Perfect Rigour

by Masha Gessen

  Did this explain Rukshin’s unprecedented coaching success? Like many insecure people, Rukshin tended to oscillate between self-effacement and self-aggrandizement, now telling me that he was no more than a mediocre mathematician himself, now telling me for the fifth time in three days that he had been offered a job with the Ministry of Education in Moscow (he turned it down). Similarly, he told me several times that his teaching methods could be reproduced, and had been, to rather spectacular results: his students made money by training math competitors all over the former Soviet bloc. But other times he told me he was a magician, and these were the times he seemed most sincere. “There are several stages of teaching,” he said. “There are the student, apprenticeship stages, like in the medieval guild. Then there are the craftsman, the master—these are the stages of mastery. Then there is the art stage. But there is a stage beyond the art stage. This is the witchcraft stage. A sort of magic. It’s a question of charisma and all sorts of other things.”

  It may also have been that Rukshin was more driven than any coach before or since. He did some research work in mathematics, but mathematics seemed to be almost a sideline of his life’s work: creating world-class mathematics competitors. That kind of single-minded passion can look and feel very much like magic.

  Magicians need willing, impressionable subjects to work their craft. Rukshin, who was so wrong for the job of mathematics teacher for so many external reasons, cast about not just for the most likely child genius but also for the best way to prove he could make a mathematician out of a child. He focused his attention not on the loudest boy, or the quickest-thinking boy, or the most fiercely competitive boy, but on the most obviously absorbent boy.

  Rukshin claims not to have appreciated the power of Perelman’s mind right away. He had helped judge some of the district competitions in Leningrad in 1976, reading through many sheets of graph paper with ten- to twelve-year-olds’ solutions to math problems. He was on the lookout for kids who might amount to something mathematically; the unwritten rules of math clubs allowed them to recruit but not poach, so an unknown like Rukshin had to look for kids early and aggressively. Perelman’s set of solutions went on the list; the child’s answers were correct, and he arrived at them in ways that were sometimes unexpected. Rukshin saw nothing in those solution sets that would have placed the child head and shoulders above the rest, but he saw solid promise. So when Professor Natanson called and said the child’s name, Rukshin recognized it. And when he finally saw the boy, he recognized in him the promise of something bigger than a good mathematician: the fulfillment of Rukshin’s ambition to be the best math coach who had ever lived. Adjusting his judgment of Perelman so quickly must have required something of a leap of faith for Rukshin, but it also promised the reward of making a singular discovery—that a child who seemed as capable as dozens of others would surpass them all.

  “When everyone is studying math and there is one person who can learn much better than others, then he inevitably receives more attention: the teacher comes to the home, he tells him things.” Alexander Golovanov spoke from experience: not only had he spent years studying mathematics alongside Perelman, but he had spent most of his adult life coaching children and teenagers for mathematics competitions. He was Rukshin’s anointed heir. And now he explained to me just what it meant to have a favorite student, or to be one. As in any human relationship, love can engender commitment, which can engender investment, which in turn deepens the commitment and perhaps even the love. “So that is one definition of a favorite pupil, and Grisha was that: a favorite pupil because he had been given more. Another aspect, a very important one, is that anyone who teaches [competitive mathematics] has a very clear idea of how much he has done—what he can and cannot take credit for. Say, there are kids who have been to the [all-Russian] olympiad three or four times—and I can say that if I hadn’t taught them, they would have made it two rather than three times. So I wasn’t the main reason. And then there are people about whom I can say that yes, I was the main reason. That doesn’t mean they were pathetic and I put a brain in their heads. What it means is love. And what I think is that Rukshin feels that way about Grisha. And I also think he is right.” There was a third aspect too, said Golovanov, one that had to do with pure closeness. Rukshin was a hypochondriac whom the erudite Golovanov compared to Voltaire. Over the months when I was in contact with Rukshin, he spent no less than a third of his time in hospitals. “So there was one time when Rukshin was going blind,” Golovanov remembered. “It was during summer camp, and he and Grisha were sharing a room.” Perelman was then a university student working as a teaching assistant to Rukshin. “And one morning Rukshin said he’d felt great joy upon awakening because he saw Grisha lying in the other bed. And there was no telling what pleased him more: that he could see in general or that he could see Grisha in particular.”

