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August 20, 2006
World's top maths genius jobless and living with mother
By Nadejda Lobastova in St Petersburg and Michael Hirst, The Telegraph
A maths genius who won fame last week for apparently spurning a million-dollar prize is living with his mother in a humble flat in St Petersburg, co-existing on her £30-a-month pension, because he has been unemployed since December.
The Sunday Telegraph tracked down the eccentric recluse who stunned the maths world when he solved a century-old puzzle known as the Poincare Conjecture.
Grigory "Grisha" Perelman's predicament stems from an acrimonious split with a leading Russian mathematical institute, the Steklov, in 2003. When the Institute in St Petersburg failed to re-elect him as a member, Dr Perelman, 40, was left feeling an "absolutely ungifted and untalented person", said a friend. He had a crisis of confidence and cut himself off.
Other friends say he cannot afford to travel to this week's International Mathematical Union's congress in Madrid, where his peers want him to receive the maths equivalent of the Nobel Prize, and that he is too modest to ask anyone to underwrite his trip.
Interviewed in St Petersburg last week, Dr Perelman insisted that he was unworthy of all the attention, and was uninterested in his windfall. "I do not think anything that I say can be of the slightest public interest," he said. "I am not saying that because I value my privacy, or that I am doing anything I want to hide. There are no top-secret projects going on here. I just believe the public has no interest in me."
He continued: "I know that self-promotion happens a lot and if people want to do that, good luck to them, but I do not regard it as a positive thing. I realised this a long time ago and nobody is going to change my mind. "Newspapers should be more discerning over who they write about. They should have more taste. As far as I am concerned, I can't offer anything for their readers.
"I don't base that on any negative experiences with the press, although they have been making up nonsense about my father being a famous physicist. It's just plain and simply that I don't care what anybody writes about me at all."
Dr Perelman has some small savings from his time as a lecturer, but is apparently reluctant to supplement them with the $1 million (£531,000) offered by the Clay Mathematics Institute in Cambridge, Massachusetts, for solving one of the world's seven "Millennium Problems".
The Poincare Conjecture was first posed by the French mathematician, Jules Henri Poincare, in 1904, and seeks to understand the shape of the universe by linking shapes, spaces and surfaces.
Friends say that evidence of Dr Perelman's innate modesty came when - having finally solved the problem after more than 10 years' work - he simply posted his conclusion on the internet, rather than publishing his explanation in a recognised journal. "If anybody is interested in my way of solving the problem, it's all there - let them go and read about it," said Dr Perelman. "I have published all my calculations. This is what I can offer the public."
Friends were not surprised to learn that he was living with his mother. The Jewish family - he has a younger sister, Elena, also a mathematician - was always close. One friend, Sergey Rukshin, head of St Petersburg Mathematical Centre for Gifted Students, gave Dr Perelman his first break as a teenager.
At 16, he won a gold medal at the 1982 International Mathematical Olympiad, with a perfect score of 42. He was also a talented violinist and played table tennis. It was after gaining his PhD from St Petersburg State University that Dr Perelman first worked at the Steklov Institute, part of the Russian Academy of Science. Later, he worked in America before returning to the Steklov in 1996. Its rejection of him, three years ago, devastated Dr Perelman, said Mr Rukshin.
Although the two old friends still discuss life, music and literature, they no longer talk about maths. "It has become a painful topic for the doctor," said Mr Rukshin.
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August 28, 2006
Manifold Destiny
A legendary problem and the battle over who solved it.
By Sylvia Nasar and David Gruber
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 Shing-Tung Yau. In the late nineteen-seventies, when Yau was in his twenties, he had made a series of breakthroughs that helped launch the string-theory revolution in physics and earned him, in addition to a Fields Medal—the most coveted award in mathematics—a 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. (More than six thousand students attended the keynote address, which was delivered by Yau’s close friend Stephen Hawking, in the Great Hall of the People.) The subject of Yau’s talk was something that few in his audience knew much about: the Poincaré conjecture, a century-old conundrum about the characteristics of three-dimensional spheres, which, because it has important implications for mathematics and cosmology and because it has eluded all attempts at solution, is regarded by mathematicians as a holy grail.
Yau, a stocky man of fifty-seven, stood at a lectern in shirtsleeves and black-rimmed glasses and, with his hands in his pockets, described how two of his students, Xi-Ping Zhu and Huai-Dong Cao, had completed a proof of the Poincaré conjecture a few weeks earlier. “I’m very positive about Zhu and Cao’s work,” Yau said. “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 Poincaré. 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. But Perelman resides in St. Petersburg and refuses to communicate with other people.”
For ninety minutes, Yau discussed some of the technical details of his students’ proof. When he was finished, no one asked any questions. 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.
Grigory Perelman is indeed reclusive. He left his job as a researcher at the Steklov Institute of Mathematics, in St. Petersburg, last December; he has few friends; 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 Poincaré conjecture with Sir John M. Ball, the fifty-eight-year-old 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 Poincaré, and Ball had gone to St. Petersburg to persuade him to accept the prize in a public ceremony at the I.M.U.’s quadrennial congress, in Madrid, on August 22nd.
The Fields Medal, like the Nobel Prize, grew, in part, out of a desire to elevate science above national animosities. German mathematicians were excluded from the first I.M.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 forty-four medals have been awarded in nearly seventy years—including three for work closely related to the Poincaré conjecture—and no mathematician has ever refused the prize. Nevertheless, Perelman told Ball that he had no intention of accepting it. “I refuse,” he said simply.
Over a period of eight months, beginning in November, 2002, Perelman posted a proof of the Poincaré 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 peer-reviewed journals; to insure fairness, reviewers are supposed to be carefully chosen by journal editors, and the identity of a scholar whose pa-per is under consideration is kept secret. Publication implies that a proof is complete, correct, and original.
By these standards, Perelman’s proof was unorthodox. 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 Poincaré 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 Poincaré. Even so, the proof’s complexity—and Perelman’s use of shorthand in making some of his most important claims—made 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. Petersburg. Since then, although he had continued to answer queries about it by e-mail, 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. As Ball planned the I.M.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. The I.M.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. Petersburg.
Ball wanted to keep his visit a secret—the names of Fields Medal recipients are announced officially at the awards ceremony—and 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 blue-green 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 walk—one 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; third, I don’t accept the prize. From the very beginning, I told him I have chosen the third one.” The Fields Medal held no interest for him, Perelman explained. “It was completely irrelevant for me,” he said. “Everybody understood that if the proof is correct then no other recognition is needed.”
Proofs of the Poincaré have been announced nearly every year since the conjecture was formulated, by Henri Poincaré, more than a hundred years ago. Poincaré was a cousin of Raymond Poincaré, the President of France during the First World War, and one of the most creative mathematicians of the nineteenth century. Slight, myopic, and notoriously absent-minded, he conceived his famous problem in 1904, eight years before he died, and tucked it as an offhand question into the end of a sixty-five-page paper.
Poincaré didn’t make much progress on proving the conjecture. “Cette question nous entraînerait trop loin” (“This question would take us too far”), he wrote. He was a founder of topology, also known as “rubber-sheet geometry,” for its focus on the intrinsic properties of spaces. From a topologist’s perspective, 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 “manifold” to describe such an abstract topological 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 soccer 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 cannot pull the slipknot closed without tearing the bagel.
Two-dimensional manifolds were well understood by the mid-nineteenth century. But it remained unclear whether what was true for two dimensions was also true for three. Poincaré proposed that all closed, simply connected, three-dimensional manifolds—those which lack holes and are of finite extent—were spheres. The conjecture was potentially important for scientists studying the largest known three-dimensional manifold: the universe. Proving it mathematically, however, was far from easy. Most attempts were merely embarrassing, but some led to important mathematical discoveries, including proofs of Dehn’s Lemma, the Sphere Theorem, and the Loop Theorem, which are now fundamental concepts in topology.
By the nineteen-sixties, topology had become one of the most productive areas of mathematics, and young topologists were launching regular attacks on the Poincaré. To the astonishment of most mathematicians, it turned out that manifolds of the fourth, fifth, and higher dimensions were more tractable than those of the third dimension. By 1982, Poincaré’s conjecture had been proved in all dimensions except the third. In 2000, the Clay Mathematics Institute, a private foundation that promotes mathematical research, named the Poincaré one of the seven most important outstanding problems in mathematics and offered a million dollars to anyone who could prove it.
“My whole life as a mathematician has been dominated by the Poincaré conjecture,” John Morgan, the head of the mathematics department at Columbia University, said. “I never thought I’d see a solution. I thought nobody could touch it.”
Grigory Perelman did not plan to become a mathematician. “There was never a decision point,” he said when we met. We were outside the apartment building where he lives, in Kupchino, a neighborhood of drab high-rises. Perelman’s father, who was an electrical engineer, encouraged his interest in math. “He gave me logical and other math problems to think about,” Perelman said. “He got a lot of books for me to read. He taught me how to play chess. He was proud of me.” Among the books his father gave him was a copy of “Physics for Entertainment,” which had been a best-seller in the Soviet Union in the nineteen-thirties. In the foreword, the book’s author describes the contents as “conundrums, brain-teasers, entertaining anecdotes, and unexpected comparisons,” adding, “I have quoted extensively from Jules Verne, H. G. Wells, Mark Twain and other writers, because, besides providing entertainment, the fantastic experiments these writers describe may well serve as instructive illustrations at physics classes.” The book’s topics included how to jump from a moving car, and why, “according to the law of buoyancy, we would never drown in the Dead Sea.”
The notion that Russian society considered worthwhile what Perelman did for pleasure came as a surprise. By the time he was fourteen, he was the star performer of a local math club. In 1982, the year that Shing-Tung Yau won a Fields Medal, Perelman earned a perfect score and the gold medal at the International Mathematical Olympiad, in Budapest. He was friendly with his teammates but not close—“I had no close friends,” he said. He was one of two or three Jews in his grade, and he had a passion for opera, which also set him apart from his peers. His mother, a math teacher at a technical college, played the violin and began taking him to the opera when he was six. By the time Perelman was fifteen, he was spending his pocket money on records. He was thrilled to own a recording of a famous 1946 performance of “La Traviata,” featuring Licia Albanese as Violetta. “Her voice was very good,” he said.
