Perfect Rigour E-Book

Masha Gessen

• Herausgeber: Icon Books
• Kategorie: Sachliteratur, Reportagen, Biografien
• Sprache: Englisch
• Veröffentlichungsjahr: 2011

Seitenzahl: 372

Beschreibung

In 2006, an eccentric Russian mathematician named Grigori Perelman solved one of the world's greatest intellectual puzzles. The Poincare conjecture is an extremely complex topological problem that had eluded the best minds for over a century. In 2000, the Clay Institute in Boston named it one of seven great unsolved mathematical problems, and promised a million dollars to anyone who could find a solution. Perelman was awarded the prize this year - and declined the money. Journalist Masha Gessen was determined to find out why. Drawing on interviews with Perelman's teachers, classmates, coaches, teammates, and colleagues in Russia and the US - and informed by her own background as a math whiz raised in Russia - she set out to uncover the nature of Perelman's astonishing abilities. In telling his story, Masha Gessen has constructed a gripping and tragic tale that sheds rare light on the unique burden of genius.

Leseprobe

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This electronic edition published in 2011 by Icon Books

ISBN: 978-1-84831-309-5 (ePub format)

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Text copyright © 2009, 2011 Masha Gessen

The author has asserted her moral rights.

No part of this book may be reproduced in any form, or by any

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Typeset by Marie Doherty

BOOKS BY MASHA GESSEN

Blood Matters: From Inherited Illness to Designer Babies, How the World and I Found Ourselves in the Future of the Gene

Ester and Ruzya: How My Grandmothers Survived Hitler’s War and Stalin’s Peace

Dead Again: The Russian Intelligentsia After Communism

In the Here and There,by Valeria Narbikova (as translator)

Half a Revolution: Contemporary Fiction by Russian Women(as editor and translator)

Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century

Contents

Prologue: A Problem for a Million Dollars

Chapter 1: Escape into the Imagination

Chapter 2: How to Make a Mathematician

Chapter 3: A Beautiful School

Chapter 4: A Perfect Score

Chapter 5: Rules for Adulthood

Chapter 6: Guardian Angels

Chapter 7: Round Trip

Chapter 8: The Problem

Chapter 9: The Proof Emerges

Chapter 10: The Madness

Chapter 11: The Million-Dollar Question

Epilogue

Acknowledgments

Notes

PROLOGUE

A Problem for a Million Dollars

Numbers cast a magic spell over all of us, but mathematicians are especially skilled at imbuing figures with meaning. In the year 2000, a group of the world’s leading mathematicians gathered in Paris for a meeting that they believed would be momentous. They would use this occasion to take stock of their field. They would discuss the sheer beauty of mathematics—a value that would be understood and appreciated by everyone present. They would take the time to reward one another with praise and, most critical, to dream. They would together try to envision the elegance, the substance, the importance of future mathematical accomplishments.

The Millennium Meeting had been convened by the Clay Mathematics Institute, a nonprofit organization founded by Boston-area businessman Landon Clay and his wife, Lavinia, for the purposes of popularizing mathematical ideas and encouraging their professional exploration. In the two years of its existence, the institute had set up a beautiful office in a building just outside Harvard Square in Cambridge, Massachusetts, and had handed out a few research awards. Now it had an ambitious plan for the future of mathematics, “to record the problems of the twentieth century that resisted challenge most successfully and that we would most like to see resolved,” as Andrew Wiles, the British number theorist who had famously conquered Fermat’s Last Theorem, put it. “We don’t know how they’ll be solved or when: it may be five years or it may be a hundred years. But we believe that somehow by solving these problems we will open up whole new vistas of mathematical discoveries and landscapes.”1

As though setting up a mathematical fairy tale, the Clay Institute named seven problems—a magic number in many folk traditions—and assigned the fantastical value of one million dollars for each one’s solution. The reigning kings of mathematics gave lectures summarizing the problems. Michael Francis Atiyah, one of the previous century’s most influential mathematicians, began by outlining the Poincaré Conjecture, formulated by Henri Poincaré in 1904. The problem was a classic of mathematical topology. “It’s been worked on by many famous mathematicians, and it’s still unsolved,” stated Atiyah. “There have been many false proofs. Many people have tried and have made mistakes. Sometimes they discovered the mistakes themselves, sometimes their friends discovered the mistakes.” The audience, which no doubt contained at least a couple of people who had made mistakes while tackling the Poincaré, laughed.

