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

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TNY—2006_08_28—PAGE 44—133SC. 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. He was a founder of topology, some of his most important claims— listened politely. He had not spoken also known as “rubber-sheet geometry,” made it vulnerable to challenge. Few English for three years, but he fluently for its focus on the intrinsic properties of mathematicians had the expertise nec- parried Ball’s entreaties, at one point spaces. From a topologist’s perspective, essary to evaluate and defend it. taking Ball on a long walk—one of there is no difference between a bagel and a coffee cup with a handle. Each has a single hole and can be manipulated to resemble the other without being torn or cut. Poincaré used the term “mani- fold” to describe such an abstract topo- logical space. The simplest possible two- dimensional manifold is the surface of a soccer ball, which, to a topologist, is a sphere—even when it is stomped on, stretched, or crumpled. The proof that an object is a so-called two-sphere, since it can take on any number of shapes, is that it is “simply connected,” meaning that no holes puncture it. Unlike a soc- cer ball, a bagel is not a true sphere. If you tie a slipknot around a soccer ball, you can easily pull the slipknot closed by sliding it along the surface of the ball. But if you tie a slipknot around a bagel through the hole in its middle you can- not pull the slipknot closed without “Should we halfheartedly try to relate?” tearing the bagel.

TNY—2006_08_28—PAGE 46—133SC—live art a11730. Two-dimensional manifolds were him was a copy of “Physics for Enter- ometry of Riemannian and Alexandrov well understood by the mid-nineteenth tainment,” which had been a best-seller spaces—extensions of traditional Eu- century. But it remained unclear whether in the Soviet Union in the nineteen- clidean geometry—and began to publish what was true for two dimensions was thirties. In the foreword, the book’s articles in the leading Russian and Amer- also true for three. Poincaré proposed author describes the contents as “co- ican mathematics journals. In 1992, that all closed, simply connected, three- nundrums, brain-teasers, entertaining Perelman was invited to spend a semes- dimensional manifolds—those which anecdotes, and unexpected compari- ter each at New York University and lack holes and are of finite extent—were sons,” adding, “I have quoted exten- Stony Brook University. By the time he spheres. The conjecture was potentially sively from Jules Verne, H. G. Wells, left for the United States, that fall, the important for scientists studying the Mark Twain and other writers, because, Russian economy had collapsed. Dan largest known three-dimensional mani- besides providing entertainment, the Stroock, a mathematician at M.I.T., re- fold: the universe. Proving it mathemat- fantastic experiments these writers de- calls smuggling wads of dollars into the ically, however, was far from easy. Most scribe may well serve as instructive illus- country to deliver to a retired mathema- attempts were merely embarrassing, but trations at physics classes.” The book’s tician at the Steklov, who, like many of some led to important mathematical topics included how to jump from a his colleagues, had become destitute. discoveries, including proofs of Dehn’s moving car, and why, “according to the Perelman was pleased to be in the Lemma, the Sphere Theorem, and the law of buoyancy, we would never drown United States, the capital of the interna- Loop Theorem, which are now funda- in the Dead Sea.” tional mathematics community. He mental concepts in topology. The notion that Russian society con- wore the same brown corduroy jacket By the nineteen-sixties, topology sidered worthwhile what Perelman did every day and told friends at N.Y.U. had become one of the most productive for pleasure came as a surprise. By the that he lived on a diet of bread, cheese, areas of mathematics, and young topol- time he was fourteen, he was the star and milk. He liked to walk to Brooklyn, ogists were launching regular attacks performer of a local math club. In 1982, where he had relatives and could buy on the Poincaré. To the astonishment the year that Shing-Tung Yau won a traditional Russian brown bread. Some of most mathematicians, it turned out Fields Medal, Perelman earned a perfect of his colleagues were taken aback by his that manifolds of the fourth, fifth, and score and the gold medal at the Interna- fingernails, which were several inches higher dimensions were more tractable tional Mathematical Olympiad, in Bu- long. “If they grow, why wouldn’t I let than those of the third dimension. By dapest. He was friendly with his team- them grow?” he would say when some- 1982, Poincaré’s conjecture had been mates but not close—“I had no close one asked why he didn’t cut them. Once proved in all dimensions except the friends,” he said. He was one of two or a week, he and a young Chinese math- third. In 2000, the Clay Mathematics three Jews in his grade, and he had a ematician named Gang Tian drove to Institute, a private foundation that pro- passion for opera, which also set him Princeton, to attend a seminar at the In- motes mathematical research, named apart from his peers. His mother, stitute for Advanced Study. the Poincaré one of the seven most im- a math teacher at a technical college, For several decades, the institute portant outstanding problems in math- played the violin and began taking him to and nearby Princeton University had ematics and offered a million dollars to the opera when he was six. By the time been centers of topological research. In anyone who could prove it. Perelman was fifteen, he was spending the late seventies, William Thurston, a “My whole life as a mathematician his pocket money on records. He was Princeton mathematician who liked to has been dominated by the Poincaré thrilled to own a recording of a famous test out his ideas using scissors and conjecture,” John Morgan, the head 1946 performance of “La Traviata,” fea- construction paper, proposed a taxon- of the mathematics department at turing Licia Albanese as Violetta. “Her omy for classifying manifolds of three Columbia University, said. “I never voice was very good,” he said. dimensions. He argued that, while the thought I’d see a solution. I thought At Leningrad University, which manifolds could be made to take on nobody could touch it.” Perelman entered in 1982, at the age of many different shapes, they nonethe- sixteen, he took advanced classes in ge- less had a “preferred” geometry, just as rigory Perelman did not plan to ometry and solved a problem posed by a piece of silk draped over a dressmak- become a mathematician. “There Yuri Burago, a mathematician at the er’s mannequin takes on the manne- wasG never a decision point,” he said Steklov Institute, who later became his quin’s form. when we met. We were outside the Ph.D. adviser. “There are a lot of stu- Thurston proposed that every three- apartment building where he lives, in dents of high ability who speak before dimensional manifold could be broken Kupchino, a neighborhood of drab thinking,” Burago said. “Grisha was down into one or more of eight types high-rises. Perelman’s father, who was different. He thought deeply. His an- of component, including a spherical an electrical engineer, encouraged his swers were always correct. He always type. Thurston’s theory—which became interest in math. “He gave me logi- checked very, very carefully.” Burago known as the geometrization conjec- cal and other math problems to think added, “He was not fast. Speed means ture—describes all possible three-dimen- about,” Perelman said. “He got a lot of nothing. Math doesn’t depend on sional manifolds and is thus a powerful books for me to read. He taught me speed. It is about deep.” generalization of the Poincaré. If it was how to play chess. He was proud of At the Steklov in the early nineties, confirmed, then Poincaré’s conjecture me.” Among the books his father gave Perelman became an expert on the ge- would be, too. Proving Thurston and