At Leningrad University, which Perelman entered in 1982, at the age of sixteen, he took advanced classes in geometry and solved a problem posed by Yuri Burago, a mathematician at the Steklov Institute, who later became his Ph.D. adviser. “There are a lot of students of high ability who speak before thinking,” Burago said. “Grisha was different. He thought deeply. His answers were always correct. He always checked very, very carefully.” Burago added, “He was not fast. Speed means nothing. Math doesn’t depend on speed. It is about deep.”
At the Steklov in the early nineties, Perelman became an expert on the geometry of Riemannian and Alexandrov spaces—extensions of traditional Euclidean geometry—and began to publish articles in the leading Russian and American mathematics journals. In 1992, Perelman was invited to spend a semester each at New York University and Stony Brook University. By the time he left for the United States, that fall, the Russian economy had collapsed. Dan Stroock, a mathematician at M.I.T., recalls smuggling wads of dollars into the country to deliver to a retired mathematician at the Steklov, who, like many of his colleagues, had become destitute.
Perelman was pleased to be in the United States, the capital of the international mathematics community. He wore the same brown corduroy jacket every day and told friends at N.Y.U. that he lived on a diet of bread, cheese, and milk. He liked to walk to Brooklyn, where he had relatives and could buy traditional Russian brown bread. Some of his colleagues were taken aback by his fingernails, which were several inches long. “If they grow, why wouldn’t I let them grow?” he would say when someone asked why he didn’t cut them. Once a week, he and a young Chinese mathematician named Gang Tian drove to Princeton, to attend a seminar at the Institute for Advanced Study.
For several decades, the institute and nearby Princeton University had been centers of topological research. In the late seventies, William Thurston, a Princeton mathematician who liked to test out his ideas using scissors and construction paper, proposed a taxonomy for classifying manifolds of three dimensions. He argued that, while the manifolds could be made to take on many different shapes, they nonetheless had a “preferred” geometry, just as a piece of silk draped over a dressmaker’s mannequin takes on the mannequin’s form.
Thurston proposed that every three-dimensional manifold could be broken down into one or more of eight types of component, including a spherical type. Thurston’s theory—which became known as the geometrization conjecture—describes all possible three-dimensional manifolds and is thus a powerful generalization of the Poincaré. If it was confirmed, then Poincaré’s conjecture would be, too. Proving Thurston and Poincaré “definitely swings open doors,” Barry Mazur, a mathematician at Harvard, said. The implications of the conjectures for other disciplines may not be apparent for years, but for mathematicians the problems are fundamental. “This is a kind of twentieth-century Pythagorean theorem,” Mazur added. “It changes the landscape.”
In 1982, Thurston won a Fields Medal for his contributions to topology. That year, Richard Hamilton, a mathematician at Cornell, published a paper on an equation called the Ricci flow, which he suspected could be relevant for solving Thurston’s conjecture and thus the Poincaré. Like a heat equation, which describes how heat distributes itself evenly through a substance—flowing from hotter to cooler parts of a metal sheet, for example—to create a more uniform temperature, the Ricci flow, by smoothing out irregularities, gives manifolds a more uniform geometry.
Hamilton, the son of a Cincinnati doctor, defied the math profession’s nerdy stereotype. Brash and irreverent, he rode horses, windsurfed, and had a succession of girlfriends. He treated math as merely one of life’s pleasures. At forty-nine, he was considered a brilliant lecturer, but he had published relatively little beyond a series of seminal articles on the Ricci flow, and he had few graduate students. Perelman had read Hamilton’s papers and went to hear him give a talk at the Institute for Advanced Study. Afterward, Perelman shyly spoke to him.
“I really wanted to ask him something,” Perelman recalled. “He was smiling, and he was quite patient. He actually told me a couple of things that he published a few years later. He did not hesitate to tell me. Hamilton’s openness and generosity—it really attracted me. I can’t say that most mathematicians act like that.
“I was working on different things, though occasionally I would think about the Ricci flow,” Perelman added. “You didn’t have to be a great mathematician to see that this would be useful for geometrization. I felt I didn’t know very much. I kept asking questions.”
Shing-Tung Yau was also asking Hamilton questions about the Ricci flow. Yau and Hamilton had met in the seventies, and had become close, despite considerable differences in temperament and background. A mathematician at the University of California at San Diego who knows both men called them “the mathematical loves of each other’s lives.”
Yau’s family moved to Hong Kong from mainland China in 1949, when he was five months old, along with hundreds of thousands of other refugees fleeing Mao’s armies. The previous year, his father, a relief worker for the United Nations, had lost most of the family’s savings in a series of failed ventures. In Hong Kong, to support his wife and eight children, he tutored college students in classical Chinese literature and philosophy.
When Yau was fourteen, his father died of kidney cancer, leaving his mother dependent on handouts from Christian missionaries and whatever small sums she earned from selling handicrafts. Until then, Yau had been an indifferent student. But he began to devote himself to schoolwork, tutoring other students in math to make money. “Part of the thing that drives Yau is that he sees his own life as being his father’s revenge,” said Dan Stroock, the M.I.T. mathematician, who has known Yau for twenty years. “Yau’s father was like the Talmudist whose children are starving.”
Yau studied math at the Chinese University of Hong Kong, where he attracted the attention of Shiing-Shen Chern, the preëminent Chinese mathematician, who helped him win a scholarship to the University of California at Berkeley. Chern was the author of a famous theorem combining topology and geometry. He spent most of his career in the United States, at Berkeley. He made frequent visits to Hong Kong, Taiwan, and, later, China, where he was a revered symbol of Chinese intellectual achievement, to promote the study of math and science.
In 1969, Yau started graduate school at Berkeley, enrolling in seven graduate courses each term and auditing several others. He sent half of his scholarship money back to his mother in China and impressed his professors with his tenacity. He was obliged to share credit for his first major result when he learned that two other mathematicians were working on the same problem. In 1976, he proved a twenty-year-old conjecture pertaining to a type of manifold that is now crucial to string theory. A French mathematician had formulated a proof of the problem, which is known as Calabi’s conjecture, but Yau’s, because it was more general, was more powerful. (Physicists now refer to Calabi-Yau manifolds.) “He was not so much thinking up some original way of looking at a subject but solving extremely hard technical problems that at the time only he could solve, by sheer intellect and force of will,” Phillip Griffiths, a geometer and a former director of the Institute for Advanced Study, said.
In 1980, when Yau was thirty, he became one of the youngest mathematicians ever to be appointed to the permanent faculty of the Institute for Advanced Study, and he began to attract talented students. He won a Fields Medal two years later, the first Chinese ever to do so. By this time, Chern was seventy years old and on the verge of retirement. According to a relative of Chern’s, “Yau decided that he was going to be the next famous Chinese mathematician and that it was time for Chern to step down.”
Harvard had been trying to recruit Yau, and when, in 1983, it was about to make him a second offer Phillip Griffiths told the dean of faculty a version of a story from “The Romance of the Three Kingdoms,” a Chinese classic. In the third century A.D., a Chinese warlord dreamed of creating an empire, but the most brilliant general in China was working for a rival. Three times, the warlord went to his enemy’s kingdom to seek out the general. Impressed, the general agreed to join him, and together they succeeded in founding a dynasty. Taking the hint, the dean flew to Philadelphia, where Yau lived at the time, to make him an offer. Even so, Yau turned down the job. Finally, in 1987, he agreed to go to Harvard.
Yau’s entrepreneurial drive extended to collaborations with colleagues and students, and, in addition to conducting his own research, he began organizing seminars. He frequently 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 Beijing. “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. “Meeting Yau changed his mathematical life,” a friend of both mathematicians said of Hamilton. “This was the first time he had been on to something extremely big. Talking to Yau gave him courage and direction.”
Yau believed that if he could help solve the Poincaré it would be a victory not just for him but also for China. In the mid-nineties, Yau and several other Chinese scholars began 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 in dire condition. According to Steve Smale, who won a Fields for proving the Poincaré in higher dimensions, and who, after retiring from Berkeley, taught in Hong Kong, Peking University had “halls filled with the smell of urine, one common room, one office for all the assistant professors,” and paid its faculty wretchedly low salaries. Yau persuaded a Hong Kong real-estate mogul to help finance a mathematics institute at the Chinese Academy of Sciences, in Beijing, and to endow a Fields-style medal for Chinese mathematicians under the age of forty-five. On his trips to China, Yau touted Hamilton and their joint work on the Ricci flow and the Poincaré as a model for young Chinese mathematicians. As he put it in Beijing, “They always say that the whole country should learn from Mao or some big heroes. So I made a joke to them, but I was half serious. I said the whole country should learn from Hamilton.”
Grigory Perelman was learning from Hamilton already. In 1993, he began a two-year fellowship at Berkeley. While he was there, Hamilton gave several talks on campus, and in one he mentioned that he was working on the Poincaré. Hamilton’s Ricci-flow strategy was extremely technical and tricky to execute. After one of his talks at Berkeley, he told Perelman about his biggest obstacle. As a space is smoothed under the Ricci flow, some regions deform into what mathematicians refer to as “singularities.” Some regions, called “necks,” become attenuated areas of infinite density. More troubling to Hamilton was a kind of singularity he called the “cigar.” If cigars formed, Hamilton worried, it might be impossible to achieve uniform geometry. Perelman realized that a paper he had written on Alexandrov spaces might help Hamilton prove Thurston’s conjecture—and the Poincaré—once Hamilton solved the cigar problem. “At some point, I asked Hamilton if he knew a certain collapsing result that I had proved but not published—which turned out to be very useful,” Perelman said. “Later, I realized that he didn’t understand what I was talking about.” Dan Stroock, of M.I.T., said, “Perelman may have learned stuff from Yau and Hamilton, but, at the time, they were not learning from him.”
By the end of his first year at Berkeley, Perelman had written several strikingly original papers. He was asked to give a lecture at the 1994 I.M.U. congress, in Zurich, and invited to apply for jobs at Stanford, Princeton, the Institute for Advanced Study, and the University of Tel Aviv. Like Yau, Perelman was a formidable problem solver. Instead of spending years constructing an intricate theoretical framework, or defining new areas of research, he focussed on obtaining particular results. According to Mikhail Gromov, a renowned Russian geometer who has collaborated with Perelman, he had been trying to overcome a technical difficulty relating to Alexandrov 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 Hamilton’s Ricci-flow equation and the problem 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 apartment were struck by how sparsely furnished 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, Perelman 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 Steklov 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 Perelman moved in with her. “I realize that in Russia I work better,” he told colleagues at the Steklov.
