Perfect Rigour

by Masha Gessen

  To be allowed to travel, a Soviet citizen had to be granted a foreign-travel passport—no ordinary person was allowed to hold one as a matter of course—and an exit visa. This required clearance by local officials, travel authorities, and the secret police. To be allowed to travel on official business, representing the country, one also had to be cleared by the Party at every level, working one’s way up from the local precinct to the district and, finally, to the federal level. At any of these stages, the documents of someone like Perelman could be stalled indefinitely by an overly cautious bureaucrat. “So Abramov and I made a pact,” recalled Rukshin. “He worked on it in Moscow. I worked in St. Petersburg, pushing his documents through. After all, you know, I’d had many students in the club who were the children of someone powerful.” Rukshin called in every chip he had; he used his connections to a secret police officer who was the father of one of his students, a local Party boss who was the father of one of his classmates, and another Party boss who was the husband of another classmate. Meanwhile, in Moscow, Abramov made regular visits to the education ministry begging officials there to keep tabs on the bureaucratic progress of the Soviet Union’s great mathematical hope.

  The six teammates—four members and two alternates—spent the month of June back in Chernogolovka. Incredibly—or, rather, it would have been incredible if they had been six regular teenagers thrown together in close quarters for a month instead of these six boys supremely gifted in mathematics—they did not socialize; they did not bond. They trained for days on end, breaking only for games of volleyball, visits with mathematical luminaries, and the inevitable Party pep talks. By July all four members of the team had their travel documents. They were Spivak; Vladimir Titenko from Belarus; Konstantin Matveev from Novosibirsk; and Perelman, the only Jewish member of the team.22

  The Soviet team arrived in Budapest on July 7. The competitors were taken to a hotel where each country’s team of four had its own room. The students were now on their own;23 their coach had arrived in Hungary a couple of days earlier to take part in the final preparations—approving the translations of competition problems and assigning points to parts of each solution—and now the ministry handler who had accompanied the boys on the flight was also gone.

  The competition lasted two days: July 9 and 10. Each day the 120 participants spent four and a half hours solving a set of three problems. Each problem was worth seven points for a complete solution, and anywhere from one to six points could be awarded for starting off in the right direction without making it to the end. The judging process24—a complicated dance of negotiation and sometimes outright haggling involving judges from the host country, adjudicators from the countries where the particular problems originated, and coaches representing the competitors’ interests—took three days following the competition.

  During that time, the competitors were left to the care of local handlers. They were charged with the task of being good guests and worthy representatives of their countries—social tasks for which they were ill-suited. They submitted to touring around Budapest, taking a boat ride down the Danube, traveling to Balaton Lake for sightseeing and a swim, and visiting Erno˝ Rubik, inventor of the cube and other torturous mathematical toys that were then enjoying worldwide popularity. For the most part, they traveled speechlessly, though Rubik managed to elicit some questions, mostly concerning the minimum number of moves required to solve his puzzle and the possibility of devising an algorithm for a universal Rubik’s Cube solution. Perelman showed no interest in the sights,25 declined to swim, and had no questions for the great Rubik.

  A final social responsibility with which the Soviet team was burdened had to do with a bag of buttons. These had been handed to the team by a ministry official, who had talked of their duty to the motherland, of their responsibility as both competitors and diplomats, and of international friendship. And then she had pulled out the bag of buttons—the tourist variety, with pictures of Moscow and Leningrad on them—and, apparently zooming in on the most vulnerable of the boys, shoved the bag into Spivak’s hands. Spivak, who had already done what he could for his country mathematically (he would be awarded a bronze medal), now had to figure out what to do with the souvenirs. He tried to draft his fellow team members into the effort but failed. So he took the bag and headed out into the hotel corridors.

  “The order had to be carried out, even if we were not being supervised,” Spivak told me. “So I went and tried to hand them out, though I barely spoke English, which made it very difficult, and then I went to the American team’s room. And the way they fled the Evil Empire. I mean, they literally climbed under their beds. You would totally get the impression I was about to open fire on them. I tried to say something about friendship and that sort of thing, but I realized it was just too hard.” Spivak left the room and disposed of the buttons someplace where he assumed they would not be found.

  On July 14, the last day of the 1982 IMO, Perelman collected his trophies: a gold medal, shaped like an elongated hexagon that year; a special award certificate sponsored by Team Kuwait (last place) given to competitors who earned the maximum number of points—forty-two out of forty-two; a giant whip, which the Hungarians gave each medal winner; and a Rubik’s Cube, which Grisha gave away when he returned to Leningrad. These were the prizes; Perelman’s actual rewards for years of single-minded training were automatic admission to a university and, more central to his needs, the right to be left alone for another five years.26