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TNY—2006_08_28—PAGE 47—133SC. Poincaré “definitely swings open doors,” spite considerable differences in tem- ing several others. He sent half of his Barry Mazur, a mathematician at Har- perament and background. A mathe- scholarship money back to his mother vard, said. The implications of the con- matician at the University of California in China and impressed his professors jectures for other disciplines may not be at San Diego who knows both men with his tenacity. He was obliged to apparent for years, but for mathemati- called them “the mathematical loves of share credit for his first major result cians the problems are fundamental. each other’s lives.” when he learned that two other math- “This is a kind of twentieth-century Py- Yau’s family moved to Hong Kong ematicians were working on the same thagorean theorem,” Mazur added. “It from mainland China in 1949, when problem. In 1976, he proved a twenty- changes the landscape.” he was five months old, along with year-old conjecture pertaining to a In 1982, Thurston won a Fields Medal hundreds of thousands of other refu- type of manifold that is now crucial to for his contributions to topology. That gees fleeing Mao’s armies. The previ- string theory. A French mathemati- year, Richard Hamilton, a mathematician ous year, his father, a relief worker for cian had formulated a proof of the at Cornell, published a paper on an equa- the United Nations, had lost most of problem, which is known as Calabi’s tion called the Ricci flow, which he sus- the family’s savings in a series of failed conjecture, but Yau’s, because it was pected could be relevant for solving Thur- ventures. In Hong Kong, to support more general, was more powerful. ston’s conjecture and thus the Poincaré. his wife and eight children, he tutored (Physicists now refer to Calabi-Yau Like a heat equation, which describes college students in classical Chinese manifolds.) “He was not so much how heat distributes itself evenly through literature and philosophy. thinking up some original way of look- a substance—flowing from hotter to When Yau was fourteen, his father ing at a subject but solving extremely cooler parts of a metal sheet, for exam- died of kidney cancer, leaving his hard technical problems that at the ple—to create a more uniform tempera- mother dependent on handouts from time only he could solve, by sheer in- ture, the Ricci flow, by smoothing out Christian missionaries and whatever tellect and force of will,” Phillip irregularities, gives manifolds a more uni- small sums she earned from selling Griffiths, a geometer and a former di- form geometry. handicrafts. Until then, Yau had been rector of the Institute for Advanced Hamilton, the son of a Cincinnati an indifferent student. But he began to Study, said. doctor, defied the math profession’s nerdy devote himself to schoolwork, tutoring In 1980, when Yau was thirty, he stereotype. Brash and irreverent, he rode other students in math to make money. became one of the youngest mathema- horses, windsurfed, and had a succession “Part of the thing that drives Yau is ticians ever to be appointed to the per- of girlfriends. He treated math as merely that he sees his own life as being his fa- manent faculty of the Institute for Ad- one of life’s pleasures. At forty-nine, he ther’s revenge,” said Dan Stroock, the vanced Study, and he began to attract was considered a brilliant lecturer, but he M.I.T. mathematician, who has known talented students. He won a Fields had published relatively little beyond a se- Yau for twenty years. “Yau’s father was Medal two years later, the first Chinese ries of seminal articles on the Ricci flow, like the Talmudist whose children are ever to do so. By this time, Chern was and he had few graduate students. Perel- starving.” seventy years old and on the verge of man had read Hamilton’s papers and Yau studied math at the Chinese retirement. According to a relative of went to hear him give a talk at the Insti- University of Hong Kong, where he Chern’s, “Yau decided that he was tute for Advanced Study. Afterward, attracted the attention of Shiing-Shen going to be the next famous Chinese Perelman shyly spoke to him. Chern, the preëminent Chinese math- mathematician and that it was time for “I really wanted to ask him some- ematician, who helped him win a Chern to step down.” thing,” Perelman recalled. “He was smil- scholarship to the University of Cali- Harvard had been trying to recruit ing, and he was quite patient. He actu- fornia at Berkeley. Chern was the au- Yau, and when, in 1983, it was about ally told me a couple of things that he thor of a famous theorem combining to make him a second offer Phillip published a few years later. He did not topology and geometry. He spent Griffiths told the dean of faculty a ver- hesitate to tell me. Hamilton’s openness most of his career in the United States, sion of a story from “The Romance of and generosity—it really attracted me. I at Berkeley. He made frequent visits the Three Kingdoms,” a Chinese clas- can’t say that most mathematicians act to Hong Kong, Taiwan, and, later, sic. In the third century A.D., a Chi- like that. China, where he was a revered symbol nese warlord dreamed of creating an “I was working on different things, of Chinese intellectual achievement, empire, but the most brilliant general though occasionally I would think about to promote the study of math and in China was working for a rival. the Ricci flow,” Perelman added. “You science. Three times, the warlord went to his didn’t have to be a great mathematician to In 1969, Yau started graduate enemy’s kingdom to seek out the gen- see that this would be useful for geome- school at Berkeley, enrolling in seven eral. Impressed, the general agreed to trization. I felt I didn’t know very much. graduate courses each term and audit- join him, and together they succeeded I kept asking questions.” in founding a dynasty. Taking the hint, the dean flew to Philadelphia, hing-Tung Yau was also asking where Yau lived at the time, to make Hamilton questions about the Ricci him an offer. Even so, Yau turned flow.S Yau and Hamilton had met in the down the job. Finally, in 1987, he seventies, and had become close, de- agreed to go to Harvard.