At twenty-nine, Perelman was firmly established as a mathematician and yet largely unburdened by professional responsibilities. He was free to pursue whatever problems he wanted to, and he knew that his work, should he choose to publish it, would be shown serious consideration. Yakov Eliashberg, a mathematician at Stanford who knew Perelman at Berkeley, thinks that Perelman returned to Russia in order to work on the Poincaré. “Why not?” Perelman said when we asked whether Eliashberg’s hunch was correct.
The Internet made it possible for Perelman to work alone while continuing to tap a common pool of knowledge. Perelman searched Hamilton’s papers for clues to his thinking and gave several seminars on his work. “He didn’t need any help,” Gromov said. “He likes to be alone. He reminds me of Newton—this obsession with an idea, working by yourself, the disregard for other people’s opinion. Newton was more obnoxious. Perelman is nicer, but very obsessed.”
In 1995, Hamilton published a paper in which he discussed a few of his ideas for completing a proof of the Poincaré. Reading the paper, Perelman realized that Hamilton had made no progress on overcoming his obstacles—the necks and the cigars. “I hadn’t seen any evidence of progress after early 1992,” Perelman told us. “Maybe he got stuck even earlier.” However, Perelman thought he saw a way around the impasse. In 1996, he wrote Hamilton a long letter outlining his notion, in the hope of collaborating. “He did not answer,” Perelman said. “So I decided to work alone.”
Yau had no idea that Hamilton’s work on the Poincaré had stalled. He was increasingly anxious about his own standing in the mathematics profession, particularly in China, where, he worried, a younger scholar could try to supplant him as Chern’s heir. More than a decade had passed since Yau had proved his last major result, though he continued to publish prolifically. “Yau wants to be the king of geometry,” Michael Anderson, a geometer at Stony Brook, said. “He believes that everything should issue from him, that he should have oversight. He doesn’t like people encroaching on his territory.” Determined to retain control over his field, Yau pushed his students to tackle big problems. At Harvard, he ran a notoriously tough seminar on differential geometry, which met for three hours at a time three times a week. Each student was assigned a recently published proof and asked to reconstruct it, fixing any errors and filling in gaps. Yau believed that a mathematician has an obligation to be explicit, and impressed on his students the importance of step-by-step rigor.
There are two ways to get credit for an original contribution in mathematics. The first is to produce an original proof. The second is to identify a significant gap in someone else’s proof and supply the missing chunk. However, 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 Fermat’s last theorem, the problem became fair game for anyone, until, the following year, Wiles fixed the error. Most mathematicians would agree that, by contrast, if a proof’s implicit steps can be made explicit by an expert, then the gap is merely one of exposition, and the proof should be considered complete and correct.
Occasionally, the difference between a mathematical gap and a gap in exposition can be hard to discern. On at least one occasion, Yau and his students have seemed to confuse the two, making claims of originality that other mathematicians believe are unwarranted. In 1996, a young geometer at Berkeley named Alexander Givental had proved a mathematical conjecture about mirror symmetry, a concept that is fundamental to string theory. Though other mathematicians found Givental’s proof hard to follow, they were optimistic that he had solved the problem. As one geometer put it, “Nobody at the time said it was incomplete and incorrect.”
In the fall of 1997, Kefeng Liu, a former student of Yau’s who taught at Stanford, gave a talk at Harvard on mirror symmetry. According to two geometers in the audience, Liu proceeded to present a proof strikingly similar to Givental’s, describing it as a paper that he had co-authored with Yau and another student of Yau’s. “Liu mentioned Givental but only as one of a long list of people who had contributed to the field,” one of the geometers said. (Liu maintains that his proof was significantly different from Givental’s.)
Around the same time, Givental received an e-mail signed by Yau and his collaborators, explaining that they had found his arguments impossible to follow and his notation baffling, and had come up with a proof of their own. They praised Givental for his “brilliant idea” and wrote, “In the final version of our paper your important contribution will be acknowledged.”
A few weeks later, the paper, “Mirror Principle I,” appeared in the Asian Journal of Mathematics, which is co-edited by Yau. In it, Yau and his coauthors describe their result as “the first complete proof” of the mirror conjecture. They mention Givental’s work only in passing. “Unfortunately,” they write, his proof, “which has been read by many prominent experts, is incomplete.” However, they did not identify a specific mathematical gap.
Givental was taken aback. “I wanted to know what their objection was,” he told us. “Not to expose them or defend myself.” In March, 1998, he published a paper that included a three-page footnote in which he pointed out a number of similarities between Yau’s 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 Chinese mathematical establishment. For years, Chern had been hoping to bring the I.M.U.’s congress to Beijing. According 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 congress in Beijing. (Yau denies that he tried to bring the congress to Hong Kong.) Among the delegates the I.M.U. appointed to a group that would be choosing speakers for the congress was Yau’s most successful student, Gang Tian, who had been at N.Y.U. with Perelman and was now a professor at M.I.T. The host committee in Beijing also asked Tian to give a plenary address.
Yau was caught by surprise. In March, 2000, he had published a survey of recent research in his field studded with glowing references to Tian and to their joint projects. He retaliated by organizing his first conference on string theory, which opened in Beijing a few days before the math congress began, in late August, 2002. He persuaded Stephen Hawking and several Nobel laureates to attend, and for days the Chinese newspapers were full of pictures of famous scientists. Yau even managed to arrange for his group to have an audience with Jiang Zemin. A mathematician who helped organize the math congress recalls that along the highway between Beijing and the airport there were “billboards with pictures of Stephen Hawking plastered everywhere.”
That summer, Yau wasn’t thinking much about the Poincaré. He had confidence in Hamilton, despite his slow pace. “Hamilton is a very good friend,” Yau told us in Beijing. “He is more than a friend. He is a hero. He is so original. We were working to finish our proof. Hamilton worked on it for twenty-five years. You work, you get tired. He probably got a little tired—and you want to take a rest.”
Then, on November 12, 2002, Yau received an e-mail message from a Russian mathematician whose name didn’t immediately register. “May I bring to your attention my paper,” the e-mail said.
On November 11th, Perelman had posted a thirty-nine-page paper entitled “The Entropy Formula for the Ricci Flow and Its Geometric Applications,” on arXiv.org, a Web site used by mathematicians to post preprints—articles awaiting publication in refereed journals. He then e-mailed an abstract of his paper to a dozen mathematicians in the United States—including Hamilton, Tian, and Yau—none of whom had heard from him for years. In the abstract, he explained that he had written “a sketch of an eclectic proof” of the geometrization conjecture.
Perelman had not mentioned the proof or shown it to anyone. “I didn’t have any friends with whom I could discuss this,” he said in St. Petersburg. “I didn’t want to discuss my work with someone I didn’t trust.” Andrew Wiles had also kept the fact that he was working on Fermat’s last theorem a secret, but he had had a colleague vet the proof before making it public. Perelman, by casually posting a proof on the Internet of one of the most famous problems in mathematics, was not just flouting academic convention but taking a considerable risk. If the proof was flawed, he would be publicly humiliated, and there would be no way to prevent another mathematician from fixing any errors and claiming victory. But Perelman said he was not particularly concerned. “My reasoning was: if I made an error and someone used my work to construct a correct proof I would be pleased,” he said. “I never set out to be the sole solver of the Poincaré.”
Gang Tian was in his office at M.I.T. when he received Perelman’s e-mail. He and Perelman had been friendly in 1992, when they were both at N.Y.U. and had attended the same weekly math seminar in Princeton. “I immediately realized its importance,” Tian said of Perelman’s paper. Tian began to read the paper and discuss it with colleagues, who were equally enthusiastic.
On November 19th, Vitali Kapovitch, a geometer, sent Perelman an e-mail:
Hi Grisha, Sorry to bother you but a lot of people are asking me about your preprint “The entropy formula for the Ricci . . .” Do I understand it correctly that while you cannot yet do all the steps in the Hamilton program you can do enough so that using some collapsing results you can prove geometrization? Vitali.
Perelman’s response, the next day, was terse: “That’s correct. Grisha.”
In fact, what Perelman had posted on the Internet was only the first installment of his proof. But it was sufficient for mathematicians to see that he had figured out how to solve the Poincaré. Barry Mazur, the Harvard mathematician, uses the image of a dented fender to describe Perelman’s achievement: “Suppose your car has a dented fender and you call a mechanic to ask how to smooth it out. The mechanic would have a hard time telling you what to do over the phone. You would have to bring the car into the garage for him to examine. Then he could tell you where to give it a few knocks. What Hamilton introduced and Perelman completed is a procedure that is independent of the particularities of the blemish. If you apply the Ricci flow to a 3-D space, it will begin to undent it and smooth it out. The mechanic would not need to even see the car—just apply the equation.” Perelman proved that the “cigars” that had troubled Hamilton could not actually occur, and he showed that the “neck” problem could be solved by performing an intricate sequence of mathematical surgeries: cutting out singularities and patching up the raw edges. “Now we have a procedure to smooth things and, at crucial points, control the breaks,” Mazur said.
Tian wrote to Perelman, asking him to lecture on his paper at M.I.T. Colleagues at Princeton and Stony Brook extended similar invitations. Perelman accepted them all and was booked for a month of lectures beginning in April, 2003. “Why not?” he told us with a shrug. Speaking of mathematicians generally, Fedor Nazarov, a mathematician at Michigan State University, said, “After you’ve solved a problem, you have a great urge to talk about it.”
Hamilton and Yau were stunned by Perelman’s announcement. “We felt that nobody else would be able to discover the solution,” Yau told us in Beijing. “But then, in 2002, Perelman said that he published something. He basically did a shortcut without doing all the detailed estimates that we did.” Moreover, Yau complained, Perelman’s proof “was written in such a messy way that we didn’t understand.”
Perelman’s April lecture tour was treated by mathematicians and by the press as a major event. Among the audience at his talk at Princeton were John Ball, Andrew Wiles, John Forbes Nash, Jr., who had proved the Riemannian embedding theorem, and John Conway, the inventor of the cellular automaton game Life. To the astonishment of many in the audience, Perelman said nothing about the Poincaré. “Here is a guy who proved a world-famous theorem and didn’t even mention it,” Frank Quinn, a mathematician at Virginia Tech, said. “He stated some key points and special properties, and then answered questions. He was establishing credibility. If he had beaten his chest and said, ‘I solved it,’ he would have got a huge amount of resistance.” He added, “People were expecting a strange sight. Perelman was much more normal than they expected.”