Atiyah suggested that the solution to the problem might come from physics. “This is a kind of clue—hint—by the teacher who cannot solve the problem to the student who is trying to solve it,” he joked. Several members of the audience were indeed working on problems that they hoped might move mathematics closer to a victory over the Poincaré. But no one thought a solution was near. True, some mathematicians conceal their preoccupations when they’re working on famous problems—as Wiles had done while he was working on Fermat’s Last—but generally they stay abreast of one another’s research. And though putative proofs of the Poincaré Conjecture had appeared more or less annually, the last major breakthrough dated back almost twenty years, to 1982, when the American Richard Hamilton laid out a blueprint for solving the problem. He had found, however, that his own plan for the solution—what mathematicians call a program—was too difficult to follow, and no one else had offered a credible alternative. The Poincaré Conjecture, like Clay’s other Millennium Problems, might never be solved.

Solving any one of these problems would be nothing short of a heroic feat. Each had claimed decades of research time, and many a mathematician had gone to the grave having failed to solve the problem with which he or she had struggled for years. “The Clay Mathematics Institute really wants to send a clear message, which is that mathematics is mainly valuable because of these immensely difficult problems, which are like the Mount Everest or the Mount Himalaya of mathematics,” said the French mathematician Alain Connes, another twentieth-century giant. “And if we reach the peak, first of all, it will be extremely difficult—we might even pay the price of our lives or something like that. But what is true is that when we reach the peak, the view from there will be fantastic.”

As unlikely as it was that anyone would solve a Millennium Problem in the foreseeable future, the Clay Institute nonetheless laid out a clear plan for giving each award. The rules stipulated that the solution to the problem would have to be presented in a refereed journal, which was, of course, standard practice. After publication, a two-year waiting period would begin, allowing the world mathematics community to examine the solution and arrive at a consensus on its veracity and authorship. Then a committee would be appointed to make a final recommendation on the award. Only after it had done so would the institute hand over the million dollars. Wiles estimated that it would take at least five years to arrive at the first solution—assuming that any of the problems was actually solved—so the procedure did not seem at all cumbersome.

Just two years later, in November 2002, a Russian mathematician posted his proof of the Poincaré Conjecture on the Internet. He was not the first person to claim he’d solved the Poincaré—he was not even the only Russian to post a putative proof of the conjecture on the Internetthat year—but his proof turned out to be right.

And then things did not go according to plan—not the Clay Institute’s plan or any other plan that might have struck a mathematician as reasonable. Grigory Perelman, the Russian, did not publish his work in a refereed journal. He did not agree to vet or even to review the explications of his proof written by others. He refused numerous job offers from the world’s best universities. He refused to accept the Fields Medal, mathematics’ highest honor, which would have been awarded to him in 2006. And then he essentially withdrew from not only the world’s mathematical conversation but also most of his fellow humans’ conversation.

Perelman’s peculiar behavior attracted the sort of attention to the Poincaré Conjecture and its proof that perhaps no other story of mathematics ever had. The unprecedented magnitude of the award that apparently awaited him helped heat up interest too, as did a sudden plagiarism controversy in which a pair of Chinese mathematicians claimed they deserved the credit for proving the Poincaré. The more people talked about Perelman, the more he seemed to recede from view; eventually, even people who had once known him well said that he had “disappeared,” although he continued to live in the St. Petersburg apartment that had been his home for many years. He did occasionally pick up the phone there—but only to make it clear that he wanted the world to consider him gone.

When I set out to write this book, I wanted to find answers to three questions: Why was Perelman able to solve the conjecture; that is, what was it about his mind that set him apart from all the mathematicians who had come before? Why did he then abandon mathematics and, to a large extent, the world? Would he refuse to accept the Clay prize money, which he deserved and most certainly could use, and if so, why?

This book was not written the way biographies usually are. I did not have extended interviews with Perelman. In fact, I had no conversations with him at all. By the time I started working on this project, he had cut off communication with all journalists and most people. That made my job more difficult—I had to imagine a person I had literally never met—but also more interesting: it was an investigation. Fortunately, most people who had been close to him and to the Poincaré Conjecture story agreed to talk to me. In fact, at times I thought it was easier than writing a book about a cooperating subject, because I had no allegiance to Perelman’s own narrative and his vision of himself—except to try to figure out what it was.

PERFECT RIGOR

ESCAPE INTO THE IMAGINATION

1

Escape into the Imagination

AS ANYONE WHOhas attended grade school knows, mathematics is unlike anything else in the universe. Virtually every human being has experienced that sense of epiphany when an abstraction suddenly makes sense. And while grade-school arithmetic is to mathematics roughly what a spelling bee is to the art of novel writing, the desire to understand patterns—and the childlike thrill of making an inscrutable or disobedient pattern conform to a set of logical rules—is the driving force of all mathematics.