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TNY—2006_08_28—PAGE 48—133SC.—live spot art r15354_C, pls inspect and report on quality Yau’s entrepreneurial drive ex- tended to collaborations with col- leagues and students, and, in addition to conducting his own research, he began organizing seminars. He fre- quently allied himself with brilliantly inventive mathematicians, including Richard Schoen and William Meeks. But Yau was especially impressed by Hamilton, as much for his swagger as for his imagination. “I can have fun with Hamilton,” Yau told us during the string-theory conference in Bei- jing. “I can go swimming with him. I go out with him and his girlfriends and all that.” Yau was convinced that Hamilton could use the Ricci-flow equation to solve the Poincaré and Thurston conjectures, and he urged him to focus on the problems. “Meet- ing Yau changed his mathematical life,” a friend of both mathematicians said of Hamilton. “This was the first time he had been on to something ex- tremely big. Talking to Yau gave him courage and direction.” Yau believed that if he could help solve the Poincaré it would be a vic- tory not just for him but also for China. In the mid-nineties, Yau and several other Chinese scholars began “I’m a local craftsperson—I make money.” meeting with President Jiang Zemin to discuss how to rebuild the country’s •• scientific institutions, which had been largely destroyed during the Cultural Revolution. Chinese universities were to them, but I was half serious. I said sible to achieve uniform geometry. in dire condition. According to Steve the whole country should learn from Perelman realized that a paper he had Smale, who won a Fields for proving Hamilton.” written on Alexandrov spaces might the Poincaré in higher dimensions, help Hamilton prove Thurston’s con- and who, after retiring from Berkeley, rigory Perelman was learning jecture—and the Poincaré—once taught in Hong Kong, Peking Uni- from Hamilton already. In 1993, Hamilton solved the cigar problem. versity had “halls filled with the smell heG began a two-year fellowship at “At some point, I asked Hamilton if of urine, one common room, one Berkeley. While he was there, Hamil- he knew a certain collapsing result office for all the assistant professors,” ton gave several talks on campus, and that I had proved but not published— and paid its faculty wretchedly low in one he mentioned that he was which turned out to be very useful,” salaries. Yau persuaded a Hong Kong working on the Poincaré. Hamilton’s Perelman said. “Later, I realized that real-estate mogul to help finance a Ricci-flow strategy was extremely he didn’t understand what I was talk- mathematics institute at the Chinese technical and tricky to execute. Af- ing about.” Dan Stroock, of M.I.T., Academy of Sciences, in Beijing, and ter one of his talks at Berkeley, he said, “Perelman may have learned stuff to endow a Fields-style medal for told Perelman about his biggest ob- from Yau and Hamilton, but, at the Chinese mathematicians under the stacle. As a space is smoothed under time, they were not learning from age of forty-five. On his trips to the Ricci flow, some regions deform him.” China, Yau touted Hamilton and into what mathematicians refer to as By the end of his first year at Berke- their joint work on the Ricci flow and “singularities.” Some regions, called ley, Perelman had written several strik- the Poincaré as a model for young “necks,” become attenuated areas of ingly original papers. He was asked to Chinese mathematicians. As he put it infinite density. More troubling to give a lecture at the 1994 I.M.U. con- in Beijing, “They always say that the Hamilton was a kind of singularity he gress, in Zurich, and invited to apply whole country should learn from Mao called the “cigar.” If cigars formed, for jobs at Stanford, Princeton, the In- or some big heroes. So I made a joke Hamilton worried, it might be impos- stitute for Advanced Study, and the

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TNY—2006_08_28—PAGE 49—133SC.—live art a11722 SKETCHBOOK BY WALTON FORD A full third of Shelter Island, at the eastern end of Long Island, is given over to the Mashomack

TNY—2006_08_28—PAGE 50—133SC.—live art r15355_RD University of Tel Aviv. Like Yau, Perelman was a formidable problem solver. Instead of spending years con- structing an intricate theoretical frame- work, or defining new areas of re- search, he focussed on obtaining par- ticular results. According to Mikhail Gromov, a renowned Russian geome- ter who has collaborated with Perel- man, he had been trying to overcome a technical difficulty relating to Alexan- drov spaces and had apparently been stumped. “He couldn’t do it,” Gromov said. “It was hopeless.” Perelman told us that he liked to work on several problems at once. At Berkeley, however, he found himself returning again and again to Hamil- ton’s Ricci-flow equation and the prob- lem that Hamilton thought he could solve with it. Some of Perelman’s friends noticed that he was becoming more and more ascetic. Visitors from St. Petersburg who stayed in his apart- ment were struck by how sparsely fur- nished it was. Others worried that he seemed to want to reduce life to a set of rigid axioms. When a member of a hiring committee at Stanford asked him for a C.V. to include with requests for letters of recommendation, Perel- man balked. “If they know my work, they don’t need my C.V.,” he said. “If they need my C.V., they don’t know my work.” Ultimately, he received several job offers. But he declined them all, and in the summer of 1995 returned to St. Petersburg, to his old job at the Stek- lov Institute, where he was paid less than a hundred dollars a month. (He told a friend that he had saved enough money in the United States to live on for the rest of his life.) His father had moved to Israel two years earlier, and his younger sister was planning to join him there after she finished college. His mother, however, had decided to remain in St. Petersburg, and Perel- man moved in with her. “I realize that in Russia I work better,” he told col- leagues at the Steklov. At twenty-nine, Perelman was YC. YC.