To Perelman’s disappointment, Hamilton did not attend that lecture or the next ones, at Stony Brook. “I’m a disciple of Hamilton’s, though I haven’t received his authorization,” Perelman told us. But John Morgan, at Columbia, where Hamilton now taught, was in the audience at Stony Brook, and after a lecture he invited Perelman to speak at Columbia. Perelman, hoping to see Hamilton, agreed. The lecture took place on a Saturday morning. Hamilton showed up late and asked no questions during either the long discussion session that followed the talk or the lunch after that. “I had the impression he had read only the first part of my paper,” Perelman said.
In the April 18, 2003, issue of Science, Yau was featured in an article about Perelman’s proof: “Many experts, although not all, seem convinced that Perelman has stubbed out the cigars and tamed the narrow necks. But they are less confident that he can control the number of surgeries. That could prove a fatal flaw, Yau warns, noting that many other attempted proofs of the Poincaré conjecture have stumbled over similar missing steps.” Proofs should be treated with skepticism until mathematicians have had a chance to review them thoroughly, Yau told us. Until then, he said, “it’s not math—it’s religion.”
By mid-July, Perelman had posted the final two installments of his proof on the Internet, and mathematicians had begun the work of formal explication, painstakingly retracing his steps. In the United States, at least two teams of experts had assigned themselves this 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 mathematicians 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 eligible, 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. Petersburg, he received a long e-mail from Tian, who said that he had just attended a two-week workshop at Princeton devoted 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 explained to us, “I didn’t worry too much myself. This was a famous problem. Some people needed time to get accustomed to the fact that this is no longer a conjecture. I personally decided for myself that it was right for me to stay away from verification and not to participate in all these meetings. It is important for me that I don’t influence this process.”
In July of that year, the National Science Foundation had given nearly a million dollars in grants to Yau, Hamilton, and several students of Yau’s to study and apply Perelman’s “breakthrough.” An entire branch of mathematics had grown up around efforts to solve the Poincaré, and now that branch appeared at risk of becoming obsolete. Michael Freedman, who won a Fields for proving the Poincaré conjecture for the fourth dimension, told the Times that Perelman’s proof was a “small sorrow for this particular branch of topology.” Yuri Burago said, “It kills the field. After this is done, many mathematicians will move to other branches of mathematics.”
Five months later, Chern died, and Yau’s efforts to insure that he-—not Tian—was recognized as his successor turned vicious. “It’s all about their primacy in China and their leadership among the expatriate Chinese,” Joseph Kohn, a former chairman of the Prince-ton mathematics department, said. “Yau’s not jealous of Tian’s mathematics, but he’s jealous of his power back in China.”
Though Yau had not spent more than a few months at a time on mainland China since he was an infant, he was convinced that his status as the only Chinese Fields Medal winner should make him Chern’s successor. In a speech he gave at Zhejiang University, in Hangzhou, during the summer of 2004, Yau reminded his listeners of his Chinese roots. “When I stepped out from the airplane, I touched the soil of Beijing and felt great joy to be in my mother country,” he said. “I am proud to say that when I was awarded the Fields Medal in mathematics, I held no passport of any country and should certainly be considered Chinese.”
The following summer, Yau returned to China and, in a series of interviews with Chinese reporters, attacked Tian and the mathematicians at Peking University. In an article published in a Beijing science newspaper, which ran under the headline “SHING-TUNG YAU IS SLAMMING ACADEMIC CORRUPTION IN CHINA,” Yau called Tian “a complete mess.” He accused him of holding multiple professorships and of collecting a hundred and twenty-five thousand dollars for a few months’ work at a Chinese university, while students were living on a hundred dollars a month. He also charged Tian with shoddy scholarship and plagiarism, and with intimidating his graduate students into letting him add his name to their papers. “Since I promoted him all the way to his academic fame today, I should also take responsibility for his improper behavior,” Yau was quoted as saying to a reporter, explaining why he felt obliged to speak out.
In another interview, Yau described how the Fields committee had passed Tian over in 1988 and how he had lobbied on Tian’s behalf with various prize committees, including one at the National Science Foundation, which awarded Tian five hundred thousand dollars in 1994.
Tian was appalled by Yau’s attacks, but he felt that, as Yau’s former student, there was little he could do about them. “His accusations were baseless,” Tian told us. But, he added, “I have deep roots in Chinese culture. A teacher is a teacher. There is respect. It is very hard for me to think of anything to do.”
While Yau was in China, he visited Xi-Ping Zhu, a protégé of his who was now chairman of the mathematics department at Sun Yat-sen University. In the spring of 2003, after Perelman completed his lecture tour in the United States, Yau had recruited Zhu and another student, Huai-Dong Cao, a professor at Lehigh University, to undertake an explication of Perelman’s proof. Zhu and Cao had studied the Ricci flow under Yau, who considered Zhu, in particular, to be a mathematician of exceptional promise. “We have to figure out whether Perelman’s paper holds together,” Yau told them. Yau arranged for Zhu to spend the 2005-06 academic year at Harvard, where he gave a seminar on Perelman’s proof and continued to work on his paper with Cao.
On April 13th of this year, the thirty-one mathematicians on the editorial board of the Asian Journal of Mathematics received a brief e-mail from Yau and the journal’s co-editor informing them that they had three days to comment on a paper by Xi-Ping Zhu and Huai-Dong Cao titled “The Hamilton-Perelman Theory of Ricci Flow: The Poincaré and Geometrization Conjectures,” which Yau planned to publish in the journal. The e-mail did not include a copy of the paper, reports from referees, or an abstract. At least one board member asked to see the paper but was told that it was not available. On April 16th, Cao received a message from Yau telling him that the paper had been accepted by the A.J.M., and an abstract was posted on the journal’s Web site.
A month later, Yau had lunch in Cambridge with Jim Carlson, the president of the Clay Institute. He told Carlson that he wanted to trade a copy of Zhu and Cao’s paper for a copy of Tian and Morgan’s book manuscript. Yau told us he was worried that Tian would try to steal from Zhu and Cao’s work, and he wanted to give each party simultaneous access to what the other had written. “I had a lunch with Carlson to request to exchange both manuscripts to make sure that nobody can copy the other,” Yau said. Carlson demurred, explaining that the Clay Institute had not yet received Tian and Morgan’s complete manuscript.
By the end of the following week, the title of Zhu and Cao’s paper on the A.J.M.’s Web site had changed, to “A Complete Proof of the Poincaré and Geometrization Conjectures: Application of the Hamilton-Perelman Theory of the Ricci Flow.” The abstract had also been revised. A new sentence explained, “This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow.”
Zhu and Cao’s paper was more than three hundred pages long and filled the A.J.M.’s entire June issue. The bulk of the paper is devoted to reconstructing many of Hamilton’s Ricci-flow results—including results that Perelman had made use of in his proof—and much of Perelman’s proof of the Poincaré. In their introduction, Zhu and Cao credit Perelman with having “brought in fresh new ideas to figure out important steps to overcome the main obstacles that remained in the program of Hamilton.” However, they write, they were obliged to “substitute several key arguments of Perelman by new approaches based on our study, because we were unable to comprehend these original arguments of Perelman which are essential to the completion of the geometrization program.” Mathematicians familiar with Perelman’s proof disputed the idea that Zhu and Cao had contributed significant new approaches to the Poincaré. “Perelman already did it and what he did was complete and correct,” John Morgan said. “I don’t see that they did anything different.”
By early June, Yau had begun to promote the proof publicly. On June 3rd, at his mathematics institute in Beijing, he held a press conference. The acting director of the mathematics institute, attempting to explain the relative contributions of the different mathematicians who had worked on the Poincaré, said, “Hamilton contributed over fifty per cent; the Russian, Perelman, about twenty-five per cent; and the Chinese, Yau, Zhu, and Cao et al., about thirty per cent.” (Evidently, simple addition can sometimes trip up even a mathematician.) Yau added, “Given the significance of the Poincaré, that Chinese mathematicians 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 Hawking. . . . Yau Shing-Tung, who organized Professor Hawking’s visit and is also Professor Cao’s teacher, said yesterday 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 Beijing, Yau told us, “We want our contribution understood. And this is also a strategy to encourage Zhu, who is in China and who has done really spectacular work. I mean, important work with a century-long problem, which will probably have another few century-long implications. If you can attach your name in any way, it is a contribution.”
E. T. Bell, the author of “Men of Mathematics,” a witty history of the discipline published in 1937, once lamented “the squabbles over priority which disfigure scientific history.” But in the days before e-mail, blogs, and Web sites, a certain decorum usually prevailed. In 1881, Poincaré, who was then at the University of Caen, had an altercation with a German mathematician in Leipzig named Felix Klein. Poincaré had published several papers in which he labelled certain functions “Fuchsian,” after another mathematician. Klein wrote to Poincaré, pointing out that he and others had done significant work on these functions, too. An exchange of polite letters between Leipzig and Caen ensued. Poincaré’s last word on the subject was a quote from Goethe’s “Faust”: “Name ist Schall und Rauch.” Loosely translated, that corresponds to Shakespeare’s “What’s in a name?”
This, essentially, is what Yau’s friends are asking themselves. “I find myself getting annoyed with Yau that he seems to feel the need for more kudos,” Dan Stroock, of M.I.T., said. “This is a guy who did magnificent things, for which he was magnificently rewarded. He won every prize to be won. I find it a little mean of him to seem to be trying to get a share of this as well.” Stroock pointed out that, twenty-five years ago, Yau was in a situation very similar to the one Perelman is in today. His most famous result, on Calabi-Yau manifolds, was hugely important for theoretical physics. “Calabi outlined a program,” Stroock said. “In a real sense, Yau was Calabi’s Perelman. Now he’s on the other side. He’s had no compunction at all in taking the lion’s share of credit for Calabi-Yau. And now he seems to be resenting Perelman getting credit for completing Hamilton’s program. I don’t know if the analogy has ever occurred to him.”
Mathematics, more than many other fields, depends on collaboration. Most problems require the insights of several mathematicians in order to be solved, and the profession has evolved a standard for crediting individual contributions that is as stringent as the rules governing math itself. As Perelman put it, “If everyone is honest, it is natural to share ideas.” Many mathematicians view Yau’s conduct over the Poincaré as a violation of this basic ethic, and worry about the damage it has caused the profession. “Politics, power, and control have no legitimate role in our community, and they threaten the integrity of our field,” Phillip Griffiths said.