Much of the thrill lies in the singular nature of the solution. There is only one right answer, which is why most mathematicians hold their field to be hard, exact, pure, and fundamental, even if it cannot precisely be called a science. The truth of science is tested by experiment. The truth of mathematics is tested by argument, which makes it more like philosophy, or, even better, the law, a discipline that also assumes the existence of a single truth. While the other hard sciences live in the laboratory or in the field, tended to by an army of technicians, mathematics lives in the mind. Its lifeblood is the thought process that keeps a mathematician turning in his sleep and waking with a jolt to an idea, and the conversation that alters, corrects, or affirms the idea.

“The mathematician needs no laboratories or supplies,”1wrote the Russian number theorist Alexander Khinchin. “A piece of paper, a pencil, and creative powers form the foundation of his work. If this is supplemented with the opportunity to use a more or less decent library and a dose of scientific enthusiasm (which nearly every mathematician possesses), then no amount of destruction can stop the creative work.” The other sciences as they have been practiced since the early twentieth century are, by their very natures, collective pursuits; mathematics is a solitary process, but the mathematician is always addressing another similarly occupied mind. The tools of that conversation—the rooms where those essential arguments take place—are conferences, journals, and, in our day, the Internet.

That Russia produced some of the twentieth century’s greatest mathematicians is, plainly, a miracle. Mathematics was antithetical to the Soviet way of everything. It promoted argument; it studied patterns in a country that controlled its citizens by forcing them to inhabit a shifting, unpredictable reality; it placed a premium on logic and consistency in a culture that thrived on rhetoric and fear; it required highly specialized knowledge to understand, making the mathematical conversation a code that was indecipherable to an outsider; and worst of all, mathematics laid claim to singular and knowable truths when the regime had staked its legitimacy on its own singular truth. All of this is what made mathematics in the Soviet Union uniquely appealing to those whose minds demanded consistency and logic, unattainable in virtually any other area of study. It is also what made mathematics and mathematicians suspect. Explaining what makes mathematics as important and as beautiful as mathematicians know it to be, the Russian algebraist Mikhail Tsfasman said, “Mathematics is uniquely suited to teaching2one to distinguish right from wrong, the proven from the unproven, the probable from the improbable. It also teaches us to distinguish that which is probable and probably true from that which, while apparently probable, is an obvious lie. This is a part of mathematical culture that the [Russian] society at large so sorely lacks.”

It stands to reason that the Soviet human rights movement was founded by a mathematician. Alexander Yesenin-Volpin, a logic theorist, organized the first demonstration in Moscow in December 1965. The movement’s slogans were based on Soviet law,3and its founders made a single demand: they called on the Soviet authorities to obey the country’s written law. In other words, they demanded logic and consistency; this was a transgression, for which Yesenin-Volpin was incarcerated in prisons and psychiatric wards for a total of fourteen years and ultimately forced to leave the country.

Soviet scholarship, and Soviet scholars, existed to serve the Soviet state. In May 1927, less than ten years after the October Revolution, the Central Committee inserted into the bylaws of the USSR’s Academy of Sciences a clause specifying just this. A member of the Academy may be stripped of his status, the clause stated, “if his activities are apparently aimed at harming the USSR.” From that point on, every member of the Academy was presumed guilty of aiming to harm the USSR. Public hearings involving historians, literary scholars, and chemists ended with the scholars publicly disgraced, stripped of their academic regalia, and, frequently, jailed on treason charges. Entire fields of study—most notably genetics—were destroyed for apparently coming into conflict with Soviet ideology. Joseph Stalin personally ruled scholarship. He even published his own scientific papers, thereby setting the research agenda in a given field for years to come. His article on linguistics,4for example, relieved comparative language study of a cloud of suspicion that had hung over it and condemned, among other things, the study of class distinctions in language as well as the whole field of semantics. Stalin personally promoted5a crusading enemy of genetics, Trofim Lysenko, and apparently coauthored Lysenko’s talk that led to an outright ban of the study of genetics in the Soviet Union.

What saved Russian mathematics from destruction by decree was a combination of three almost entirely unrelated factors. First, Russian mathematics happened to be uncommonly strong right when it might have suffered the most. Second, mathematics proved too obscure for the sort of meddling the Soviet leader most liked to exercise. And third, at a critical moment it proved immensely useful to the State.