firmly established as a mathematician N and yet largely unburdened by profes- Y, ALLER

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pursue whatever problems he wanted SM A to, and he knew that his work, should K Preserve, which has protected wetlands and ten miles of natural coast. he choose to publish it, would be PAUL

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TNY—2006_08_28—PAGE 51—133SC.—live art r15355_RD construct it, fixing any errors and filling in gaps. Yau believed that a mathema- tician has an obligation to be explicit, and impressed on his students the im- portance of step-by-step rigor. There are two ways to get credit for an original contribution in mathemat- ics. The first is to produce an original proof. The second is to identify a sig­ nificant gap in someone else’s proof and supply the missing chunk. How- ever, only true mathematical gaps— missing or mistaken arguments—can be the basis for a claim of originality. Filling in gaps in exposition—shortcuts and abbreviations used to make a proof more efficient—does not count. When, in 1993, Andrew Wiles revealed that a gap had been found in his proof of Fer- mat’s last theorem, the problem be- came fair game for anyone, until, the following year, Wiles fixed the error. Most mathematicians would agree “Over here, Billingsley.” that, by contrast, if a proof ’s implicit steps can be made explicit by an expert, •• then the gap is merely one of exposi- tion, and the proof should be consid- ered complete and correct. shown serious consideration. Yakov man thought he saw a way around the Occasionally, the difference be- Eliashberg, a mathematician at Stan- impasse. In 1996, he wrote Hamilton a tween a mathematical gap and a gap in ford who knew Perelman at Berkeley, long letter outlining his notion, in the exposition can be hard to discern. On thinks that Perelman returned to Rus- hope of collaborating. “He did not an- at least one occasion, Yau and his stu- sia in order to work on the Poincaré. swer,” Perelman said. “So I decided to dents have seemed to confuse the two, “Why not?” Perelman said when we work alone.” making claims of originality that other asked whether Eliashberg’s hunch was mathematicians believe are unwar- correct. au had no idea that Hamilton’s ranted. In 1996, a young geometer at The Internet made it possible for work on the Poincaré had stalled. Berkeley named Alexander Givental Perelman to work alone while continu- HeY was increasingly anxious about his had proved a mathematical conjec- ing to tap a common pool of knowl- own standing in the mathematics pro- ture about mirror symmetry, a concept edge. Perelman searched Hamilton’s fession, particularly in China, where, that is fundamental to string theory. papers for clues to his thinking and he worried, a younger scholar could try Though other mathematicians found gave several seminars on his work. “He to supplant him as Chern’s heir. More Givental’s proof hard to follow, they didn’t need any help,” Gromov said. than a decade had passed since Yau were optimistic that he had solved the “He likes to be alone. He reminds me had proved his last major result, though problem. As one geometer put it, “No- of Newton—this obsession with an he continued to publish prolifically. body at the time said it was incomplete idea, working by yourself, the disregard “Yau wants to be the king of geome- and incorrect.” for other people’s opinion. Newton was try,” Michael Anderson, a geometer at In the fall of 1997, Kefeng Liu, a more obnoxious. Perelman is nicer, but Stony Brook, said. “He believes that former student of Yau’s who taught at very obsessed.” everything should issue from him, that Stanford, gave a talk at Harvard on In 1995, Hamilton published a he should have oversight. He doesn’t mirror symmetry. According to two paper in which he discussed a few of his like people encroaching on his terri- geometers in the audience, Liu pro- ideas for completing a proof of the tory.” Determined to retain control ceeded to present a proof strikingly Poincaré. Reading the paper, Perelman over his field, Yau pushed his students similar to Givental’s, describing it as a realized that Hamilton had made no to tackle big problems. At Harvard, he paper that he had co-authored with progress on overcoming his obstacles— ran a notoriously tough seminar on Yau and another student of Yau’s. the necks and the cigars. “I hadn’t seen differential geometry, which met for “Liu mentioned Givental but only as any evidence of progress after early three hours at a time three times a one of a long list of people who had 1992,” Perelman told us. “Maybe he week. Each student was assigned a re- contributed to the field,” one of the got stuck even earlier.” However, Perel- cently published proof and asked to re- geometers said. (Liu maintains that