Perelman likes to attend opera performances at the Mariinsky Theatre, in St. Petersburg. Sitting high up in the back of the house, he can’t make out the singers’ expressions or see the details of their costumes. But he cares only about the sound of their voices, and he says that the acoustics are better where he sits than anywhere else in the theatre. Perelman views the mathematics community—and much of the larger world—from a similar remove.
Before we arrived in St. Petersburg, on June 23rd, we had sent several messages to his e-mail address at the Steklov Institute, hoping to arrange a meeting, but he had not replied. We took a taxi to his apartment building and, reluctant to intrude on his privacy, left a book—a collection of John Nash’s papers—in his mailbox, along with a card saying that we would be sitting on a bench in a nearby playground the following afternoon. The next day, after Perelman failed to appear, we left a box of pearl tea and a note describing some of the questions we hoped to discuss with him. We repeated this ritual a third time. Finally, believing that Perelman was out of town, we pressed the buzzer for his apartment, hoping at least to speak with his mother. A woman answered and let us inside. Perelman met us in the dimly lit hallway of the apartment. It turned out that he had not checked his Steklov e-mail address for months, and had not looked in his mailbox all week. He had no idea who we were.
We arranged to meet at ten the following morning on Nevsky Prospekt. From there, Perelman, dressed in a sports coat and loafers, took us on a four-hour walking tour of the city, commenting on every building and vista. After that, we all went to a vocal competition at the St. Petersburg Conservatory, which lasted for five hours. Perelman repeatedly said that he had retired from the mathematics community and no longer considered himself a professional mathematician. He mentioned a dispute that he had had years earlier with a collaborator over how to credit the author of a particular proof, and said that he was dismayed by the discipline’s lax ethics. “It is not people who break ethical standards who are regarded as aliens,” he said. “It is people like me who are isolated.” We asked him whether he had read Cao and Zhu’s paper. “It is not clear to me what new contribution did they make,” he said. “Apparently, Zhu did not quite understand the argument and reworked it.” As for Yau, Perelman said, “I can’t say I’m outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest.”
The prospect of being awarded a Fields Medal had forced him to make a complete break with his profession. “As long as I was not conspicuous, I had a choice,” Perelman explained. “Either to make some ugly thing”—a fuss about the math community’s lack of integrity—“or, if I didn’t do this kind of thing, to be treated as a pet. Now, when I become a very conspicuous person, I cannot stay a pet and say nothing. That is why I had to quit.” We asked Perelman whether, by refusing the Fields and withdrawing from his profession, he was eliminating any possibility of influencing the discipline. “I am not a politician!” he replied, angrily. Perelman would not say whether his objection to awards extended to the Clay Institute’s million-dollar prize. “I’m not going to decide whether to accept the prize until it is offered,” he said.
Mikhail Gromov, the Russian geometer, said that he understood Perelman’s logic: “To do great work, you have to have a pure mind. You can think only about the mathematics. Everything else is human weakness. Accepting prizes is showing weakness.” Others might view Perelman’s refusal to accept a Fields as arrogant, Gromov said, but his principles are admirable. “The ideal scientist does science and cares about nothing else,” he said. “He wants to live this ideal. Now, I don’t think he really lives on this ideal plane. But he wants to.”
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August 16, 2006
Meet the cleverest man in the world (who's going to say no to a $1m prize)
By James Randerson
He is possibly the cleverest person on the planet: an enigmatic and reclusive genius who shocked the academic world with his claim to have solved one of the hardest problems in maths. He is tipped to win a "maths Nobel" for his work on possible shapes of the universe. But rumours are rife that the brilliant Russian mathematician will spurn the greatest accolade his peers can bestow.
Since Grigory "Grisha" Perelman revealed his solution in 2002 to a century-old maths problem, it has been subjected to unparalleled scrutiny by the best academic minds. But no one has been able to find a mistake and there is a growing consensus that he has cracked the problem.
So, next Tuesday he is tipped to win a Fields medal. But even by the standards of troubled maths virtuosos such as John Nash, portrayed in the film A Beautiful Mind, Dr Perelman is described as "unconventional".
He has said he will refuse a $1m prize offered by a private maths research institute in the US that would be his if his claim is proved correct. And upper echelons of the maths world are buzzing with rumours that even if he is offered the gong he will not accept it. The medals are open to mathematicians under 40 years old at the beginning of the prize year. Dr Perelman turned 40 in June so this is the last year that he can win.
He has also refused a major European maths prize, supposedly on the grounds that he did not believe the committee awarding the prize was sufficiently qualified to judge his work.
"I just don't see him turning up in a stretch limo with four over-endowed women and waving his cheque in the air. It's not his style," said Jeremy Gray, a maths historian at the Open University.
"I think he's a very unconventional person. He's against being involved in pageantry and idolatry," said Arthur Jaffe at Harvard University. "But he carries it to extreme which people might describe as a little crazy."
Little is known about Dr Perelman, who refuses to talk to the media. He was born on June 13 1966 and his prodigious talent led to his early enrolment at a St Petersburg school specialising in advanced mathematics and physics. At the age of 16, he won a gold medal with a perfect score at the 1982 International Mathematical Olympiad, a competition for gifted schoolchildren.
After receiving his PhD from the St Petersburg State University, he worked at the Steklov Institute of Mathematics before moving to the US in the late 80s to take posts at various universities. He returned to the Steklov about 10 years ago to work on his proof of the universe's shape.
The maths world was set humming in 2002 by the first instalment of his ground-breaking work on the problem which was set out by the French mathematician, physicist and philosopher Jules Henri Poincaré in 1904. The conjecture, which is difficult for most non-mathematicians even to understand, exercised some of the greatest minds of the 20th century.
It concerns the geometry of multidimensional spaces and is key to the field of topology. Dr Perelman claims to have solved a more general version of the problem called Thurston's geometrisation conjecture, of which the Poincaré conjecture is a special case.
"It's a central problem both in maths and physics because it seeks to understand what the shape of the universe can be," said Marcus Du Sautoy at Oxford University, who will be giving this year's Royal Institution Christmas Lectures. "It is very tricky to pin down. A lot of people have announced false proofs of this thing."
The obsession with the problem, shared by several great mathematicians, has been dubbed Poincaritis.
But Dr Perelman seems to have succeeded where so many failed. "I think for many months or even years now people have been saying they were convinced by the argument," said Nigel Hitchin, professor of mathematics at Oxford University. "I think it's a done deal."
Even the way he announced his proof - which took eight years to complete - was unusual. Rather than publishing in a peer-reviewed journal, he posted three manuscripts in an online archive of maths and physics papers.
"He placed the papers on the web archive and basically said 'that's it'," Prof Hitchin said. "A lot of details needed to be filled in. And there's a bit of squabbling in the background actually about who was first to fill in the details." The most recent of the papers fleshing out his proof runs to a mind-numbing 473 pages.
There is more than just professional acclaim at stake. In 2000, the Clay Institute in Boston, a private maths research organisation, established seven "millennium problems", each with a million-dollar reward for a solution. The Poincaré conjecture is one, but Dr Perelman has said he is not interested in the money. "There are all sorts of jokes going round the community that having a million dollars in St Petersburg is quite dangerous," Prof Hitchin said.
No one is quite sure what will happen if the Russian spurns the medal. "If he were to win it and turn it down it would be slightly insulting," said Prof Du Sautoy. But it seems unlikely that Dr Perelman, who recently relinquished his academic position, will care much about offending his peers. "He has sort of alienated himself from the maths community," Prof Du Sautoy added. "He has become disillusioned with mathematics, which is quite sad. He's not interested in money. The big prize for him is proving his theorem."
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August 15, 2006
Elusive Proof, Elusive Prover: A New Mathematical Mystery
By Dennis Overbye
Grisha Perelman, where are you?
Three years ago, a Russian mathematician by the name of Grigory Perelman, a k a Grisha, in St. Petersburg, announced that he had solved a famous and intractable mathematical problem, known as the Poincaré conjecture, about the nature of space.
After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world’s mathematicians to pick up the pieces and decide if he was right.
Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose between them.
As a result there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought.
“It’s really a great moment in mathematics,” said Bruce Kleiner of Yale, who has spent the last three years helping to explicate Dr. Perelman’s work. “It could have happened 100 years from now, or never.”
In a speech at a conference in Beijing this summer, Shing-Tung Yau of Harvard said the understanding of three-dimensional space brought about by Poincaré’s conjecture could be one of the major pillars of math in the 21st century.
Quoting Poincaré himself, Dr.Yau said, “Thought is only a flash in the middle of a long night, but the flash that means everything.”
But at the moment of his putative triumph, Dr. Perelman is nowhere in sight. He is an odds-on favorite to win a Fields Medal, math’s version of the Nobel Prize, when the International Mathematics Union convenes in Madrid next Tuesday. But there is no indication whether he will show up.
Also left hanging, for now, is $1 million offered by the Clay Mathematics Institute in Cambridge, Mass., for the first published proof of the conjecture, one of seven outstanding questions for which they offered a ransom back at the beginning of the millennium.
“It’s very unusual in math that somebody announces a result this big and leaves it hanging,” said John Morgan of Columbia, one of the scholars who has also been filling in the details of Dr. Perelman’s work.
Mathematicians have been waiting for this result for more than 100 years, ever since the French polymath Henri Poincaré posed the problem in 1904. And they acknowledge that it may be another 100 years before its full implications for math and physics are understood. For now, they say, it is just beautiful, like art or a challenging new opera.
Dr. Morgan said the excitement came not from the final proof of the conjecture, which everybody felt was true, but the method, “finding deep connections between what were unrelated fields of mathematics.”
William Thurston of Cornell, the author of a deeper conjecture that includes Poincaré’s and that is now apparently proved, said, “Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide,” explaining that curiosity is tied in some way with intuition.
“You don’t see what you’re seeing until you see it,” Dr. Thurston said, “but when you do see it, it lets you see many other things.”
Depending on who is talking, Poincaré’s conjecture can sound either daunting or deceptively simple. It asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere.
The conjecture is fundamental to topology, the branch of math that deals with shapes, sometimes described as geometry without the details. To a topologist, a sphere, a cigar and a rabbit’s head are all the same because they can be deformed into one another. Likewise, a coffee mug and a doughnut are also the same because each has one hole, but they are not equivalent to a sphere.