  PERFECT RIGOR

  RULES FOR ADULTHOOD

  5

  Rules for Adulthood

  THE UNIVERSITY, FOR PERELMAN, egan with long train trips, long lines, and paperwork. Roughly ten members of Rukshin’s math club traveled as a pack. As Rukshin saw it, the path to Mathmech had been blazed by Perelman, whose right to be admitted without entrance exams had either forced or allowed the university to exceed its usual two-Jews-per-year quota and accept at least three people who, for the purposes of admissions discrimination policies, were Jewish in every way: their surnames sounded Jewish, and their identity documents stated they were Jewish. One additional Jewish student in an entering class of roughly three hundred and fifty may seem like a drop in the bucket, but for Rukshin, who got to send three rather than two of his Jewish students off to Mathmech, it felt like a victory and even, perhaps—if he was to be believed when he spoke about it a quarter of a century later—a revolution. The other members of the math club who made it to the prestigious mathematics department were either ethnic Russians or, like Golovanov, Jews who through marriage or other circumstances had lucked into Russian surnames and Russian identity documents.

  The large entering class was split into groups of about twenty-five people each. Perelman and several others from Rukshin’s math club and from Leningrad’s other specialized math schools were assigned to the same group, and those who were not arranged to be transferred to it. In the end the group represented a sort of elite learning center1 within Mathmech, singled out much as its members had been when they were schoolchildren. Most of them traveled daily from the city; in the 1970s Leningrad University had moved its science departments2 to Petrodvorets, a suburb about twenty miles west of the city. What had been conceived as an ambitious project, a campus that was a city unto itself like a sort of Russian Cambridge University, had fizzled, turning the newly built glass-and-concrete math, physics, and science buildings into a very inconveniently located commuter school (the rest of the university remained in Leningrad). The students took unheated suburban trains with wooden seats, invariably having to run to catch the one that would deliver them to school in time for the day’s first lecture and often risking missing the last city-bound train, which left before midnight.

  Russian universities offered a highly specialized education. Mathmech was geared toward producing professional mathematicians or, if that failed, mathematics instructors and computer programmers. Detours into what might be considered li
beral arts were minimal, while detours into Marxist theory, though not as demanding as they were at the humanities departments, still included required courses in dialectical materialism, historical materialism, scientific communism, scientific atheism, the political economics of capitalism, and an entire course entitled A Critique of Certain Strands of Contemporary Bourgeois Philosophy and Anti-Communist Ideology, which was taught by a young philosophy professor who managed to sing all the requisite praises of Marxist-Leninist philosophy, brand other contemporary philosophers rotten, and then proceed to tell the students what they had always wanted to know but were afraid to ask about Nietzsche and Kierkegaard. “So this was a class we actually attended,” Golovanov told me. Otherwise, most students attempted to devise ways to avoid showing up not only to the ideological classes but also to the large lecture courses and, in most cases, courses that fell outside the area in which they planned to specialize. There was, naturally, one exception: Grisha Perelman attended everything, including the large lectures from which he was exempt because his grades never dipped below a four on a five-point scale.

  Golovanov called the Marxism courses “the crazy disciplines.” Perelman accepted them as part of the learning package and used the great compacting brain of his to the benefit of all his classmates. “Grisha’s clarity of mind was very helpful here,” recalled Golovanov. “The thing about all this stream-of-unconsciousness is that you either have to process it all or ignore it completely. The former is impossible for ordinary humans, and the latter is fraught with danger. Grisha somehow managed to find the strands of thought, if you can call it that, in those disciplines. So his notes on all the crazy disciplines were of great value to us all.”

  What no doubt helped Perelman plow through the dense nonsense of Marxist theory as it was then being taught was his genuine disregard for politics of any sort. “In Grisha’s lexicon politics was always a swearword,” said Golovanov. “Say, if I wanted to organize something to make things better, some campaign aimed at helping our beloved Sergei Rukshin even, he would say, ‘That’s politics, let’s focus on solving problems instead.’ And you have to understand that this was a genuine position: he disliked all sorts and directions of politics equally.” The traditional Russian intellectual’s queasiness at the political process had less to do with Perelman’s position than did the fact that he was truly uninterested in anything that was not mathematics. While other students may have felt insulted or excited, Perelman remained dispassionate; none of the issues discussed in these courses had a connection to anything that mattered. His notes on Marxist theory were purely systematizing exercises, performed with his unique efficiency.

  The ideology courses notwithstanding—and they were, after all, fewer than at many other departments—Mathmech was what in the Soviet Union passed for a liberal institution of higher learning. Those who wanted to get through its five-year course with minimal effort and minimal knowledge had to suffer through the first year with a heavy learning load and afterward could proceed to coast. Those who wanted to specialize early could tune much of the rest of mathematics out. Perelman represented the rarest breed of Mathmech student: one who sought to be universally educated in mathematics.

