced enough. From then on he practiced ceaselessly. While for other kids life was divided into school and leisure, for Perelman it was split into time devoted to solving problems without disruption and the rest of the time.11

  The 1982 IMO team was to have four members; that meant six would be chosen, so there would be two alternates. In January 1982 Abramov collected a dozen potential members of the team at a school in the science town of Chernogolovka, about fifty miles north of Moscow. The national chemistry and physics coaches were gathering their potential competitors at the same time and in the same place, so about forty of the country’s brightest high-school seniors were there, bunking four to a room in the school’s dormitory, located in the same building as the school. They were fifteen- to seventeen-year-olds—seventeen being the standard age for a graduating student. But several of these competitors were, like Perelman, precocious; at fifteen and a half, Grisha was not the youngest. So they were not quite grownups, and though several of them lived away from home at specialized schools, they later remembered the odd sensation of being on their own in Chernogolovka. One student recalled waking up in the morning12 and seeing that water in a jar on the windowsill had frozen because a pane of glass was broken; though the room was nonetheless adequately heated, he felt shocked and depressed by the sight. Another recalled arriving by bus in Chernogolovka13 in the dark evening—which in January is any time after four in the afternoon—and then, unable to find the school, wandering the empty and poorly lit streets of the town carrying a suitcase with clothes and books and a mesh bag of food supplies that were so heavy they hurt his gloveless hands. Grisha Perelman certainly remembered nothing so traumatic because he traveled to Chernogolovka with his mother. Other trainees thought that was odd and slightly humiliating for a male adolescent, even if he was a math prodigy, but Perelman was apparently oblivious.

  As he was oblivious to the grueling physical routine to which the trainees were subjected. In full accordance with Kolmogorov’s ideals, the boys were expected to train not only in their chosen sciences but also in athletics—a custom that set the Soviet math-competition training system sharply apart from those of Western countries, which also gathered potential team members for training sessions. “They would collect all the mathematicians, physicists, and chemists—that’s more than thirty people right there—in one gym,” recalled Alexander Spivak, who eventually made the team. He was a student at the Kolmogorov boarding school in Moscow, where athletics was stressed as an important part of the study program, but as he recalled, he had never been subjected to anything so physically taxing. “To give us all something to do, first they made us run around the perimeter of the gym, and run, and run. And then there were these long benches there, and there was the gym coach and his imagination, which determined what could be done with them. You could do pushups off them. You could lift them over your head. You could jump over them back and forth. And you do all this. And all you see is this bench in front of your eyes. The whole time it’s the bench, the bench, the bench.”

  Spivak recalled that one of the boys fainted, and at one point the others simply stopped and sat down on a bench, all of them in a row. What he remembered about Grisha Perelman was that he was “heroic,” which in that case meant that, unlike the other boys, he did not protest, stage a sit-down strike, or generally show any dissatisfaction with the proceedings. He could not have enjoyed the exercise or found it easy: Perelman had a terrible time in gym class at school, and despite everyone’s best efforts, he never managed to fulfill the Preparedness for Labor and Defense of the USSR requirements,14 which called for an upperclassman to run, swim, perform pull-ups, and shoot a small-caliber rifle. Nor did he manage to get above a C-level grade in physical education, which accounted for the only nonperfect grade on his graduating transcript.15 But rules were rules, and if Grisha was told to hop back and forth over a bench as part of his training for the international mathematics competition, hop he did.

  His behavior at the gym may partly explain why some of his fellow trainees remembered Perelman as athletic. “He wasn’t formally athletic, as if he had trained in tennis or something like that,” recalled Sergei Samborsky, who made the team reserve. “But we all tended to ignore gym class and be shapeless while he was fit, in shape. And if you asked me what kind of sport I would associate with him, I’d say it was boxing.” Over the course of a quarter of a century, Samborsky’s memory had probably melded the deep impression left by Perelman’s competitiveness and confidence with the recollection of Perelman’s physical being. Perelman was pale, slightly overweight, and much shorter than his teammates; he was no boxer. But he was a math fighter, certain he would never again be defeated.

  He was cocky. “One time one of the coaches reproached him by saying, ‘You know, Grisha, everyone else knows derivatives and you don’t,’” recalled Samborsky. “That was a part of mathematical analysis, and strictly speaking, as a secondary-school student
, he wasn’t required to know. But he responded, ‘So what, I’ll solve the problems without it.’ It sounded brazen, but in essence, he was right.” And then Samborsky added something that showed he remembered Grisha Perelman perhaps more accurately than he himself realized: “I suspect he knew a lot more than he let on.” In fact, he probably knew derivatives. But he left this information out because he was there to solve problems, not to prove anything to the coaches.

