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MANIFOLD DESTINY A legendary problem and the battle over who solved it. by SYLVIA NASAR AND DAVID GRUBER Issue of 20060828 Posted 20060821 On the evening of June 20th, several hundred physicists, including a Nobel laureate, assembled in an auditorium at the Friendship Hotel in Beijing for a lecture by the Chinese mathematician ShingTung Yau. In the late nineteenseventies, when Yau was in his twenties, he had made a series of breakthroughs that helped launch the stringtheory revolution in physics and earned him, in addition to a Fields Medalthe most coveted award in mathematicsa reputation in both disciplines as a thinker of unrivalled technical power. Yau had since become a professor of mathematics at Harvard and the director of mathematics institutes in Beijing and Hong Kong, dividing his time between the United States and China. His lecture at the Friendship Hotel was part of an international conference on string theory, which he had organized with the support of the Chinese government, in part to promote the country's recent advances in theoretical physics. Yau, a stocky man of fiftyseven, stood at a lectern in shirtsleeves and blackrimmed glasses and, with his hands in his pockets, described how two of his students, XiPing Zhu and HuaiDong Cao, had completed a proof of the Poincar conjecture a few weeks earlier. "Chinese mathematicians should have every reason to be proud of such a big success in completely solving the puzzle." He said that Zhu and Cao were indebted to his longtime American collaborator Richard Hamilton, who deserved most of the credit for solving the Poincar. He also mentioned Grigory Perelman, a Russian mathematician who, he acknowledged, had made an important contribution. Nevertheless, Yau said, "in Perelman's work, spectacular as it is, many key ideas of the proofs are sketched or outlined, and complete details are often missing." He added, "We would like to get Perelman to make comments. Petersburg and refuses to communicate with other people." For ninety minutes, Yau discussed some of the technical details of his students' proof. That night, however, a Brazilian physicist posted a report of the lecture on his blog. "Looks like China soon will take the lead also in mathematics," he wrote. He left his job as a researcher at the Steklov Institute of Mathematics, in St. and he lives with his mother in an apartment on the outskirts of the city. Although he had never granted an interview before, he was cordial and frank when we visited him, in late June, shortly after Yau's conference in Beijing, taking us on a long walking tour of the city. "I'm looking for some friends, and they don't have to be mathematicians," he said. The week before the conference, Perelman had spent hours discussing the Poincar conjecture with Sir John M Ball, the fiftyeightyearold president of the International Mathematical Union, the discipline's influential professional association. The meeting, which took place at a conference center in a stately mansion overlooking the Neva River, was highly unusual. At the end of May, a committee of nine prominent mathematicians had voted to award Perelman a Fields Medal for his work on the Poincar, and Ball had gone to St. The Fields Medal, like the Nobel Prize, grew, in part, out of a desire to elevate science above national animosities. U congress, in 1924, and, though the ban was lifted before the next one, the trauma it caused led, in 1936, to the establishment of the Fields, a prize intended to be "as purely international and impersonal as possible." However, the Fields Medal, which is awarded every four years, to between two and four mathematicians, is supposed not only to reward past achievements but also to stimulate future research; for this reason, it is given only to mathematicians aged forty and younger. In recent decades, as the number of professional mathematicians has grown, the Fields Medal has become increasingly prestigious. Only fortyfour medals have been awarded in nearly seventy yearsincluding three for work closely related to the Poincar conjectureand no mathematician has ever refused the prize. Nevertheless, Perelman told Ball that he had no intention of accepting it. Over a period of eight months, beginning in November, 2002, Perelman posted a proof of the Poincar on the Internet in three installments. Like a sonnet or an aria, a mathematical proof has a distinct form and set of conventions. It begins with axioms, or accepted truths, and employs a series of logical statements to arrive at a conclusion. If the logic is deemed to be watertight, then the result is a theorem. Unlike proof in law or science, which is based on evidence and therefore subject to qualification and revision, a proof of a theorem is definitive. Judgments about the accuracy of a proof are mediated by peerreviewed journals; to insure fairness, reviewers are supposed to be carefully chosen by journal editors, and the identity of a scholar whose paper is under consideration is kept secret. Publication implies that a proof is complete, correct, and original. It was astonishingly brief for such an ambitious piece of work; logic sequences that could have been elaborated over many pages were often severely compressed. Moreover, the proof made no direct mention of the Poincar and included many elegant results that were irrelevant to the central argument. But, four years later, at least two teams of experts had vetted the proof and had found no significant gaps or errors in it. A consensus was emerging in the math community: Perelman had solved the Poincar. Even so, the proof's complexityand Perelman's use of shorthand in making some of his most important claimsmade it vulnerable to challenge. Few mathematicians had the expertise necessary to evaluate and defend it. After giving a series of lectures on the proof in the United States in 2003, Perelman returned to St. Since then, although he had continued to answer queries about it by email, he had had minimal contact with colleagues and, for reasons no one understood, had not tried to publish it. Still, there was little doubt that Perelman, who turned forty on June 13th, deserved a Fields Medal. U's 2006 congress, he began to conceive of it as a historic event. More than three thousand mathematicians would be attending, and King Juan Carlos of Spain had agreed to preside over the awards ceremony. U's newsletter predicted that the congress would be remembered as "the occasion when this conjecture became a theorem." Ball, determined to make sure that Perelman would be there, decided to go to St. Ball wanted to keep his visit a secretthe names of Fields Medal recipients are announced officially at the awards ceremonyand the conference center where he met with Perelman was deserted. For ten hours over two days, he tried to persuade Perelman to agree to accept the prize. Perelman, a slender, balding man with a curly beard, bushy eyebrows, and bluegreen eyes, listened politely. He had not spoken English for three years, but he fluently parried Ball's entreaties, at one point taking Ball on a long walkone of Perelman's favorite activities. As he summed up the conversation two weeks later: "He proposed to me three alternatives: accept and come; accept and don't come, and we will send you the medal later; From the very beginning, I told him I have chosen the third one." The Fields Medal held no interest for him, Perelman explained. "Everybody understood that if the proof is correct then no other recognition is needed." Proofs of the Poincar have been announced nearly every year since the conjecture was formulated, by Henri Poincar, more than a hundred years ago. Poincar was a cousin of Raymond Poincar, the President of France during the First World War, and one of the most creative mathematicians of the nineteenth century. Slight, myopic, and notoriously absentminded, he conceived his famous problem in 1904, eight years before he died, and tucked it as an offhand question into the end of a sixtyfivepage paper. Poincar didn't make much progress on proving the conjecture. "Cette question nous entranerait trop loin" ("This question wou...