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TNY—2006_08_28—PAGE 52—133SC.52—133SC—live opi art A11737. his proof was significantly different ing speakers for the congress was Yau’s more than a friend. He is a hero. He is from Givental’s.) most successful student, Gang Tian, so original. We were working to finish Around the same time, Givental re- who had been at N.Y.U. with Perelman our proof. Hamilton worked on it for ceived an e-mail signed by Yau and his and was now a professor at M.I.T. The twenty-five years. You work, you get collaborators, explaining that they had host committee in Beijing also asked tired. He probably got a little tired— found his arguments impossible to fol- Tian to give a plenary address. and you want to take a rest.” low and his notation baffling, and had Yau was caught by surprise. In Then, on November 12, 2002, Yau come up with a proof of their own. March, 2000, he had published a survey received an e-mail message from a They praised Givental for his “brilliant of recent research in his field studded Russian mathematician whose name idea” and wrote, “In the final version of with glowing references to Tian and to didn’t immediately register. “May I our paper your important contribution their joint projects. He retaliated by or- bring to your attention my paper,” the will be acknowledged.” ganizing his first conference on string e-mail said. A few weeks later, the paper, “Mir- theory, which opened in Beijing a few ror Principle I,” appeared in the Asian days before the math congress began, in n November 11th, Perelman had Journal of Mathematics, which is co- late August, 2002. He persuaded Ste- posted a thirty-nine-page paper edited by Yau. In it, Yau and his co- phen Hawking and several Nobel lau- entitledO “The Entropy Formula for the authors describe their result as “the first reates to attend, and for days the Chi- Ricci Flow and Its Geometric Applica- complete proof ” of the mirror conjec- nese newspapers were full of pictures of tions,” on arXiv.org, a Web site used by ture. They mention Givental’s work famous scientists. Yau even managed to mathematicians to post preprints—ar- only in passing. “Unfortunately,” they arrange for his group to have an audi- ticles awaiting publication in refereed write, his proof, “which has been read ence with Jiang Zemin. A mathemati- journals. He then e-mailed an abstract by many prominent experts, is incom- cian who helped organize the math of his paper to a dozen mathematicians plete.” However, they did not identify congress recalls that along the highway in the United States—including Ham- a specific mathematical gap. between Beijing and the airport there ilton, Tian, and Yau—none of whom Givental was taken aback. “I wanted were “billboards with pictures of Ste- had heard from him for years. In the to know what their objection was,” he phen Hawking plastered everywhere.” abstract, he explained that he had writ- told us. “Not to expose them or defend That summer, Yau wasn’t thinking ten “a sketch of an eclectic proof ” of myself.” In March, 1998, he published much about the Poincaré. He had the geometrization conjecture. a paper that included a three-page confidence in Hamilton, despite his Perelman had not mentioned the footnote in which he pointed out a slow pace. “Hamilton is a very good proof or shown it to anyone. “I didn’t number of similarities between Yau’s friend,” Yau told us in Beijing. “He is have any friends with whom I could proof and his own. Several months later, a young mathematician at the University of Chicago who was asked by senior colleagues to investigate the dispute concluded that Givental’s proof was complete. Yau says that he had been working on the proof for years with his students and that they achieved their result independently of Givental. “We had our own ideas, and we wrote them up,” he says. Around this time, Yau had his first serious conflict with Chern and the Chi- nese mathematical establishment. For years, Chern had been hoping to bring the I.M.U.’s congress to Beijing. Ac- cording to several mathematicians who were active in the I.M.U. at the time, Yau made an eleventh-hour effort to have the congress take place in Hong Kong instead. But he failed to persuade a sufficient number of colleagues to go along with his proposal, and the I.M.U. ultimately decided to hold the 2002 con- gress in Beijing. (Yau denies that he tried to bring the congress to Hong Kong.) Among the delegates the I.M.U. ap- pointed to a group that would be choos-

TNY—2006_08_28—PAGE 53—133SC.—live opi art A11721 discuss this,” he said in St. Petersburg. rage for him to examine. Then he could ian embedding theorem, and John “I didn’t want to discuss my work with tell you where to give it a few knocks. Conway, the inventor of the cellular someone I didn’t trust.” Andrew Wiles What Hamilton introduced and Perel- automaton game Life. To the astonish- had also kept the fact that he was work- man completed is a procedure that is ment of many in the audience, Perel- ing on Fermat’s last theorem a secret, independent of the particularities of the man said nothing about the Poincaré. but he had had a colleague vet the proof blemish. If you apply the Ricci flow to “Here is a guy who proved a world-fa- before making it public. Perelman, by a 3-D space, it will begin to undent mous theorem and didn’t even mention casually posting a proof on the Internet it and smooth it out. The mechanic it,” Frank Quinn, a mathematician at of one of the most famous problems in would not need to even see the car— Virginia Tech, said. “He stated some mathematics, was not just flouting aca- just apply the equation.” Perelman key points and special properties, and demic convention but taking a consid- proved that the “cigars” that had trou- then answered questions. He was es- erable risk. If the proof was flawed, he bled Hamilton could not actually occur, tablishing credibility. If he had beaten would be publicly humiliated, and there and he showed that the “neck” problem his chest and said, ‘I solved it,’ he would would be no way to prevent another could be solved by performing an intri- have got a huge amount of resistance.” mathematician from fixing any errors cate sequence of mathematical surger- He added, “People were expecting a and claiming victory. But Perelman said ies: cutting out singularities and patch- strange sight. Perelman was much more he was not particularly concerned. “My ing up the raw edges. “Now we have a normal than they expected.” reasoning was: if I made an error and procedure to smooth things and, at To Perelman’s disappointment, someone used my work to construct a crucial points, control the breaks,” Hamilton did not attend that lecture or correct proof I would be pleased,” he Mazur said. the next ones, at Stony Brook. “I’m a said. “I never set out to be the sole solver Tian wrote to Perelman, asking him disciple of Hamilton’s, though I haven’t of the Poincaré.” to lecture on his paper at M.I.T. Col- received his authorization,” Perelman Gang Tian was in his office at leagues at Princeton and Stony Brook told us. But John Morgan, at Colum- M.I.T. when he received Perelman’s extended similar invitations. Perelman bia, where Hamilton now taught, was e-mail. He and Perelman had been accepted them all and was booked for a in the audience at Stony Brook, and friendly in 1992, when they were both month of lectures beginning in April, after a lecture he invited Perelman to at N.Y.U. and had attended the same 2003. “Why not?” he told us with a speak at Columbia. Perelman, hoping weekly math seminar in Princeton. “I shrug. Speaking of mathematicians to see Hamilton, agreed. The lecture immediately realized its importance,” generally, Fedor Nazarov, a mathema- took place on a Saturday morning. Ham­ Tian said of Perelman’s paper. Tian tician at Michigan State University, ilton showed up late and asked no ques- began to read the paper and discuss said, “After you’ve solved a problem, tions during either the long discussion it with colleagues, who were equally you have a great urge to talk about it.” session that followed the talk or the enthusiastic. lunch after that. “I had the impression On November 19th, Vitali Kapo- amilton and Yau were stunned by he had read only the first part of my vitch, a geometer, sent Perelman an Perelman’s announcement. “We paper,” Perelman said. e-mail: feltH that nobody else would be able to In the April 18, 2003, issue of Sci- Hi Grisha, Sorry to bother you but a lot discover the solution,” Yau told us in ence, Yau was featured in an article of people are asking me about your preprint “The entropy formula for the Ricci . . .” Do Beijing. “But then, in 2002, Perelman about Perelman’s proof: “Many ex- I understand it correctly that while you can- said that he published something. He perts, although not all, seem convinced not yet do all the steps in the Hamilton basically did a shortcut without doing that Perelman has stubbed out the ci- program you can do enough so that using some collapsing results you can prove ge- all the detailed estimates that we did.” gars and tamed the narrow necks. But ometrization? Vitali. Moreover, Yau complained, Perelman’s they are less confident that he can proof “was written in such a messy way control the number of surgeries. That Perelman’s response, the next day, that we didn’t understand.” could prove a fatal flaw, Yau warns, was terse: “That’s correct. Grisha.” Perelman’s April lecture tour was noting that many other attempted In fact, what Perelman had posted treated by mathematicians and by the proofs of the Poincaré conjecture have on the Internet was only the first in- press as a major event. Among the au- stumbled over similar missing steps.” stallment of his proof. But it was dience at his talk at Princeton were Proofs should be treated with skepti- sufficient for mathematicians to see John Ball, Andrew Wiles, John Forbes cism until mathematicians have had a that he had figured out how to solve the Nash, Jr., who had proved the Riemann­ chance to review them thoroughly, Poincaré. Barry Mazur, the Harvard Yau told us. Until then, he said, “it’s mathematician, uses the image of a not math—it’s religion.” dented fender to describe Perelman’s By mid-July, Perelman had posted achievement: “Suppose your car has a the final two installments of his proof dented fender and you call a mechanic on the Internet, and mathematicians to ask how to smooth it out. The me- had begun the work of formal explica- chanic would have a hard time telling tion, painstakingly retracing his steps. you what to do over the phone. You In the United States, at least two teams would have to bring the car into the ga- of experts had assigned themselves this