  Everyone got the point anyway. Coach Abramov remembered Perelman as being the only student who had never seen a competition problem he could not solve. And Samborsky put it simply: “He was better at solving problems—so much better, in fact, that one could say he was better than the rest of us put together. There was Grisha, and then there were the rest of us.”

  Out of the rest of them, at the end of the winter training camp, five more members of the team were tentatively chosen. The trainees were ranked according to the number of problems solved in the course of the camp. Number six was fifteen-year-old Spivak. An ethnic Russian who came to Moscow from a village in the Urals to study at the Kolmogorov boarding school, he hadn’t known he had a Jewish-sounding last name. So he had no way of making sense of things when he was suddenly bumped off the list in favor of an ethnic Ukrainian who ranked seventh.

  To the trainees, the winter camp was a succession of problem-solving competitions designed to resemble the actual olympiad; grueling gym sessions; lectures by renowned mathematicians, many of whom were living legends in the boys’ world; and a nagging but tolerably quiet buzz produced by various education ministry and Party officials who hovered around the camp and occasionally cornered the trainees to remind them that it was the honor of the great Union they would be defending at the IMO. To the coaches, however, the camp was equal parts training and evaluating the boys and neutralizing the buzzing officials. They chose their battles. Even the obvious, inevitable inclusion of the extraordinary Perelman on the team required that the coaches put up a fight, for a competitor with a surname like that spelled trouble for the ministry minders; the coaches used up all of their fighting points, and the sixth-ranked Spivak, with his suspicious last name, was sacrificed.

  When I met Spivak a quarter of a century later, he was an overgrown math boy: huge, with a disheveled head of graying hair, dressed in mismatched multicolored knits, he pleaded with me to relieve him of the social discomfort of a café, and he came to be interviewed at my apartment instead. He was working as a math instructor at one of Moscow’s specialized schools, and he had spent much of his life putting together collections of math problems for gifted children. His manner of answering questions was disarmingly direct:

  “So do you remember arriving in Chernogolovka?” I asked. “Was it morning, daytime, or evening?”

  “I don’t see why that’s interesting,” he responded. “It would be so much more interesting to ask me where everyone was now.”

  “Indeed it would be,” I admitted. “Where is everyone now?”

  “I don’t know,” he answered simply.

  I fared barely better with questions regarding the connections that team members had made with one another: Spivak claimed he didn’t see what was so special about the experience that it would have made the boys bond. When I argued that stress was a great unifier, he launched into a discussion of the comparative levels of complexity of the problems in different competitions. But he had a striking, emotionally charged memory of his experience of trying to get on the team. He had known that he had to make it in order to gain admission to a university. Even if he was unaware of the suspicious sound of his surname, he had judged—rightly, in all likelihood—that he would be unable to write the essay that was part of the entrance exams. “I just knew that I would be spending two years in the army, and I didn’t know what would happen to me there,” he told me. He had to claw his way to the IMO. He begged and pleaded, and he caused the coaches and ministry officials to scream at one another, and in the end, while he remained the seventh-ranking competitor, he was allowed to work on the take-home problem set, a small book that filled the potential competitors’ time between the January camp and the All-Soviet Olympiad in April.

  April saw all the boys in Odessa, a once-grand city on the Black Sea. They spent two days at a seaside resort solving the hardest problems they had ever faced: the consensus was that the All-Soviet problem sets were harder than those at the IMO. Spivak, who felt the rest of his life was at stake, took nothing for granted—he worked frantically, desperately, filling two entire composition books with textbook proofs that formed merely a part of the basis of his solutions and that he should have claimed were well established. Had Perelman perceived the world as the unfair place it was, he also would have had reason to think the rest of his life was at stake. But his confidence in himself and in the order of things was unshakable. He did what he always did: he read the problem, closed his eyes, leaned back, rubbed his pant legs with his palms with growing intensity, then rubbed his hands together, opened his eyes, and wrote down a very precise and very succinct solution to the problem. When solving the more difficult problems, he hummed softly. He filled only a couple of pages with his solutions. Both he and Spivak had perfect scores.16

