ANNAls of mathematicS mANifOld destiNY A legendary problem and the battle over who solved it. BY SYlvia Nasar and David Gruber n the evening of June 20th, sev- Zhu and Cao’s work,” Yau said. “Chi- fessional association. The meeting, eral hundred physicists, including nese mathematicians should have every which took place at a conference center aO Nobel laureate, assembled in an audi- reason to be proud of such a big success in a stately mansion overlooking the torium at the Friendship Hotel in Bei- in completely solving the puzzle.” He Neva River, was highly unusual. At the jing for a lecture by the Chinese math- said that Zhu and Cao were indebted end of May, a committee of nine prom- ematician Shing-Tung Yau. In the late to his longtime American collaborator inent mathematicians had voted to nineteen-seventies, when Yau was in Richard Hamilton, who deserved most award Perelman a Fields Medal for his his twenties, he had made a series of of the credit for solving the Poincaré. work on the Poincaré, and Ball had breakthroughs that helped launch the He also mentioned Grigory Perelman, gone to St. Petersburg to persuade him string-theory revolution in physics and a Russian mathematician who, he ac- to accept the prize in a public ceremony earned him, in addition to a Fields knowledged, had made an important at the I.M.U.’s quadrennial congress, in Medal—the most coveted award in contribution. Nevertheless, Yau said, Madrid, on August 22nd. mathematics—a reputation in both “in Perelman’s work, spectacular as it The Fields Medal, like the Nobel disciplines as a thinker of unrivalled is, many key ideas of the proofs are Prize, grew, in part, out of a desire to technical power. sketched or outlined, and complete de- elevate science above national animos- Yau had since become a professor of tails are often missing.” He added, “We ities. German mathematicians were ex- mathematics at Harvard and the direc- would like to get Perelman to make cluded from the first I.M.U. congress, tor of mathematics institutes in Beijing comments. But Perelman resides in in 1924, and, though the ban was lifted and Hong Kong, dividing his time be- St. Petersburg and refuses to commu- before the next one, the trauma it tween the United States and China. nicate with other people.” caused led, in 1936, to the establish- His lecture at the Friendship Hotel was For ninety minutes, Yau discussed ment of the Fields, a prize intended to part of an international conference on some of the technical details of his stu- be “as purely international and imper- string theory, which he had organized dents’ proof. When he was finished, no sonal as possible.” with the support of the Chinese gov- one asked any questions. That night, However, the Fields Medal, which ernment, in part to promote the coun- however, a Brazilian physicist posted a is awarded every four years, to between try’s recent advances in theoretical report of the lecture on his blog. “Looks two and four mathematicians, is sup- physics. (More than six thousand stu- like China soon will take the lead also posed not only to reward past achieve- dents attended the keynote address, in mathematics,” he wrote. ments but also to stimulate future re- which was delivered by Yau’s close search; for this reason, it is given only friend Stephen Hawking, in the Great rigory Perelman is indeed reclu- to mathematicians aged forty and Hall of the People.) The subject of sive. He left his job as a researcher younger. In recent decades, as the num- Yau’s talk was something that few in atG the Steklov Institute of Mathemat- ber of professional mathematicians has his audience knew much about: the ics, in St. Petersburg, last December; grown, the Fields Medal has become Poincaré conjecture, a century-old co- he has few friends; and he lives with his increasingly prestigious. Only forty- nundrum about the characteristics of mother in an apartment on the out- four medals have been awarded in three-dimensional spheres, which, be- skirts of the city. Although he had nearly seventy years—including three cause it has important implications for never granted an interview before, he for work closely related to the Poincaré mathematics and cosmology and be- was cordial and frank when we visited conjecture—and no mathematician has cause it has eluded all attempts at solu- him, in late June, shortly after Yau’s ever refused the prize. Nevertheless, tion, is regarded by mathematicians as conference in Beijing, taking us on a Perelman told Ball that he had no in- a holy grail. long walking tour of the city. “I’m look- tention of accepting it. “I refuse,” he Yau, a stocky man of fifty-seven, ing for some friends, and they don’t said simply. stood at a lectern in