In the 1920s and ’30s, Moscow boasted a robust mathematical community; groundbreaking work was being done in topology, probability theory, number theory, functional analysis, differential equations, and other fields that formed the foundation of twentieth-century mathematics. Mathematics is cheap, and this helped: when the natural sciences perished for lack of equipment and even of heated space in which to work, the mathematicians made do with their pencils and their conversations. “A lack of contemporary literature was, to some extent, compensated by ceaseless scientific communication, which it was possible to organize and support in those years,” wrote Khinchin about that period. An entire crop of young mathematicians, many of whom had received part of their education abroad, became fast-track professors and members of the Academy in those years.

The older generation of mathematicians—those who had made their careers before the revolution—were, naturally, suspect. One of them, Dimitri Egorov,6the leading light of Russian mathematics at the turn of the twentieth century, was arrested and in 1931 died in internal exile. His crimes: he was religious and made no secret of it, and he resisted attempts to ideologize mathematics—for example, trying (unsuccessfully) to sidetrack a letter of salutation sent from a mathematicians’ congress to a Party congress. Egorov’s vocal supporters were cleansed from the leadership of Moscow mathematical institutions, but by the standards of the day, this was more of a warning than a purge: no area of study was banned, and no general line was imposed by the Kremlin. Mathematicians would have been well advised to brace for a bigger blow.

In the 1930s, a mathematical show trial was all set to go forward. Egorov’s junior partner in leading the Moscow mathematical community was his first student, Nikolai Luzin, a charismatic teacher himself whose numerous students called their circle Luzitania, as though it were a magical country, or perhaps a secret brotherhood united by a common imagination. Mathematics, when taught by the right kind of visionary, does lend itself to secret societies. As most mathematicians are quick to point out, there are only a handful of people in the world who understand what the mathematicians are talking about. When these people happen to talk to one another—or, better yet, form a group that learns and lives in sync—it can be exhilarating.

“Luzin’s militant idealism,” wrote a colleague who denounced Luzin, “is amply expressed by the following quote from his report to the Academy on his trip abroad: ‘It seems the set of natural numbers is not an absolutely objective formation. It seems it is a function of the mind of the mathematician who happens to be speaking of a set of natural numbers at the given moment. It seems there are, among the problems of arithmetic, those that absolutely cannot be solved.’”

The denunciation was masterful: the addressee did not need to know anything about mathematics and would certainly know that solipsism, subjectivity, and uncertainty were utterly un-Soviet qualities. In July 1936 a public campaign against the famous mathematician was launched in the dailyPravda, where Luzin was exposed as “an enemy wearing a Soviet mask.”

The campaign against Luzin continued with newspaper articles, community meetings, and five days of hearings by an emergency committee formed by the Academy of Sciences. Newspaper articles exposed Luzin and other mathematicians as enemies because they published their work abroad. In other words, events unfolded in accordance with the standard show-trial scenario. But then the process seemed to fizzle out: Luzin publicly repented and was severely reprimanded although allowed to remain a member of the Academy. A criminal investigation into his alleged treason was quietly allowed to die.

Researchers who have studied the Luzin case7believe it was Stalin himself who ultimately decided to stop the campaign. The reason, they think, is that mathematics is useless for propaganda. “The ideological analysis of the case would have devolved to a discussion of the mathematician’s understanding of a natural number set, which seemed like a far cry from sabotage, which, in the Soviet collective consciousness, was rather associated with coal mine explosions or killer doctors,” wrote Sergei Demidov and Vladimir Isakov, two mathematicians who teamed up to study the case when this became possible, in the 1990s. “Such a discussion would better be conducted using material more conducive to propaganda, such as, say, biology and Darwin’s theory of evolution, which the great leader himself was fond of discussing. That would have touched on topics that were ideologically charged and easily understood: monkeys, people, society, and life itself. That’s so much more promising than the natural number set or the function of a real variable.”

Luzin and Russian mathematics were very, very lucky.

Mathematics survived the attack but was permanently hobbled. In the end, Luzin was publicly disgraced and dressed down for practicing mathematics: publishing in international journals, maintaining contacts with colleagues abroad, taking part in the conversation that is the life of mathematics. The message of the Luzin hearings, heeded by Soviet mathematicians well into the 1960s and, to a significant extent, until the collapse of the Soviet Union, was this: Stay behind the Iron Curtain. Pretend Soviet mathematics is not just the world’s most progressive mathematics—this was its official tag line—but the world’s only mathematics. As a result, Soviet and Western mathematicians,8unaware of one another’s endeavors, worked on the same problems, resulting in a number of double-named concepts such as the Chaitin-Kolmogorov complexities and the Cook-Levin theorem. (In both cases the eventual coauthors worked independently of each other.) A top Soviet mathematician,9Lev Pontryagin, recalled in his memoir that during his first trip abroad, in 1958—five years after Stalin’s death—when he was fifty years old and world famous among mathematicians, he had had to keep asking colleagues if his latest result was actually new; he did not really have another way of knowing.