In effect, what Poincaré suggested was that anything without holes has to be a sphere. The one qualification was that this “anything” had to be what mathematicians call compact, or closed, meaning that it has a finite extent: no matter how far you strike out in one direction or another, you can get only so far away before you start coming back, the way you can never get more than 12,500 miles from home on the Earth.
In the case of two dimensions, like the surface of a sphere or a doughnut, it is easy to see what Poincaré was talking about: imagine a rubber band stretched around an apple or a doughnut; on the apple, the rubber band can be shrunk without limit, but on the doughnut it is stopped by the hole.
With three dimensions, it is harder to discern the overall shape of something; we cannot see where the holes might be. “We can’t draw pictures of 3-D spaces,” Dr. Morgan said, explaining that when we envision the surface of a sphere or an apple, we are really seeing a two-dimensional object embedded in three dimensions. Indeed, astronomers are still arguing about the overall shape of the universe, wondering if its topology resembles a sphere, a bagel or something even more complicated.
Poincaré’s conjecture was subsequently generalized to any number of dimensions, but in fact the three-dimensional version has turned out to be the most difficult of all cases to prove. In 1960 Stephen Smale, now at the Toyota Technological Institute at Chicago, proved that it is true in five or more dimensions and was awarded a Fields Medal. In 1983, Michael Freedman, now at Microsoft, proved that it is true in four dimensions and also won a Fields.
“You get a Fields Medal for just getting close to this conjecture,” Dr. Morgan said.
In the late 1970’s, Dr. Thurston extended Poincaré’s conjecture, showing that it was only a special case of a more powerful and general conjecture about three-dimensional geometry, namely that any space can be decomposed into a few basic shapes.
Mathematicians had known since the time of Georg Friedrich Bernhard Riemann, in the 19th century, that in two dimensions there are only three possible shapes: flat like a sheet of paper, closed like a sphere, or curved uniformly in two opposite directions like a saddle or the flare of a trumpet. Dr. Thurston suggested that eight different shapes could be used to make up any three-dimensional space.
“Thurston’s conjecture almost leads to a list,” Dr. Morgan said. “If it is true,” he added, “Poincaré’s conjecture falls out immediately.” Dr. Thurston won a Fields in 1982.
Topologists have developed an elaborate set of tools to study and dissect shapes, including imaginary cutting and pasting, which they refer to as “surgery,” but they were not getting anywhere for a long time.
In the early 1980’s Richard Hamilton of Columbia suggested a new technique, called the Ricci flow, borrowed from the kind of mathematics that underlies Einstein’s general theory of relativity and string theory, to investigate the shapes of spaces.
Dr. Hamilton’s technique makes use of the fact that for any kind of geometric space there is a formula called the metric, which determines the distance between any pair of nearby points. Applied mathematically to this metric, the Ricci flow acts like heat, flowing through the space in question, smoothing and straightening all its bumps and curves to reveal its essential shape, the way a hair dryer shrink-wraps plastic.
Dr. Hamilton succeeded in showing that certain generally round objects, like a head, would evolve into spheres under this process, but the fates of more complicated objects were problematic. As the Ricci flow progressed, kinks and neck pinches, places of infinite density known as singularities, could appear, pinch off and even shrink away. Topologists could cut them away, but there was no guarantee that new ones would not keep popping up forever.
“All sorts of things can potentially happen in the Ricci flow,” said Robert Greene, a mathematician at the University of California, Los Angeles. Nobody knew what to do with these things, so the result was a logjam.
It was Dr. Perelman who broke the logjam. He was able to show that the singularities were all friendly. They turned into spheres or tubes. Moreover, they did it in a finite time once the Ricci flow started. That meant topologists could, in their fashion, cut them off, and allow the Ricci process to continue to its end, revealing the topologically spherical essence of the space in question, and thus proving the conjectures of both Poincaré and Thurston.
Dr. Perelman’s first paper, promising “a sketch of an eclectic proof,” came as a bolt from the blue when it was posted on the Internet in November 2002. “Nobody knew he was working on the Poincaré conjecture,” said Michael T. Anderson of the State University of New York in Stony Brook.
Dr. Perelman had already established himself as a master of differential geometry, the study of curves and surfaces, which is essential to, among other things, relativity and string theory Born in St. Petersburg in 1966, he distinguished himself as a high school student by winning a gold medal with a perfect score in the International Mathematical Olympiad in 1982. After getting a Ph.D. from St. Petersburg State, he joined the Steklov Institute of Mathematics at St. Petersburg.
In a series of postdoctoral fellowships in the United States in the early 1990’s, Dr. Perelman impressed his colleagues as “a kind of unworldly person,” in the words of Dr. Greene of U.C.L.A. — friendly, but shy and not interested in material wealth.
“He looked like Rasputin, with long hair and fingernails,” Dr. Greene said.
Asked about Dr. Perelman’s pleasures, Dr. Anderson said that he talked a lot about hiking in the woods near St. Petersburg looking for mushrooms.
Dr. Perelman returned to those woods, and the Steklov Institute, in 1995, spurning offers from Stanford and Princeton, among others. In 1996 he added to his legend by turning down a prize for young mathematicians from the European Mathematics Society.
Until his papers on Poincaré started appearing, some friends thought Dr. Perelman had left mathematics. Although they were so technical and abbreviated that few mathematicians could read them, they quickly attracted interest among experts. In the spring of 2003, Dr. Perelman came back to the United States to give a series of lectures at Stony Brook and the Massachusetts Institute of Technology, and also spoke at Columbia, New York University and Princeton.
But once he was back in St. Petersburg, he did not respond to further invitations. The e-mail gradually ceased.
“He came once, he explained things, and that was it,” Dr. Anderson said. “Anything else was superfluous.”
Recently, Dr. Perelman is said to have resigned from Steklov. E-mail messages addressed to him and to the Steklov Institute went unanswered.
In his absence, others have taken the lead in trying to verify and disseminate his work. Dr. Kleiner of Yale and John Lott of the University of Michigan have assembled a monograph annotating and explicating Dr. Perelman’s proof of the two conjectures.
Dr. Morgan of Columbia and Gang Tian of Princeton have followed Dr. Perelman’s prescription to produce a more detailed 473-page step-by-step proof only of Poincaré’s Conjecture. “Perelman did all the work,” Dr. Morgan said. “This is just explaining it.”
Both works were supported by the Clay institute, which has posted them on its Web site, claymath.org. Meanwhile, Huai-Dong Cao of Lehigh University and Xi-Ping Zhu of Zhongshan University in Guangzhou, China, have published their own 318-page proof of both conjectures in The Asian Journal of Mathematics (www.ims.cuhk.edu.hk/).
Although these works were all hammered out in the midst of discussion and argument by experts, in workshops and lectures, they are about to receive even stricter scrutiny and perhaps crossfire. “Caution is appropriate,” said Dr. Kleiner, because the Poincaré conjecture is not just famous, but important.
James Carlson, president of the Clay Institute, said the appearance of these papers had started the clock ticking on a two-year waiting period mandated by the rules of the Clay Millennium Prize. After two years, he said, a committee will be appointed to recommend a winner or winners if it decides the proof has stood the test of time.
“There is nothing in the rules to prevent Perelman from receiving all or part of the prize,” Dr. Carlson said, saying that Dr. Perelman and Dr. Hamilton had obviously made the main contributions to the proof.
In a lecture at M.I.T. in 2003, Dr. Perelman described himself “in a way” as Dr. Hamilton’s disciple, although they had never worked together. Dr. Hamilton, who got his Ph.D. from Princeton in 1966, is too old to win the Fields medal, which is given only up to the age of 40, but he is slated to give the major address about the Poincaré conjecture in Madrid next week. He did not respond to requests for an interview.
Allowing that Dr. Perelman, should he win the Clay Prize, might refuse the honor, Dr. Carlson said the institute could decide instead to use award money to support Russian mathematicians, the Steklov Institute or even the Math Olympiad.
Dr. Anderson said that to some extent the new round of papers already represented a kind of peer review of Dr. Perelman’s work. “All these together make the case pretty clear,” he said. “The community accepts the validity of his work. It’s commendable that the community has gotten together.”
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August 22, 2006
Maths genius declines top prize
Grigory Perelman, the Russian who seems to have solved one of the hardest problems in mathematics, has declined one of the discipline's top awards.
Dr Perelman was to have been presented with the prestigious Fields Medal by King Juan Carlos of Spain, at a ceremony in Madrid on Tuesday.
In 2002, the mathematician claimed to have solved a century-old problem called the Poincare Conjecture.
So far, experts working to verify his proof have found no significant flaws.
There had been considerable speculation that Grigory "Grisha" Perelman would decline the award. He has been described as an "unconventional" and "reclusive" genius who spurns self-promotion.
The medals were presented to three other winners at the International Congress of Mathematicians (ICM) in Madrid.
John Ball, outgoing president of the International Mathematical Union, said he had travelled to St Petersburg to meet Perelman in person to try to understand his reasons for declining the award.
Professor Ball said he had spoken to Dr Perelman of personal experiences with the mathematical community during his career that had caused him to remain at a distance.
"However, I am unable to disclose these comments in public," he said, adding: "He has a different psychological make up, which makes him see life differently."
Prestigious honour
The Fields Medals come with prize money of 15,000 Canadian dollars (£7,000) for each recipient. They are awarded every four years, when the ICM meets. Founded at the behest of Canadian mathematician John Charles Fields, the medal was first presented in 1936.
In 1996, Perelman turned down a prize awarded to him by the European Congress of Mathematicians.
Observers suspect he will refuse a $1m (£529,000) prize offered by the Clay Mathematics Institute in Massachusetts, US, if his proof of the Poincare Conjecture stands up to scrutiny.
The Fields Medals are regarded as the equivalent of the Nobel Prize for mathematics. They are awarded to mathematicians under the age of 40 for an outstanding body of work and are decided by an anonymous committee. The age limit is designed to encourage future endeavour.
The winners are Andrei Okounkov of Princeton University; Terence Tao from the University of California, Los Angeles; and Wendelin Werner of the University of Paris-Sud in Orsay, France.
Exemplary behaviour
"It's quite an honour - very different to anything that's happened to me before. This prize is the highest in mathematics," Terence Tao told the BBC News website.
"Most prizes are specific to a single field, but this recognises achievement across the whole of mathematics."
Tao received the award for a diverse body of work that, amongst other things, has shed light on the properties of prime numbers. Despite being the youngest of the winners at 31, he has a variety of mathematical proofs to his name and has published over 80 papers.