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TNY—2006_08_28—PAGE 54—133SC.—live spot art r15354_D, pls inspect and report on quality task: Gang Tian (Yau’s rival) and John Morgan; and a pair of researchers at the University of Michigan. Both projects were supported by the Clay Institute, which planned to publish Tian and Morgan’s work as a book. The book, in addition to providing other mathema- ticians with a guide to Perelman’s logic, would allow him to be considered for the Clay Institute’s million-dollar prize for solving the Poincaré. (To be eligi- ble, a proof must be published in a peer- reviewed venue and withstand two years of scrutiny by the mathematical community.) On September 10, 2004, more than a year after Perelman returned to St. Pe- tersburg, he received a long e-mail from Tian, who said that he had just attended “Great, but how is he in bed?” a two-week workshop at Princeton de- voted to Perelman’s proof. “I think that we have understood the whole paper,” •• Tian wrote. “It is all right.” Perelman did not write back. As he Kohn, a former chairman of the Prince­ while students were living on a hun- explained to us, “I didn’t worry too ton mathematics department, said. dred dollars a month. He also charged much myself. This was a famous prob- “Yau’s not jealous of Tian’s mathemat- Tian with shoddy scholarship and pla- lem. Some people needed time to get ics, but he’s jealous of his power back giarism, and with intimidating his accustomed to the fact that this is no in China.” graduate students into letting him longer a conjecture. I personally de- Though Yau had not spent more add his name to their papers. “Since I cided for myself that it was right for me than a few months at a time on main- promoted him all the way to his aca- to stay away from verification and not land China since he was an infant, he demic fame today, I should also take to participate in all these meetings. It is was convinced that his status as the responsibility for his improper behav- important for me that I don’t influence only Chinese Fields Medal winner ior,” Yau was quoted as saying to a re- this process.” should make him Chern’s successor. In porter, explaining why he felt obliged In July of that year, the National a speech he gave at Zhejiang Univer- to speak out. Science Foundation had given nearly a sity, in Hangzhou, during the summer In another interview, Yau described million dollars in grants to Yau, Ham- of 2004, Yau reminded his listeners of how the Fields committee had passed ilton, and several students of Yau’s to his Chinese roots. “When I stepped out Tian over in 1988 and how he had study and apply Perelman’s “break- from the airplane, I touched the soil of lobbied on Tian’s behalf with various through.” An entire branch of mathe- Beijing and felt great joy to be in my prize committees, including one at the matics had grown up around efforts mother country,” he said. “I am proud National Science Foundation, which to solve the Poincaré, and now that to say that when I was awarded the awarded Tian five hundred thousand branch appeared at risk of becoming Fields Medal in mathematics, I held no dollars in 1994. obsolete. Michael Freedman, who won passport of any country and should cer- Tian was appalled by Yau’s attacks, a Fields for proving the Poincaré con- tainly be considered Chinese.” but he felt that, as Yau’s former stu- jecture for the fourth dimension, told The following summer, Yau re- dent, there was little he could do about the Times that Perelman’s proof was a turned to China and, in a series of in- them. “His accusations were base- “small sorrow for this particular branch terviews with Chinese reporters, at- less,” Tian told us. But, he added, “I of topology.” Yuri Burago said, “It kills tacked Tian and the mathematicians at have deep roots in Chinese culture. A the field. After this is done, many Peking University. In an article pub- teacher is a teacher. There is respect. It mathematicians will move to other lished in a Beijing science newspaper, is very hard for me to think of anything branches of mathematics.” which ran under the headline “SHING- to do.” TUNG YAU IS SLAMMING ACADEMIC While Yau was in China, he visited ive months later, Chern died, and CORRUPTION IN CHINA,” Yau called Xi-Ping Zhu, a protégé of his who was Yau’s efforts to insure that he—not Tian “a complete mess.” He accused now chairman of the mathematics de- Tian—wasF recognized as his successor him of holding multiple professor- partment at Sun Yat-sen University. turned vicious. “It’s all about their pri- ships and of collecting a hundred and In the spring of 2003, after Perelman macy in China and their leadership twenty-five thousand dollars for a few completed his lecture tour in the United among the expatriate Chinese,” Joseph months’ work at a Chinese university, States, Yau had recruited Zhu and an-