“It was in the 1960s10that a couple of people were allowed to go to France for half a year or a year,” recalled Sergei Gelfand, a Russian mathematician who now runs the American Mathematics Society’s publishing program. “When they went and came back, it was very useful for all of Soviet mathematics, because they were able to communicate there and to realize, and make others realize, that even the most talented of people, when they keep cooking in their own pot behind the Iron Curtain, they don’t have the full picture. They have to speak with others, and they have to read the work of others, and it cut both ways: I know American mathematicians who studied Russian just to be able to read Soviet mathematics journals.” Indeed, there is a generation of American mathematicians who are more likely than not to possess a reading knowledge of mathematical Russian—a rather specialized skill even for a native Russian speaker; Jim Carlson, president of the Clay Mathematics Institute, is one of them. Gelfand himself left Russia in the early 1990s because he was drafted by the American Mathematics Society to fill the knowledge gap that had formed during the years of the Soviet reign over mathematics: he coordinated the translation and publication in the United States of Russian mathematicians’ accumulated work.

So some of what Khinchin described as the tools of a mathematician’s labor—“a more or less decent library” and “ceaseless scientific communication”—were stripped from Soviet mathematicians. They still had the main prerequisites, though—“a piece of paper, a pencil, and creative powers”—and, most important, they had one another: mathematicians as a group slipped by the first rounds of purges because mathematics was too obscure for propaganda. Over the nearly four decades of Stalin’s reign, however, it would turn out that nothing was too obscure for destruction. Mathematics’ turn would surely have come if it weren’t for the fact that at a crucial point in twentieth-century history, mathematics left the realm of abstract conversation and suddenly made itself indispensable. What ultimately saved Soviet mathematicians and Soviet mathematics was World War II and the arms race that followed it.

Nazi Germany invaded the Soviet Union on June 22, 1941. Three weeks later, the Soviet air force was gone:11bombed out of existence in the airfields before most of the planes ever took off. The Russian military set about retrofitting civilian airplanes for use as bombers. The problem was, the civilian airplanes were significantly slower than the military ones, rendering moot everything the military knew about aim. A mathematician was needed to recalculate speeds and distances so the air force could hit its targets. In fact, a small army of mathematicians was needed. The greatest Russian mathematician of the twentieth century, Andrei Kolmogorov,12returned to Moscow from the academics’ wartime haven in Tatarstan and led a classroom full of students armed with adding machines in recalculating the Red Army’s bombing and artillery tables. When this work was done, he set about creating a new system of statistical control and prediction for the Soviet military.

At the beginning of World War II, Kolmogorov was thirty-eight years old, already a member of the Presidium of the Soviet Academy of Sciences—making him one of a handful of the most influential academics in the empire—and world famous for his work in probability theory. He was also an unusually prolific teacher: by the end of his life he had served as an adviser on seventy-nine dissertations13and had spearheaded both the math olympiads system and the Soviet mathematics-school culture. But during the war, Kolmogorov put his scientific career on hold to serve the Soviet state directly—proving in the process that mathematicians were essential to the State’s very survival.

The Soviet Union declared victory—and the end of what it called the Great Patriotic War—on May 9, 1945. In August, the United States dropped atomic bombs on the Japanese cities of Hiroshima and Nagasaki. Stalin kept his silence for months afterward. When he finally spoke publicly, following his so-called reelection in February 1946, it was to promise the people of his country that the Soviet Union would surpass the West14in developing its atomic capability. The effort to assemble an army of physicists and mathematicians15to match the Manhattan Project’s had by that time been under way for at least a year; young scholars had been recalled from the frontlines and even released from prisons in order to join the race for the bomb.

Following the war, the Soviet Union invested heavily in high-tech military research, building more than forty entire cities where scientists and mathematicians worked in secret. The urgency of the mobilization indeed recalled the Manhattan Project—only it was much, much bigger and lasted much longer. Estimates of the number of people engaged in the Soviet arms effort16in the second half of the century are notoriously inaccurate, but they range as high as twelve million, with a couple million of them employed by military research institutions. For many years, a newly graduated young mathematician or physicist was more likely to be assigned to defense-related research than to a civilian institution. These jobs spelled nearly total scientific isolation: for defense employees, burdened by security clearances whether or not they actually had access to sensitive military information, any contact with foreigners was considered not just suspect but treasonous. In addition, some of these jobs required moving to the research towns, which provided comfortably cloistered social environments but no possibility for outside intellectual contact. The mathematician’s pencil and paper could be useless tools in the absence of an ongoing mathematical conversation. So the Soviet Union managed to hide some of its best mathematical minds away, in plain sight.