Fellow winner Wendelin Werner, whose work straddles the intersection between maths and physics, commented: "We are all around 40 years old - so still relatively young. It's a big honour but also quite a lot of pressure for the future."
Andrei Okounkov, who works on probability theory, commented: "I suppose we will have to exhibit exemplary behaviour from now on, because a lot of people will be watching."
A spokesperson for the Clay Mathematics Institute said it would put off making a decision on an award for the Poincare Conjecture for two years. The $1m prize money could be split between Perelman and US mathematician Richard Hamilton who devised the "Ricci flow" equation that forms the basis for the Russian's solution.
Grigory Perelman was born in Leningrad (St Petersburg) in 1966 in what was then the Soviet Union. Aged 16, he won the top prize at the International Mathematical Olympiad in Budapest.
Having received his doctorate from St Petersburg State University, he taught at various US universities during the 1990s before returning home to take up a post at the Steklov Mathematics Institute.
Century-old problem
He resigned from the institute suddenly on 1 January, and has reportedly been unemployed since, living at home with his mother.
"He was very polite but he didn't talk very much," said Natalya Stepanovna, a former colleague at the Steklov Mathematics Institute in St Petersburg. On his decision to resign his post, she speculated: "Maybe he wanted to be free to do his research."
Dr Perelman gained international recognition in 2002 and 2003 when he published two papers online that purported to solve the Poincare Conjecture.
The riddle had perplexed mathematicians since it was first posited by Frenchman Henri Poincare in 1904.
It is a central question in topology, the study of the geometrical properties of objects that do not change when they are stretched, distorted or shrunk.
The hollow shell of the surface of the Earth is what topologists call a two-dimensional sphere. If one were to encircle it with a lasso of string, it could be pulled tight to a point.
On the surface of a doughnut, however, a lasso passing through the hole in the centre cannot be shrunk to a point without cutting through the surface.
More dimensions
Since the 19th Century, mathematicians have known that the sphere is the only enclosed two-dimensional space with this property. But they were uncertain about objects with more dimensions.
The Poincare Conjecture says that a three-dimensional sphere is the only enclosed three-dimensional space with no holes. But proof of the conjecture has so far eluded mathematicians.
Two other maths prizes were awarded at the meeting in Madrid. The Nevanlinna Prize is awarded for advances in mathematics made in the field of information technology. It went to Jon Kleinberg, a professor of computer science at Cornell University. His work into link-related web searching has influenced Google.
The newly created Carl Friedrich Gauss prize for applications of mathematics was awarded to the Japanese mathematician Kiyoshi Ito. Ill health meant the 90-year-old could not receive the prize - worth $11,500 - in person. It was picked up by his youngest daughter, Junko.
The award honoured his achievements in the mathematical modelling of random events.
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August 26, 2006
Making apples from oranges; Mathematics
All the same, to a mathematician. Proof of the truth of the Poincare conjecture.
For the past century mathematicians have struggled to prove the conjecture of a late-19th-century French polymath, Henri Poincare. Three years ago a possible proof was posted on the internet by Grigori Perelman, a Russian who lives with his mother in a Saint Petersburg flat and is as reclusive as Poincare was feted. On August 22nd the International Congress of Mathematicians awarded Dr Perelman its highest honour, the Fields medal, for his solution which, having stood for three years, is now taken as correct.
Poincare's conjecture is important not for its practical applications, but because a vast quantity of mathematical work assumes that it is true. Proving the conjecture false would have cast doubt on much of modern mathematics--and everything that depends on it. Dr Perelman's work, therefore, is reason to sigh with relief.
To understand the Poincare conjecture, start by thinking of any object existing in a three-dimensional world. Although it is usual to think of the object as three-dimensional, mathematicians consider only the surface of these objects--which are two-dimensional. All objects in a three-dimensional world can be simplified by smoothing out their shape to look like either a two-dimensional sphere (otherwise known as a circle) or a two-dimensional torus with however many holes necessary. To mathematicians, a chair is equivalent to an apple; a mug--at least, one with a handle--is like a doughnut.
Whether the simplified shape is a sphere or a torus is determined by the behaviour of one-dimensional curves on its surface. Imagine an elastic band stretched over the surface of an apple. The band can slowly shrink, moving as though it is slipping from the surface, until it becomes just a dot on the apple's skin. It can do this without tearing itself and without ever leaving the surface. In this example, mathematicians would say that the surface of the apple is "simply connected". Any object with a simply connected surface can be smoothed out to look like a sphere.
Imagine, by contrast, an elastic band that passes through the hole in a doughnut. If this band is slowly shrunk, it becomes necessary to cut the doughnut or break the band. In this example, the surface is not simply connected and any smoothed-out object looks like a torus with at least one hole.
On the next rung up the ladder of difficult mathematics comes doing the same thing in four-dimensional space. The surface of an object in four-dimensional space would look like a three-dimensional "surface" that curves in on itself. More than a century ago, Poincare wanted to calculate a way of classifying such three-dimensional surfaces that live in four-dimensional spaces. His conjecture, made in 1904, was that in this four-dimensional world, all closed three-dimensional surfaces that are simply connected could be transformed to look like a three-dimensional sphere. As mathematicians say, "Every simply connected closed 3-manifold is homeomorphic to a 3-sphere."
Dr Perelman proved Poincare's conjecture by taking an equation normally used to model heat dissipation in three-dimensional objects, known as Ricci flow. The equation smoothes out the irregularities of an object, transforming it mathematically into something that looks like a uniform three-dimensional sphere. This does not change the object's essential properties: the transformed shape is equivalent to the starting shape.
Richard Hamilton of Columbia University had previously realised the usefulness of this heat-flow model but became unstuck when he found that, under this transformation, the object may stretch out to form singularities--spikes that could not be easily manipulated into a sphere. Dr Perelman overcame this difficulty by cutting off the singularities, continuing with the Ricci-flow application and then rejoining the transformed objects later. He then applied the Ricci flow to the rejoined piece and this smoothed the complete object into a sphere--a piece of lateral thinking that would surely have delighted Poincare himself.
Fields medals are only awarded once every four years but the organisers give four medals at each ceremony. This year, the other medals went to Andrei Okhounkov of Princeton University, Terence Tao of the University of California, Los Angeles, and Wendelin Werner of the University of Paris. These winners accepted their medals but Dr Perelman stayed at home, reportedly because he has no desire to be a figurehead of the mathematics community. He also appears to have turned down the $1m prize money offered by the Clay Mathematics Institute for solving the problem. For Dr Perelman, transforming a conjecture into a theorem appears to have been prize enough.
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Lundi 10 Septembre 2007
Le génie des maths retiré du monde
Par Alban Traquet, envoyé spécial à Saint-Pétersbourg, Le Journal du Dimanche
C'est un homme hirsute, qui vit reclus dans un quartier de Saint-Pétersbourg à la mauvaise réputation. Grigori Perelman est un mystère. Du jour au lendemain, ce génie a abadonné les mathématiques et la recherche, pour une vie de rien. Il venait de refuser la médaille Fields - l'équivalent du Nobel - après avoir été le premier à résoudre l'une des sept énigmes du siècles, la conjecture Poincaré.
Au milieu d'une forêt d'immeubles soviétiques, dans un dédale de béton parsemé de bouleaux, le 98-3, rue Budapestskaïa, dans la banlieue sud de Saint-Pétersbourg. Nous sommes dans le quartier populaire de Kouptchino, à vingt bonnes minutes, en marchant, du terminus de la ligne de métro n° 2. Kouptchino, une cité-dortoir construite à la fin des années 1960 sur d'anciens marais asséchés, près de l'aéroport. Au cinquième étage d'un bloc sinistre qui en compte huit, l'appartement 131 et sa vieille porte en bois. Difficile de croire qu'un génie du siècle se terre au coeur de cette zone qui, la nuit tombée, traîne une mauvaise réputation. C'est pourtant là que vit, à la réglette de ses maigres besoins, cet illustre hirsute de 41 ans à l'allure de vagabond insaisissable: le grand mathématicien russe Grigori Iakovlevitch Perelman.
Le 11 novembre 2002, un article mis en ligne par ce chercheur obstiné sur arxiv.org allait changer l'histoire des mathématiques. En 39 pages, "Grisha" Perelman parvient à résoudre l'un des "sept problèmes du millénaire" listés en 2000 par l'Institut Clay, qui a mis à prix leur résolution à un million de dollars chacun: la conjecture de Poincaré. Avancée en 1904 par le mathématicien français Henri Poincaré, cette conjecture est "une hypothèse audacieuse concernant rien moins que la nature et la forme de notre Univers", raconte, dans un résumé gourmand et vulgarisateur, le professeur américain Donal O'Shea, récent auteur d'un livre sur le sujet*.
Fort de sa démonstration, au printemps 2003 Perelman effectue une "tournée" aux Etats-Unis pour présenter ses travaux. Et emporte l'adhésion de ses pairs. Trois ans plus tard à Madrid, il est alors logiquement récompensé par la prestigieuse médaille Fields, l'équivalent du prix Nobel dans la discipline. Mais, comme annoncé, le "héros" n'est pas là, ce 22 août 2006. Deux mois avant la cérémonie, il avait précisé qu'il refuserait la médaille, une première, et qu'il ne se rendrait pas en Espagne. Il renonce également à la prime d'un million de dollars qui lui est promise.
Coupé du monde extérieur
En fait, le mystérieux Grisha a largué les amarres depuis six mois, quittant brusquement son poste au célèbre Institut Steklov de mathématiques où il travaillait depuis quinze ans. Reste seulement cette missive aride de deux lignes que déterre la secrétaire dans son dossier: "Je vous demande de prendre en considération ma démission pour raisons personnelles au 1er janvier 2006", peut-on y lire. Le revirement est sans appel: Perelman fait également savoir qu'il a complètement arrêté les maths. Fin de partie pour l'idéaliste, retourné à l'austérité de sa vie d'ermite pour esquiver l'insignifiance du cirque qui l'entoure.