  At some point, the care and teaching of Perelman became the thing that gave meaning to Rukshin’s life; Rukshin, for his part, strove to insert meanings into Perelman’s head. He got Grisha to quit the violin—and the derision with which he spoke of it almost thirty years later impressed me. “It’s the shtetl dream.” He scowled. “Learn the fiddle and play at weddings and funerals.”

  Like every competitive sports coach, Rukshin disliked it when his boys spent their time doing anything else. He claimed that he’d kicked Alexander Khalifman, the future chess world champion, out of his club for failing to choose math over chess. And like many coaches, he claimed his sport was the fairest, truest, and most beautiful sport of all. Also like many coaches, he saw it as his mission to shape not only his students’ competitive skills but their entire personalities. When they grew older, Rukshin hounded10 any boy who was sighted doing something as undignified and distracting as kissing a girl—and he caught them with such regularity that the boys began to suspect he had spies shadowing them. Perelman never disappointed his teacher in this way; as Rukshin repeatedly told me, “He was never interested in girls.”

  Two evenings a week Rukshin, accompanied by his math boys and a couple of girls, walked from the Palace of Pioneers to the Vitebsk Railroad Station, where he and Grisha boarded the same train. Rukshin, who had married very early, lived with his wife and mother-in-law outside the city in the historic town of Pushkin; Grisha lived with his mother, father, and baby sister on the far southern outskirts of the city, in a dreary concrete apartment block in the Kupchino neighborhood. Rukshin and his pupil rode the subway together to Kupchino, which was the last stop, where Grisha would get off and walk home and Rukshin would switch to a commuter train with hard wooden seats and ride another twenty minutes to Pushkin. Along the way, Rukshin discovered things about Grisha. He learned, for example, that Grisha would not untie the earpieces of his fur hat while riding the subway. “It’s not just that he would not take the hat off,” recalled Rukshin. “He would not even untie the ears, saying that his mother would kill him because she told him never to untie the hat or he’d catch cold.” The subway car was generally heated to normal room temperature, but the compactor in Grisha’s brain left no room for the nuance of circumstance. Rules were rules.

  When Rukshin criticized Grisha for not reading enough—Rukshin saw it as his duty to introduce the children not only to mathematics but to literature and music—Grisha asked why he should be reading books. To Rukshin’s argument that reading was “interesting,” Grisha responded that anything that needed to be read would be included on the school’s required-reading list. Rukshin had better luck with music. When Grisha came to the club, his taste was limited to clear and precise classical instrumental music, generally with a violin solo. While solving a problem, he often engaged in what his club mates alternately called “howling” and “acoustic terror,” but when he was asked, Grisha explained that he was humming Camille Saint-Saëns’ Introduction and Rondo Capriccioso, a composition for violin and orchestra remarkable for both its clarity and the prominence of a virtuoso violin soloist. However,
at one of the summer camps, Rukshin succeeded in interesting his pupil in vocal music, through which Grisha proceeded to move systematically: he accepted the lower-range voices first, then gradually moved through to the sopranos, but he drew the line at Rukshin’s attempt to introduce him to the singing of castrati, which he deemed “unnatural” and therefore “uninteresting.”