Following Stalin’s death, in 1953, the country shifted its stance on its relationship to the rest of the world: now the Soviet Union was to be not only feared but respected. So while it fell to most mathematicians to help build bombs and rockets, it fell to a select few to build prestige. Very slowly, in the late 1950s, the Iron Curtain began to open a tiny crack—not quite enough to facilitate much-needed conversation between Soviet and non-Soviet mathematicians but enough to show off some of Soviet mathematics’ proudest achievements.

By the 1970s, a Soviet mathematics establishment had taken shape. It was a totalitarian system within a totalitarian system. It provided its members with not only work and money but also apartments, food, and transportation; it determined where they lived and when, where, and how they traveled for work or pleasure. To those in the fold, it was a controlling and strict but caring mother: her children were well nourished and nurtured, an undeniably privileged group compared with the rest of the country. When basic goods were scarce, official mathematicians and other scientists could shop at specially designated stores,17which tended to be better stocked and less crowded than those open to the general public. Since for most of the Soviet century there was no such thing as a private apartment, regular Soviet citizens received their dwellings from the State; members of the science establishment were assigned apartments by their institutions, and these apartments tended to be larger and better located than their compatriots’. Finally, one of the rarest privileges in the life of a Soviet citizen—foreign travel—was available to members of the mathematics establishment. It was the Academy of Sciences, with the Party and the State security organizations watching over it, that decided if a mathematician could accept, say, an invitation to address a scholarly conference, who would accompany him on the trip, how long the trip would last, and, in many instances, where he would stay. For example, in 1970, the first Soviet winner of the Fields Medal, Sergei Novikov, was not allowed to travel to Nice to accept his award.18He received it a year later, when the International Mathematical Union met in Moscow.

Even for members of the mathematical establishment, though, resources were always scarce. There were always fewer good apartments than there were people who desired them, and there were always more people wanting to travel to a conference than would be allowed to go. So it was a vicious, backstabbing little world, shaped by intrigue, denunciations, and unfair competition. The barriers to entry into this club were prohibitively high: a mathematician had to be ideologically reliable and personally loyal not only to the Party but to existing members of the establishment, and Jews and women had next to no chance of getting in.

One could easily be expelled by the establishment for misbehaving. This happened with Kolmogorov’s student Eugene Dynkin, who fostered an atmosphere of unconscionable liberalism at a specialized mathematics school he ran in Moscow. Another of Kolmogorov’s students, Leonid Levin, describes being ostracized19for associating with dissidents. “I became a burden for everyone to whom I was connected,” he wrote in a memoir. “I would not be hired by any serious research institution, and I felt I didn’t even have the right to attend seminars, since participants had been instructed to inform [the authorities] whenever I appeared. My Moscow existence began to seem pointless.” Both Dynkin and Levin emigrated. It must have been soon after Levin’s arrival in the United States that he learned that a problem he had been describing at Moscow mathematics seminars (building in part on Kolmogorov’s work on complexities) was the same problem U.S. computer scientist Stephen Cook had defined. Cook and Levin, who became a professor at Boston University, are considered coinventors20of the NP-completeness theorem, also known as the Cook-Levin theorem; it forms the foundation of one of the seven Millennium Problems that the Clay Mathematics Institute is offering a million dollars to solve. The theorem says, in essence, that some problems are easy to formulate but require so many computations that a machine capable of solving them cannot exist.

And then there were those who almost never became members of the establishment: those who happened to be born Jewish or female, those who had had the wrong advisers at their universities, and those who could not force themselves to join the Party. “There were people who realized that they would never be admitted to the Academy and that the most they could hope for was being able to defend their doctoral dissertation at some institute in Minsk, if they could secure connections there,” said Sergei Gelfand, the American Mathematics Society publisher, who happens to be the son of one of Russia’s top twentieth-century mathematicians, Israel Gelfand, a student of Kolmogorov’s. “These people attended seminars at the university and were officially on the staff of some research institute, say, of the timber industry. They did very good math, and at a certain point they even started having contacts abroad and could even get published occasionally in the West—it was hard, and they had to prove that they were not divulging state secrets, but it was possible. Some mathematicians came from the West, some even came for an extended stay because they realized there were a lot of talented people. This was unofficial mathematics.”