  Most mathematically ambitious students had for years known their specialization was preordained: they had one sort of brain or the other. The algebraists might then look for the most promising problems of algebra while the geometers might cast about for the most interesting geometer with whom to study, but in general, their directions were set. Perelman’s brain was made to embrace all of mathematics. In retrospect, one might suppose that topology ultimately attracted him as the quintessence of mathematics—the province of pure categories and clear systems, with no informational interference—but as a first-year student, he was barely exposed to topology. Most mathematicians remember their one freshman course in topology for teaching them the mental exercise of turning an inner tube inside out using a tiny hole; it is that mind-bending quality of topology that most recall, not its streamlined clarity. Perelman did not have the other usual motivation to specialize early: he had no reason to try to save time by studying only the mathematics in which he planned to work. He was not rushing anywhere. He was living for mathematics and by doing mathematics.

  He attended lectures and seminars across mathematical disciplines without much apparent concern for the quality of instruction offered. The effect could be comical. In his fourth year at the university Perelman attended a course in computer science taught by an instructor who had earned the reputation of being one of the department’s worst lecturers. “Normal people did not attend this,” said Golovanov. Perelman did. And he generally sat at the front of the room, which was probably why he caught the eye of the instructor, who at one particular moment became agitated about the state of mathematical knowledge among Mathmech students in general. “Our fourth-year students can’t even solve the simple Cauchy Problem,” he declared. He wrote out the classic differential-equation problem on the board and turned to Perelman. “Can you tell me how this problem is solved?”

  Perelman approached the board calmly and wrote out the solution.

  “Yes,” said the instructor. “This student solved the problem correctly.”

  Where Perelman and his crowd came from, a high-school student who could not produce the solution to the Cauchy Problem on request would be disdained as an imbecile—“and rightly so,” commented Golovanov. Still, when the instructor was in a position of authority, Perelman seemed willing to submit to ridiculous exercises without protestation. Later, what he perceived as the need to prove his worthiness to his peers or to academic authorities infuriated him instantly, but within the confines of the university, he apparently gave professors almost unlimited license. This particular computer-science instructor also had the bizarre custom of nailing his students’ notes to their desks—to ensure that students actually attended the sessions rather than borrowed one another’s notes. Perelman tolerated this indignity too and helped the rest of his group by verbally summarizing the notes.

  He was loyal to his group as long as no one broke the rules as he perceived them. A Mathmech custom dictated that students help peers who found themselves stuck during a written test. Outright cheating was impossible, since every student had an individual problem set, drawn at random from a large pool. But if one was stalled desperately, one could generally pass a note to another student that briefly summarized the issue. The response was never a solution but often something along the lines of “Try this tack.” Perelman, the universal problem-solver, the fastest thinker in his age group in the Soviet Union and perhaps the world, would have been the best person to answer these sorts of questions. He was, however, unwilling to entertain them,3 and he let his disapproval of the practice be known: everyone had to solve his own problem for himself.

  Somewhere in the transition from adolescence to adulthood, Perelman seemed to have found a way to relieve the tension between prevailing social mores, which he perceived as illogical, internally inconsistent, and perpetually shifting—and they certainly were all of these things—and his idea of how the world should work. He derived a set of his own rules based on the few values he knew to be absolute and proceeded to follow them. As new situations presented themselves, he figured out the rules that applied to them—this too may have seemed inconsistent and shifting to an observer, but only because the observer did not know the algorithm. Naturally, Perelman expected the rest of the world to follow his rules; it would not have occurred to him that other people did not know them. After all, the rules were based on universal values, honesty being primary among them. Honesty meant always telling the whole truth, which is to say, all of the available accurate information—much as Perelman did when he supplied his proofs with information extraneous to the actual solution. Clearly, in the case of a student taking a Mathmech test, supplying all of the available information would have included naming the person with whom the idea for the solution had ori
ginated, and that would truly have been inconsistent with the rule that every student must do his own work. Later, he would view, say, sloppy footnoting, as practiced by many mathematicians, as plagiarism. It is possible too that a bit of the competitor’s habit shaped his perception of the written tests; after all, they did look and, perhaps for Perelman, feel a bit like the olympiad, and it would have been inconceivable for a competitor to ask his fellow problem-solvers for hints.

  In the third year, each Mathmech student chose a specialty that would presumably take him through graduate school and into a research career. Golovanov chose number theory. It was a natural choice for a boy who could be knocked out of competitions upon encountering a geometry problem and who seemed to relate to numbers as others did to people. Perelman chose his own destiny. He had picked geometry, he told his group cryptically, because he wanted to go into a field populated by a few remaining dinosaurs so that he might also become one of them. In the 1980s in Leningrad, geometry seemed like an anachronism: it had none of the flair of computer science and none of the romance of numbers, and its practitioners were indeed a few larger-than-life old men. One of his classmates, Mehmet Muslimov, remembered that Perelman’s declaration had not sounded pretentious. If anything, it sounded logical: here was a person from another time and place, odd and differently minded even in an environment as full of eccentrics as a university mathematics department; it was only reasonable that he would consciously fashion himself into a dinosaur. What Perelman may also have been telling his classmates was that he felt quite exasperated with his fellow humans and their ways, and his chosen field seemed to attract the few people whose internal codes of conduct were as strict as his own.