  On the final day of the competition, as the jury gathered to grade the results, the top seven contenders—now including Spivak—were chosen to accompany Kolmogorov, who was visiting the national competition for the last time, on a walk through Odessa. Neither Spivak nor Samborsky remembered what Kolmogorov discussed with them—in any case, he was already afflicted with Parkinson’s, and making out what he said must have been difficult—but both recalled that at a certain point he commanded the entire group to head for the beach. “The wind from the sea was piercing,” recalled Samborsky. “We had to stay by his side because we’d been warned never to leave him alone since he couldn’t see well. And Kolmogorov decided to go swimming. He undressed and went into the sea, and I was scared even to look at it; it was so cold it was almost like there were slabs of ice still floating. Waves the color of lead, foaming, wind so strong it could knock you off your feet. None of us followed him.” Presently a guard emerged and told the boys to “rescue the grandpa,” who surely could not fare well in the sea in this weather. The boys refused—either because none of them could swim well enough, as Spivak remembered, or because none of them dared confront Kolmogorov, as Samborsky recalled.

  In either case, the following picture emerges. On a cold gray afternoon in the second half of April 1982, the greatest Russian mathematician of the twentieth century, making his last mathematical journey, went for a swim in the freezing water of the Black Sea while the greatest Russian mathematician of the twenty-first century sat impassively on shore and looked on. He had come because he was instructed to watch over “grandpa”; he had little use for all the walking and small-talking that was tacked onto the body of mathematics, and he had a distinct dislike for the water in which Kolmogorov was now enjoying what was left of his physical strength. The exuberant, expansive era of Russian mathematics was ending; a time of closed, secretive, concentrated individualism was beginning. Of course, no one could know this yet.

  While Perelman waited for Kolmogorov on the beach, the All-Soviet Mathematical Olympiad jury worked out the final results of the competition, and Rukshin, Abramov, and several others began the final leg of the long and arduous process of ensuring Perelman would travel to Budapest for the IMO. The previous year, the IMO had been held in Washington, D.C. The Soviet Union’s number one that year had been a Kiev high-school senior named Natalia Grinberg, a Jewish girl. This was a year after the United States had boycotted the Olympic Games (not the mathematics variety) held in Moscow. It was a year when Ronald Reagan’s Evil Empire rhetoric defined U.S. policy toward Moscow. It was also the year when the Soviet Union de facto ended Jewish emigration. There was no way Soviet officials were going to let a Jewish girl represent the country at an IMO held in Washington: U.S. media coverag
e of her participation as envisioned by Moscow, as well as the possibility that she would defect—and the publicity surrounding that—added up to unacceptable risks. Grinberg was picked for the team—she had to be—but shortly before the planned trip she was told that her travel documents could not be processed in time.17 The USSR fielded six competitors instead of the eight required that year—another member of the team also had so-called problems with his documents—and took ninth place with 230 points; every country that beat the Soviets that year had fielded eight competitors.18 Abramov was proud of that achievement: he had made sure that the Soviet team was set back no more than the 84 points the two missing members could have brought it.

  Natalia Grinberg emigrated to Germany and became a professor of mathematics at Karlsruhe University.19 Her son, Darij Grinberg, represented Germany at the IMO three times between 2004 and 2006,20 winning two silver medals and one gold. Upon learning, during the judging of the IMO, that her son had apparently won the gold, Natalia Grinberg congratulated him and the team on a math forum and signed her post, “Natalia Grinberg, former number 121 in the 1981 USSR team, who was not allowed (in the last minute) to quit the beloved motherland to participate at IMO in Washington.” For this professor, twenty-five years had clearly not assuaged the pain and insult of having been denied a prize for which she had worked most of her childhood and young adulthood.

  As usual, Perelman was lucky and unaware of it. After placing ninth in Washington, the Soviet Union needed to restore its IMO status. The 1982 competition would be held in Budapest, the capital of Hungary, which was a part of the Soviet bloc and so, from a Soviet official’s perspective, posed fewer publicity and security concerns than Washington. Nonetheless, competitors would still have contact with students from other parts of the world, including the United States. Further, the IMO was set up in such a way that competitors had next to no adult supervision: since all coaches were engaged in the judging process, teams and their adults had to have separate accommodations and keep contact to a minimum. To ensure that the Soviet competitors performed appropriately in every way, the boys were subjected to regular pep talks by ministry officials reminding them that they were representing the honor of their great land, and the adults were forced to prove to a dozen different officials that their charges were ideologically reliable. And still the risks, in the eyes of the officials, were formidable. Just four years earlier, when the IMO was held in Communist Romania, the Soviet Union had fielded no team at all—because, rumor had it, every single member of the team would have been Jewish.