  Far from being disappointed in his student, Rukshin seemed to rejoice in Perelman’s lopsided nature. In this love pairing of teacher and student, each continuously got to be the other’s better half. Perelman could be the competitor Rukshin never was, while Rukshin could interact with the outside world on Perelman’s behalf and shield his student from it at the same time. They—or, rather, Rukshin—created situations in which they complemented each other in more practical ways too. At summer camp, where fifteen-year-old Perelman lived away from his mother for the first time in his life, Rukshin took care of his day-to-day needs. Personal hygiene was tricky, but Rukshin occasionally managed to get Perelman to change his socks and underwear and pack the soiled items away in a plastic bag, since he refused to wash them—as, often, he refused to wash himself. He also refused to go swimming with the rest of the boys, both because he disliked the water and, more important, because he did not see the point of such a nonintellectual and noncompetitive pastime (he did play Ping-Pong, and was very good at it, and very competitive). So Rukshin used him as an extension of himself: Rukshin got in the water with the children and swam in the deep end, using his own body to mark the line the children were not allowed to cross; Perelman sat on the shore and kept a constant head count, making sure no one went missing. As time went on, Rukshin found other ways to use Perelman’s brain as a more efficient extension of his own. As a university student, for example, Perelman would sift through thousands of math problems to select problem sets for training. “It’s work that I could have done and spent, say, t amount of time on it,” Rukshin told me. “Grisha did it in t over five. Now these problem sets are club classics and no one remembers at this point what was done by me and what was done by Perelman.”

  It was a match made in mathematical heaven.

  PERFECT RIGOR

  A BEAUTIFUL SCHOOL

  3

  A Beautiful School

  AS PERELMAN MATURED, he learned to take the words that bunched up in his mouth and combine them to form sentences—beautiful, precise, correct sentences—but his narrative remained tangled and personal. The reigning star of the club for the first three or four years, a boy named Alexander Levin, would, said Rukshin, “explain his solution with the idea of helping people understand how to solve these sorts of problems. Perelman told the story of his own personal communication with this particular problem. Imagine the difference between a doctor filling out a medical history and the patient’s mother talking about sitting by her child’s bedside, wiping his brow and listening to his labored breathing. So did Grisha tell the story of his own journey through the problem. And if the solution could have been different or even shorter, Grisha would still only tell the story of how he had solved it. After he talked, I often had to go up to the blackboard and point out what was important and what could have been cut or simplified—not because he did not see it himself but because he was not the one who would do it.”

  It is remarkable that Perelman learned to explain as well as he did. Imagine how unmanageable everyday language is for someone given to understanding things literally. Language is not just a frustratingly imprecise way of trying to navigate the world but also a willfully and outrageously inaccurate one. The psychologist and linguist Steven Pinker observed1 that “language describes space in a way that is unlike anything known to geometry, and it can sometimes leave listeners up in the air, at sea, or in the dark as to where things are.” In speech, noted Pinker, objects have primary and secondary dimensions, ranked by importance. A road is imagined as one-dimensional, as is a river or a ribbon—all of them consist of length only, like a segment in geometry. “A layer or a slab has two primary dimensions,2 defining a surface,” continued Pinker, “and a bounded secondary dimension, its thickness. A tube or a beam has a single primary dimension, its length, and two secondary dimensions, plumping out its cross-section.”

  Even greater trouble with language begins when we split up objects into their contents and their boundaries. We describe a stripe as the boundary of a plate, and we portray both objects as two-dimensional, and to a literal mind, all of this is wrong: the stripe is not the actual boundary of the plate (the plate’s edge is), and the plate has three dimensions. At the same time, words like end and edge are used3 to denote shapes that have anywhere from zero to three dimensions. What is worse, the sloppy way of describing objects coexists in language with an extreme wealth of names for actual shapes. There may be as many as ten thousand shape-names in English; and in all human languages, the number of shape nouns far exceeds the ability to define them. To a literal mind, this is an outrage: how can we use words for things that we not only cannot define properly but insist on defining incorrectly?