One of the people who came for an extended stay was Dusa McDuff,21then a British algebraist (and now a professor emeritus at the State University of New York at Stony Brook). She studied with the older Gelfand for six months and credits this experience with opening her eyes to both the way mathematics ought to be practiced—in part through continuous conversation with other mathematicians—and to what mathematics really is. “It was a wonderful education,22in which reading Pushkin’sMozart and Salieriplayed as important a role as learning about Lie groups or reading Cartan and Eilenberg. Gelfand amazed me by talking of mathematics as though it were poetry. He once said about a long paper bristling with formulas that it contained the vague beginnings of an idea which he could only hint at and which he had never managed to bring out more clearly. I had always thought of mathematics as being much more straightforward: a formula is a formula, and an algebra is an algebra, but Gelfand found hedgehogs lurking in the rows of his spectral sequences!”

On paper, the jobs that members of the mathematical counterculture held were generally undemanding and unrewarding, in keeping with the best-known formula of Soviet labor: “We pretend to work, and they pretend to pay us.” The mathematicians received modest salaries that grew little over a lifetime but that were enough to cover basic needs and allow them to spend their time on real research. “There was no such thing as thinking that you had to focus your work in some one narrow area because you have to write faster because you had to get tenure,” said Gelfand. “Mathematics was almost a hobby. So you could spend your time doing things that would not be useful to anyone for the nearest decade.” Mathematicians called it “math for math’s sake,”23intentionally drawing a parallel between themselves and artists who toiled for art’s sake. There was no material reward in this—no tenure, no money, no apartments, no foreign travel; all they stood to gain by doing brilliant work was the respect of their peers. Conversely, if they competed unfairly, they stood to lose the respect of their colleagues while gaining nothing. In other words, the alternative mathematics establishment in the Soviet Union was very much unlike anything else anywhere in the real world: it was a pure meritocracy where intellectual achievement was its own reward.

In after-hours lectures and seminars, the mathematical conversation in the Soviet Union was reborn, and the appeal of mathematics to a mind in search of challenge, logic, and consistency once again became evident. “In the post-Stalin Soviet Union it was one of the most natural ways for a freethinking intellectual to seek self-realization,” said Grigory Shabat, a well-known Moscow mathematician. “If I had been free to choose any profession,24I would have become a literary critic. But I wanted to work, not spend my life fighting the censors.” Mathematics held out the promise that one could not only do intellectual work without State interference (if also without its support) but also find something not available anywhere else in late-Soviet society: a knowable singular truth. “Mathematicians are people possessed of a special intellectual honesty,” Shabat continued. “If two mathematicians are making contradictory claims, then one of them is right and the other one is wrong. And they will definitely figure it out, and the one who was wrong will definitely admit that he was mistaken.” The search for that truth could take long years—but in the late Soviet Union, time stood still, which meant that the inhabitants of the alternative mathematics universe had all the time they needed.

PERFECT RIGOR

HOW TO MAKE A MATHEMATICIAN

2

How to Make a Mathematician

IN THE MID-1960SProfessor Garold Natanson offered a graduate-study spot to a student of his, a woman named Lubov. One did not make this sort of offer lightly: female graduate students were notoriously unreliable, prone to pregnancy and other distracting pursuits. In addition, this particular student was Jewish, which meant that securing a spot for her would have required Professor Natanson to scheme, strategize, and call in favors: in the eyes of the system, Jews were even more unreliable than women, and convoluted discriminatory anti-Semitic practices carried the force of unwritten law. Natanson, a Jew himself, taught at the Herzen Pedagogical Institute, which ranked second to Leningrad State University and so was allowed to accept Jews as students and teachers—within reason, or what passed for it in the postwar Soviet Union. The student was older—she was nearing thirty, which placed her well beyond the usual Russian marrying-and-having-children threshold, so Natanson could be justified in assuming that she had resolved to devote her life entirely to mathematics.

Natanson was not entirely off the mark: the woman was indeed wholly devoted to mathematics. But she turned down his generous offer. She explained that she had recently married and planned to start a family, and with that she accepted a job teaching mathematics at a trade school and disappeared from the Leningrad mathematical scene for more than ten years.

Ten or twelve years was nothing in Soviet time. There was a bit of new housing construction in Leningrad, and some families were able to leave the crowded and crumbling city center for the new concrete towers on its outskirts. Clothing and food continued to be in short supply and of regrettable quality, but industrial production picked up a bit, so some of the new suburban dwellers could actually buy basic semiautomatic washing machines and television sets for their apartments. The televisions claimed to be black-and-white but showed mostly shades of gray, thereby providing an accurate visual reflection of reality. Other than that, little changed. Natanson continued to teach at the Herzen, which itself grew only more crowded and crumbling. His former student Lubov found him in his office. She was older and a bit heavier. She reported that she had indeed had a baby all those years ago, and now this baby was a schoolboy who exhibited a talent for mathematics. He had taken part in a district math competition in one of those newly constructed concrete suburbs where they now lived, and he had done well. In the timeless scheme of Russian mathematics, he was ready to take up where his mother had left off.