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TNY—2006_08_28—PAGE 55—133SC—live opi art A11705. other student, Huai-Dong Cao, a pro- ident of the Clay Institute. He told “brought in fresh new ideas to figure fessor at Lehigh University, to under- Carlson that he wanted to trade a copy out important steps to overcome the take an explication of Perelman’s proof. of Zhu and Cao’s paper for a copy of main obstacles that remained in the Zhu and Cao had studied the Ricci Tian and Morgan’s book manuscript. program of Hamilton.” However, they flow under Yau, who considered Zhu, Yau told us he was worried that Tian write, they were obliged to “substitute in particular, to be a mathematician of would try to steal from Zhu and Cao’s several key arguments of Perelman by exceptional promise. “We have to figure work, and he wanted to give each party new approaches based on our study, out whether Perelman’s paper holds to- simultaneous access to what the other because we were unable to comprehend gether,” Yau told them. Yau arranged had written. “I had a lunch with Carl- these original arguments of Perelman for Zhu to spend the 2005-06 academic son to request to exchange both manu- which are essential to the completion year at Harvard, where he gave a semi- scripts to make sure that nobody can of the geometrization program.” Math- nar on Perelman’s proof and continued copy the other,” Yau said. Carlson de- ematicians familiar with Perelman’s to work on his paper with Cao. murred, explaining that the Clay Insti- proof disputed the idea that Zhu and tute had not yet received Tian and Cao had contributed significant new n April 13th of this year, the Morgan’s complete manuscript. approaches to the Poincaré. “Perelman thirty-one mathematicians on the By the end of the following week, already did it and what he did was editorialO board of the Asian Journal of the title of Zhu and Cao’s paper on the complete and correct,” John Morgan Mathematics received a brief e-mail A.J.M.’s Web site had changed, to “A said. “I don’t see that they did anything from Yau and the journal’s co-editor Complete Proof of the Poincaré and different.” informing them that they had three Geometrization Conjectures: Applica- By early June, Yau had begun to days to comment on a paper by Xi- tion of the Hamilton-Perelman The- promote the proof publicly. On June Ping Zhu and Huai-Dong Cao titled ory of the Ricci Flow.” The abstract 3rd, at his mathematics institute in “The Hamilton-Perelman Theory of had also been revised. A new sentence Beijing, he held a press conference. Ricci Flow: The Poincaré and Geom- explained, “This proof should be con- The acting director of the mathematics etrization Conjectures,” which Yau sidered as the crowning achievement institute, attempting to explain the rel- planned to publish in the journal. The of the Hamilton-Perelman theory of ative contributions of the different e-mail did not include a copy of the Ricci flow.” mathematicians who had worked on paper, reports from referees, or an ab- Zhu and Cao’s paper was more than the Poincaré, said, “Hamilton contrib- stract. At least one board member three hundred pages long and filled the uted over fifty per cent; the Russian, asked to see the paper but was told that A.J.M.’s entire June issue. The bulk of Perelman, about twenty-five per cent; it was not available. On April 16th, the paper is devoted to reconstructing and the Chinese, Yau, Zhu, and Cao et Cao received a message from Yau tell- many of Hamilton’s Ricci-flow re- al., about thirty per cent.” (Evidently, ing him that the paper had been ac- sults—including results that Perelman simple addition can sometimes trip up cepted by the A.J.M., and an abstract had made use of in his proof—and even a mathematician.) Yau added, was posted on the journal’s Web site. much of Perelman’s proof of the Poin- “Given the significance of the Poin- A month later, Yau had lunch in caré. In their introduction, Zhu and caré, that Chinese mathematicians Cambridge with Jim Carlson, the pres- Cao credit Perelman with having played a thirty-per-cent role is by no means easy. It is a very important contribution.” On June 12th, the week before Yau’s conference on string theory opened in Beijing, the South China Morning Post reported, “Mainland mathematicians who helped crack a ‘millennium math problem’ will present the methodology and findings to physicist Stephen Hawk- ing. . . . Yau Shing-Tung, who orga- nized Professor Hawking’s visit and is also Professor Cao’s teacher, said yester- day he would present the findings to Professor Hawking because he believed the knowledge would help his research into the formation of black holes.” On the morning of his lecture in Bei- jing, Yau told us, “We want our contri- bution understood. And this is also a strategy to encourage Zhu, who is in “Call me ‘dude’ again! I dare you! I double-dare you, Alexander! China and who has done really spectac- Call me ‘dude’ one more goddam time!” ular work. I mean, important work with