  Take the Möbius strip, the length of ribbon famously twisted before being reconnected to itself. Language is stumped by the Möbius strip. Does something move along the strip, as with a one-dimensional object; around the strip, as with a two-dimensional object; or, as in the title of a 2006 animated film, “thru” the strip—suggesting a three-dimensional object? For a literal mind, salvation lies in the geometry that lives in the imagination—where every shape is clearly defined. In fact, geometry as it is studied in secondary school, with its basic theorems and its precise measurements, represents a marked improvement over everyday speech, but it is topology that is the quintessence of geometrical clarity. Not coincidentally, the Möbius strip, which evades casual understanding, is among the earliest known objects of topological inquiry.4 Clearly defined, in the case of topology, does not mean that every shape can easily be visualized. Quite the opposite: it means that every shape has only those qualities that its definition grants it. A shape has a certain number of dimensions; it may be bounded; it may or may not be smooth; and it may or may not be simply connected, which is to say, it may or may not have holes. An object in topology may be a sphere—that is, all of its constituting points are an equal distance from the center—but a topologist notes that the essential qualities of a sphere do not change if the sphere is dented; the sphere can easily be reshaped, so its temporary change in imaginary appearance may be disregarded. Not so if a hole appears in the sphere: the sphere is no longer a sphere but a torus, an object with a different relationship to that which surrounds it and one that cannot easily be reconstituted as a sphere. The topological universe has no use for silly riddles like those of which Pinker is fond: “What can you put in a bucket to make it lighter?” “A hole!” This is not funny to the literal mind. You cannot put a hole anywhere. Moreover, a hole—or an additional hole—means the shape is no longer what it was; the bucket would not be made lighter because it would no longer be a bucket.

  Normally, even mathematicians do not begin to study topology until they have entered college; the discipline has traditionally been considered too abstract to present to children. But a mind like Grisha Perelman’s, an undeniably mathematical mind that was at the same time neither visual nor numeric—a mind that thought in systems, that traded in definitions—was a mind born for topology. Starting roughly when Perelman was in eighth grade (when he was around thirteen), visiting lecturers at the math club sometimes taught a class in topology. Topology called to Perelman from beyond the more traditional geometry he had already navigated, the same way the lights of Broadway call to the child who moves the audience to tears in a middle-school production of Annie. Grisha Perelman would grow up to live in the universe of topology. He would master all its rules and definitions. He would be a lawyer in the court of shapes, eventually able to argue precisely and articulately why a three-dimensional, simply connected closed object would always be a sphere. Rukshin wo
uld light Perelman’s way there; he came to Perelman as an emissary from his mathematical future, and his implicit promise was that he would make Perelman’s life in Leningrad as safe and as ordered as his life in the imagination.

  For this, there was Leningrad’s Specialized Mathematics School Number 239.

  The summer Grisha Perelman turned fourteen, he took the train from Kupchino to Pushkin every morning and spent the day being tutored by Rukshin in the English language. The plan was to cover four years’ worth of English in one summer so that in September Perelman could enter Leningrad’s Specialized Mathematics and Physics School Number 239. This was the shortest path to engaging with mathematics fully, with minimal outside disturbances.

  The strange story of the specialized math schools goes back to Andrei Kolmogorov. Having been so essential to the war effort during World War II, Kolmogorov alone among the top Soviet mathematicians avoided being drafted into the postwar military effort. His students always wondered why5—and the only likely explanation seems to be Kolmogorov’s homosexuality. His lifelong partner, with whom he shared a home starting in 1929,6 was the topologist Pavel Alexandrov. Five years after the couple started living together, the Soviet Union criminalized male homosexuality, but Kolmogorov and Alexandrov, who exercised minimal discretion—they called each other “friends” but made no secret of the life-shaping nature of their relationship—apparently had no trouble with the law. The academic world accepted them as a pair, if not a couple: they generally requested academic appointments together, booked their accommodations together7 at Academy of Sciences resorts, and made donations to military relief efforts together. In his last interview, recorded for a documentary film about his life, the eighty-year-old Kolmogorov asked the filmmaker to use Johann Sebastian Bach’s Double Violin Concerto8—a baroque composition based on the interplay of two violins—when showing the home he had made with Alexandrov.