It all must have made perfect sense to Natanson. He himself hailed from a mathematical dynasty: his father, Isidor Natanson, was the author of the definitive Russian calculus textbook and had also taught at the Herzen, until his death, in 1963. Lubov’s boy was entering fifth grade—the age at which he could begin appropriately rigorous mathematical study in a system that had been constructed over the years for the making of mathematicians. Natanson had his eye on a young mathematics coach to whom he could direct the boy and his mother.

So began the education of Grigory Perelman.

Competitive mathematics is more like a sport than most people imagine. It has its coaches, its clubs, its practice sessions, and, of course, its competitions. Natural ability is necessary but entirely insufficient for success: the talented child needs to have the right coach, the right team, the right kind of family support, and, most important, the will to win. At the beginning, it is nearly impossible to tell the difference between future stars and those who will be good but never great.

Grisha Perelman arrived at the math club of the Leningrad Palace of Pioneers in the fall of 1976, an ugly duckling among ugly ducklings. He was pudgy and awkward. He played the violin; his mother, who had studied not only mathematics but also the violin when she was a child, had engaged a private teacher when Grisha was very young. When he tried to explain a solution to a math problem, words seemed to get tangled at the tip of his tongue, where too many of them collected too quickly, froze momentarily, and then tumbled out, all jumbled up. He was precocious—a year younger than the other children at his grade level—but one of the other kids at the club was even younger: Alexander Golovanov1had packed two grades into every year of school and would be finishing high school at thirteen. Three other boys beat Grisha in competitions2for the first few years in the club. At least one more—Boris Sudakov,3a round, animated, curious boy whose parents happened to know Grisha’s family—showed more natural ability than Grisha. Sudakov and Golovanov both carried the marks of brilliance: they seemed always to be rushing forward and bubbling over. They naturally fought for dominance in any room, and mathematics was simply one of many things that got them excited, one of the ways to apply their excellent minds, and one of the tools to showcase their uniqueness. Next to them, Grisha was the interested but quiet partner, almost a mirror; he was a joy for them to bounce their ideas off, but he himself rarely seemed to exhibit the same need. He formed relationships with the math problems; these relationships were deep but also, it seemed, deeply private: most of his conversations appeared to be mathematical and to take place inside his head. A casual visitor to the club would not have singled him out from the other boys. Indeed, even among the people who met him many years later, not one that I encountered described him as brilliant; no one thought he sparkled or shone. People described him, rather, as very, very smart and very, very precise in his thinking.

Just what manner of thinking this was remained something of a mystery. Crudely speaking, mathematicians fall into two categories: the algebraists, who find it easiest to reduce all problems to sets of numbers and variables, and the geometers, who understand the world through shapes. Where one group sees this:

a2+ b22

the other sees this:

Golovanov, who studied and occasionally competed alongside Perelman for more than ten years, tagged him as an unambiguous geometer: Perelman had a geometry problem solved in the time it took Golovanov to grasp the question. This was because Golovanov was an algebraist. Sudakov, who spent about six years studying and occasionally competing with Perelman, claimed Perelman reduced every problem to a formula. This, it appears, was because Sudakov was a geometer: his favorite proof of the classic theorem above was an entirely graphical one, requiring no formulas and no language to demonstrate. In other words, each of them was convinced Perelman’s mind was profoundly different from his own. Neither had any hard evidence. Perelman did his thinking almost entirely inside his head, neither writing nor sketching on scrap paper. He did a lot of other things—he hummed, moaned, threw a Ping-Pong ball against the desk,4rocked back and forth, knocked out a rhythm on the desk with his pen, rubbed his thighs until his pant legs shone, and then rubbed his hands together—a sign that the solution would now be written down, fully formed. For the rest of his career, even after he chose to work with shapes, he never dazzled colleagues with his geometric imagination, but he almost never failed to impress them5with the single-minded precision with which he plowed through problems. His brain seemed to be a universal math compactor, capable of compressing problems to their essence. Club mates eventually dubbed whatever it was he had inside his head the “Perelman stick”—a very large imaginary instrument with which he sat quietly before striking an always-fatal blow.

Practice sessions at mathematics clubs the world over look roughly the same. Kids come in to find a set of problems written on the blackboard or handed to them. They sit down and attempt to solve them. The coach spends most of his time sitting quietly; teaching assistants check in with the students occasionally, sometimes prodding them with questions, sometimes trying to nudge them in different directions.