TNY—2006_08_28—PAGE 56—133SC—live opi art A11574. a century-long problem, which will tributions that is as stringent as the for five hours. Perelman repeatedly said probably have another few century-long rules governing math itself. As Perel- that he had retired from the mathemat- implications. If you can attach your man put it, “If everyone is honest, it is ics community and no longer considered name in any way, it is a contribution.” natural to share ideas.” Many mathe- himself a professional mathematician. maticians view Yau’s conduct over the He mentioned a dispute that he had had T. Bell, the author of “Men of Poincaré as a violation of this basic years earlier with a collaborator over how Mathematics,” a witty history of ethic, and worry about the damage it to credit the author of a particular proof, theE discipline. published in 1937, once has caused the profession. “Politics, and said that he was dismayed by the dis- lamented “the squabbles over priority power, and control have no legitimate cipline’s lax ethics. “It is not people who which disfigure scientific history.” But in role in our community, and they break ethical standards who are regarded the days before e-mail, blogs, and Web threaten the integrity of our field,” as aliens,” he said. “It is people like me sites, a certain decorum usually pre- Phillip Griffiths said. who are isolated.” We asked him whether vailed. In 1881, Poincaré, who was then he had read Cao and Zhu’s paper. “It is at the University of Caen, had an alter- erelman likes to attend opera per- not clear to me what new contribution cation with a German mathematician in formances at the Mariinsky The- did they make,” he said. “Apparently, Leipzig named Felix Klein. Poincaré atre,P in St. Petersburg. Sitting high up Zhu did not quite understand the argu- had published several papers in which he in the back of the house, he can’t make ment and reworked it.” As for Yau, labelled certain functions “Fuchsian,” out the singers’ expressions or see the Perelman said, “I can’t say I’m outraged. after another mathematician. Klein details of their costumes. But he cares Other people do worse. Of course, there wrote to Poincaré, pointing out that he only about the sound of their voices, and are many mathematicians who are more and others had done significant work on he says that the acoustics are better or less honest. But almost all of them are these functions, too. An exchange of po- where he sits than anywhere else in the conformists. They are more or less hon- lite letters between Leipzig and Caen theatre. Perelman views the mathemat- est, but they tolerate those who are not ensued. Poincaré’s last word on the sub- ics community—and much of the larger honest.” ject was a quote from Goethe’s “Faust”: world—from a similar remove. The prospect of being awarded a “Name ist Schall und Rauch.” Loosely Before we arrived in St. Petersburg, Fields Medal had forced him to make a translated, that corresponds to Shake- on June 23rd, we had sent several mes- complete break with his profession. “As speare’s “What’s in a name?” sages to his e-mail address at the Steklov long as I was not conspicuous, I had a This, essentially, is what Yau’s Institute, hoping to arrange a meeting, choice,” Perelman explained. “Either to friends are asking themselves. “I find but he had not replied. We took a taxi to make some ugly thing”—a fuss about the myself getting annoyed with Yau that his apartment building and, reluctant to math community’s lack of integrity— he seems to feel the need for more intrude on his privacy, left a book—a “or, if I didn’t do this kind of thing, to be kudos,” Dan Stroock, of M.I.T., said. collection of John Nash’s papers—in his treated as a pet. Now, when I become a “This is a guy who did magnificent mailbox, along with a card saying that very conspicuous person, I cannot stay a things, for which he was magnificently we would be sitting on a bench in a pet and say nothing. That is why I had rewarded. He won every prize to be nearby playground the following after- to quit.” We asked Perelman whether, won. I find it a little mean of him to noon. The next day, after Perelman by refusing the Fields and withdrawing seem to be trying to get a share of this failed to appear, we left a box of pearl tea from his profession, he was eliminating as well.” Stroock pointed out that, and a note describing some of the ques- any possibility of influencing the disci- twenty-five years ago, Yau was in a sit- tions we hoped to discuss with him. We pline. “I am not a politician!” he replied, uation very similar to the one Perelman repeated this ritual a third time. Finally, angrily. Perelman would not say whether is in today. His most famous result, on believing that Perelman was out of town, his objection to awards extended to the Calabi-Yau manifolds, was hugely im- we pressed the buzzer for his apartment, Clay Institute’s million-dollar prize. “I’m portant for theoretical physics. “Calabi hoping at least to speak with his mother. not going to decide whether to accept outlined a program,” Stroock said. “In A woman answered and let us inside. the prize until it is offered,” he said. a real sense, Yau was Calabi’s Perel- Perelman met us in the dimly lit hallway Mikhail Gromov, the Russian geom- man. Now he’s on the other side. He’s of the apartment. It turned out that he eter, said that he understood Perelman’s had no compunction at all in taking the had not checked his Steklov e-mail ad- logic: “To do great work, you have to lion’s share of credit for Calabi-Yau. dress for months, and had not looked in have a pure mind. You can think only And now he seems to be resenting his mailbox all week. He had no idea about the mathematics. Everything else Perelman getting credit for completing who we were. is human weakness. Accepting prizes is Hamilton’s program. I don’t know if We arranged to meet at ten the fol- showing weakness.” Others might view the analogy has ever occurred to him.” lowing morning on Nevsky Prospekt. Perelman’s refusal to accept a Fields as Mathematics, more than many From there, Perelman, dressed in a sports arrogant, Gromov said, but his princi- other fields, depends on collaboration. coat and loafers, took us on a four-hour ples are admirable. “The ideal scientist Most problems require the insights of walking tour of the city, commenting on does science and cares about nothing several mathematicians in order to be every building and vista. After that, we else,” he said. “He wants to live this ideal. solved, and the profession has evolved all went to a vocal competition at the St. Now, I don’t think he really lives on this a standard for crediting individual con- Petersburg Conservatory, which lasted ideal plane. But he wants to.” 

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