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 ele gance, the substance, the importance of future mathematical accomplishments.
The Millennium Meeting had been convened by the Clay Mathematics Institute, a non profit 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.”
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 in ?u en tial 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 dec ades 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 ac tually 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 Inter net. 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 Inter net that 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 e...