shirtsleeves and have to be mathematicians,” he said. Over a period of eight months, be- black-rimmed glasses and, with his The week before the conference, Perel- ginning in November, 2002, Perelman hands in his pockets, described how man had spent hours discussing the posted a proof of the Poincaré on the two of his students, Xi-Ping Zhu and Poincaré conjecture with Sir John M. Internet in three installments. Like a Huai-Dong Cao, had completed a Ball, the fifty-eight-year-old president sonnet or an aria, a mathematical proof proof of the Poincaré conjecture a few of the International Mathematical has a distinct form and set of conven- weeks earlier. “I’m very positive about Union, the discipline’s influential pro- tions. It begins with axioms, or ac- pierre le-tan 44 THE NEW YORKER, AUGUST 28, 2006 tnY—2006_08_28—PAGe 44—133SC. Grigory Perelman (right) says, “If the proof is correct, then no other recognition is needed.” Shing-Tung Yau isn’t so sure. tnY—2006_08_28—PAGe 45—133SC.—live art r15383_rD2 (plS uSe Color Sent Down, labeleD r15383_rD) cepted truths, and employs a series of After giving a series of lectures on Perelman’s favorite activities. As he logical statements to arrive at a conclu- the proof in the United States in 2003, summed up the conversation two weeks sion. If the logic is deemed to be water- Perelman returned to St. Petersburg. later: “He proposed to me three alter- tight, then the result is a theorem. Un- Since then, although he had continued natives: accept and come; accept and like proof in law or science, which is to answer queries about it by e-mail, don’t come, and we will send you the based on evidence and therefore subject he had had minimal contact with col- medal later; third, I don’t accept the to qualification and revision, a proof leagues and, for reasons no one under- prize. From the very beginning, I told of a theorem is definitive. Judgments stood, had not tried to publish it. Still, him I have chosen the third one.” The about the accuracy of a proof are medi- there was little doubt that Perelman, Fields Medal held no interest for him, ated by peer-reviewed journals; to in- who turned forty on June 13th, de- Perelman explained. “It was completely sure fairness, reviewers are supposed to served a Fields Medal. As Ball planned irrelevant for me,” he said. “Everybody be carefully chosen by journal editors, the I.M.U.’s 2006 congress, he began understood that if the proof is correct and the identity of a scholar whose pa- to conceive of it as a historic event. then no other recognition is needed.” per is under consideration is kept se- More than three thousand mathemati- cret. Publication implies that a proof is cians would be attending, and King roofs of the Poincaré have been an- complete, correct, and original. Juan Carlos of Spain had agreed to pre- nounced nearly every year since the By these standards, Perelman’s proof side over the awards ceremony. The conjectureP was formulated, by Henri was unorthodox. It was astonishingly I.M.U.’s newsletter predicted that the Poincaré, more than a hundred years brief for such an ambitious piece of congress would be remembered as “the ago. Poincaré was a cousin of Raymond work; logic sequences that could have occasion when this conjecture became Poincaré, the President of France dur- been elaborated over many pages were a theorem.” Ball, determined to make ing the First World War, and one of often severely compressed. Moreover, sure that Perelman would be there, de- the most creative mathematicians of the proof made no direct mention of cided to go to St. Petersburg. the nineteenth century. Slight, myopic, the Poincaré and included many ele- Ball wanted to keep his visit a se- and notoriously absent-minded, he gant results that were irrelevant to the cret—the names of Fields Medal re- conceived his famous problem in 1904, central argument. But, four years later, cipients are announced officially at the eight years before he died, and tucked at least two teams of experts had vetted awards ceremony—and the conference it as an offhand question into the end the proof and had found no signifi- center where he met with Perelman of a sixty-five-page paper. cant gaps or errors in it. A consensus was deserted. For ten hours over two Poincaré didn’t make much progress was emerging in the math community: days, he tried to persuade Perelman to on proving the conjecture. “Cette ques- Perelman had solved the Poincaré. agree to accept the prize. Perelman, a tion nous entraînerait trop loin” (“This Even so, the proof ’s complexity—and slender, balding man with a curly beard, question would take us too far”), he Perelman’s use of shorthand in making bushy eyebrows, and blue-green eyes, wrote.