Depuis, il est quasiment coupé du monde extérieur. Pour remonter le chemin de son renoncement, il faut retourner au centre-ville. Rue Kirostchnaïa, à l'école 239, un lycée scientifique de renommée mondiale. Là, son nom est inscrit au tableau d'honneur - année 1981 - face à l'escalier central. "Grigori Perelman, vainqueur des Olympiades internationales de mathématiques-physique". Avec un score parfait: 42 points sur 42 possibles. "Il avait la note maximale dans toutes les matières sauf en sport, note Sergueï Roukchine, un de ses anciens professeurs. Pas parce qu'il n'aimait pas le sport, mais parce que cela lui prenait du temps sur les maths..."
La genèse d'un parcours exceptionnel: de 1982 (il a 16 ans !) à 1987, il étudie à l'Université de Leningrad (ex-Saint-Pétersbourg), d'où il sort diplômé avec mention d'excellence. Il entre comme doctorant à l'Institut Steklov et soutient sa thèse en novembre 1990. Mais déjà, il s'illustre par ses réticences aux honneurs, refusant un prix accordé par la Société européenne des mathématiques... Son CV lui ouvre toutefois les portes des grandes universités américaines: NYU, Stony Brook et Berkeley, où il travaille deux ans, de 1993 à 1995. C'est lors de son séjour californien qu'il rencontre l'éminent professeur de mathématiques américain Richard Hamilton. Lui aussi s'est penché sur la conjecture de Poincaré et a déblayé le terrain. Perelman va finir le boulot. Il entre à "Saint-Pet'" à l'été 1995 et s'enferme, pour sept ans, dans ses équations... Jusqu'au fameux courriel du 11 novembre 2002. Et à cette renommée qu'il a préféré fuir.
Objet d'un culte sur la Toile
Paradoxe des temps modernes, cette quête farouche d'anonymat a fait de Grisha une vedette involontaire, jusque dans les pages de la jolty press (la presse jaune, les journaux à scandales russes). Le rebelle au regard habité, qui fuit argent et distinctions, étonne, détonne et intrigue. Un tee-shirt a été édité à son effigie. Côté pile, son visage superposé sur celui du Christ. Côté face, un slogan: "L'argent ne peut pas tout acheter". Le 19 juin, le tabloïd Komsomolskaïa Pravda a publié les dernières photos en date de Perelman, tirées d'une vidéo faite au téléphone portable, dans le métro. Un film d'une totale banalité: "le génie" monte dans un wagon à la station Kouptchino, reste debout près des portes, l'air ailleurs, et descend trois stations plus loin. Depuis l'année dernière, ce Diogène moderne est également devenu une vedette du net. Sa radicalité enflamme les forums et les sites satiriques. On délire sur son look de "moujik" et la longueur de ses ongles, qu'il laisse pousser à hauteur de trois centimètres. Un petit film amateur, délirant, intitulé Life after Poincaré: a Perelman adventure, a même été mis en ligne sur un site de vidéos en partage. On y voit un faux Grisha rencontrer et faire disparaître un faux Russell Crowe, l'acteur qui incarnait le mathématicien John Nash - un autre original - dans le film A beautiful mind (Un homme d'exception).
S'il n'est pas schizophrène comme Nash, le vrai Perelman tendrait plutôt vers la misanthropie. Difficile de débusquer la bête, qui fuit farouchement toutes les sollicitations. "Je ne pense pas que mes paroles puissent avoir le moindre intérêt public, a-t-il brièvement répondu, en août dernier, au quotidien anglais The Daily Telegraph. L'autopromotion existe, et si certains veulent en faire, bonne chance à eux. Mais je ne conçois pas ça comme une chose positive. Autant que je sache, je n'ai rien à offrir aux lecteurs." Le seul entretien qu'il ait accordé de bonne grâce remonte au début de l'été 2006, avant la tempête de Madrid: le magazine américain The New Yorker lui avait consacré un imposant article, en pointant certaines querelles responsables de son exil intérieur. Perelman y évoquait son amertume face à une polémique née en Chine (où est contestée sa prépondérance dans la résolution de la conjecture de Poincaré) et, au-delà, le manque d'intégrité de la communauté scientifique, les querelles oiseuses de son microcosme. La persécution médiatique dont il fit l'objet, dans les jours suivant sa consécration involontaire à Madrid, parachèvera son splendide isolement.
Plus d'un an après, Galina Prytkova, sa voisine de palier de l'appartement 130, subit encore les dommages de cette effervescence. Après un accueil plutôt glacé, elle profite de la présence d'une amie, aperçue au pied de l'immeuble, pour sortir de chez elle et entrer dans la discussion. "Dès le moment où il a résolu l'énigme, tout le monde s'est précipité chez lui et chez sa mère, où il est souvent [elle vit avec la soeur de Perelman à proximité, son père a émigré en Israël]. Alors ils se sont cloîtrés. Mais quelle est cette modestie qui pousse à refuser une telle somme ? C'est un mystère pour tout le monde... Cette mathématique-là, personne ne peut la résoudre !"
Il refuse les sollicitations, n'a "rien à dire"
Un adjectif revient, effectivement, à l'évocation de son caractère et de son comportement; "skromno": humble, modeste, discret. Un reportage de la chaîne NTV, rediffusé en juillet, le montre au retour des courses, démarche lente, un vieil anorak sur le dos, un bonnet râpé sur la tête et un sac en plastique blanc à la main. Filmé en caméra cachée, on l'entend ronchonner sur l'augmentation de deux roubles du prix du kilo de pommes (1 euro vaut 35 roubles). Pas vraiment le style "nouveau Russe", à l'heure du capitalisme sauvage et sans complexes qui sévit dans le pays. En fait, Perelman "subsisterait" grâce à l'argent qu'il a gagné comme maître de conférences à Berkeley, il y a une douzaine d'années. Il a ses habitudes en fin de matinée dans les rayons du supermarché du coin. Et selon une vendeuse, "il ne parle avec personne. Je ne l'ai jamais vu accompagné". Une confirmation, aussi: "Son comportement n'a absolument pas changé depuis ces histoires."
A l'image de ses résolutions. Le mathématicien français Gérard Besson, directeur de recherches au CNRS, est un spécialiste des travaux de Perelman. Il a écrit plusieurs courriels au "génie" pour obtenir des détails sur ses recherches. "Il ne m'a jamais répondu, raconte-t-il. C'est frustant car j'avais des questions à lui poser. Intellectuellement, il est hors norme. Sa méthode est originale et il a apporté des idées nouvelles. En revanche, je n'ai pas de jugement à porter sur sa vie..."
Même silence radio à l'Institut Steklov. Son actuel directeur, Sergueï Kisliakov, a pris ses fonctions deux semaines après la démission de Perelman, fin décembre 2005. En souriant, il cite à la volée d'illustres "prédécesseurs" pour démystifier le cas Perelman: Alexandre Grothendieck, un autre grand mathématicien reclus, lauréat rebelle de la médaille Fields 1966, ou le compositeur finlandais Jean Sibelius (1865-1957), qui a brutalement arrêté de composer dans les années 1930. Il raconte qu'en juin une invitation lancée à l'occasion d'un hommage rendu à son directeur de thèse, dont Perelman fut pourtant le dernier élève, est restée, une fois de plus, sans réponse. "Il y a encore du courrier qui arrive pour lui ici, explique Kisliakov, blasé. Si on lui demande quoi faire de ces lettres, Grisha nous répond: 'Gardez-les. Ou jetez-les à la poubelle.' En partant d'ici, il a dit qu'il trouverait un autre métier. Que fait-il à présent ? Je n'en sais rien."
*Grigori Perelman face à la conjecture de Poincaré, de Donal O'Shea, Quai des sciences, Editions Dunod.
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23 août 2006 23h01
Fidèle à son habitude, l'ours Perelman refuse la Fields
Par Sylvestre Huet
Un ours russe et trois matheux «normaux»... socialement parlant. C'est le quatuor majeur récompensé par les médailles Fields les Russes Grigory Perelman et Andreï Odonkov, l'Australien Terence Tao et le Français Wendelin Werner décernées hier par l'Union mathématique internationale qui tient son congrès à Madrid.
Squelette. L'ours russe fait évidemment le miel des médias, tant il joue à ravir le rôle caricatural du matheux génial mais asocial. Grigory Perelman, 40 ans, tête de pope (cheveux clairsemés, barbe abondante) n'a pas tardé à faire savoir qu'il refusait sa médaille. Snobant ainsi ses collègues, comme lorsqu'il avait refusé, en 1996, le prix que la société mathématique européenne voulait lui remettre. Au début de l'année, il a même démissionné de son poste à l'Institut Steklov de Saint-Pétersbourg. Refusant toute interview, envoyant ses textes sur le Net, réduits au squelette de la démonstration et laissant le soin aux collègues de vérifier eux-mêmes les chemins qui relient ses étapes principales, Perelman ne fait donc rien dans les règles. Pas à cheval sur l'étiquette, les matheux, qui ont vu d'autres zèbres dans le genre et savent reconnaître le génie sous la gangue, considéreront donc Perelman comme récipiendaire de leur médaille fétiche. Non, d'ailleurs, pour avoir résolu la conjecture de Poincaré (1) il y a trois ans, mais «pour ses contributions à la géométrie et ses vues révolutionnaires sur la structure du flot de Ricci»... Précision qui ira droit à l'esprit des initiés.
Les trois autres lauréats, souligne Jean-Pierre Bourguignon, le directeur de l'Institut des hautes études scientifiques (Bures-sur-Yvette, Essonne), allient la «virtuosité technique à une socialité normale».
Le plus jeune, l'Australien Terence Tao, 31 ans, «très agréable et vibrionnant, est un authentique petit génie : il aurait même pu décrocher la médaille au congrès de Pékin, il y a quatre ans», indique-t-il.
Le second Russe, Andreï Odonkov, 37 ans, est bien connu puisqu'il travaille souvent avec un des physiciens théoriciens de l'IHES, Nikita Nekrassov. Comme souvent chez les «grands mathématiciens, c'est un tisseur de liens entre des domaines les probabilités et la géométrie algébrique qui semblent séparés».
Excellence. Le dernier récipiendaire, Wendelin Werner, est le neuvième Français à recevoir la Fields, sur les 48 attribuées depuis sa création en 1936. Un signe clair de l'excellence de l'école française, qui n'est dépassée que par les Etats-Unis au palmarès. «En outre, c'est la première fois qu'un spécialiste des probabilités reçoit la médaille», se réjouit Bourguignon.
Pur produit du système français élève à Normale supérieure, thèse à Pierre-et-Marie-Curie, professeur à l'université Paris-Sud (Orsay) et à l'ENS Wendelin Werner est récompensé pour ses travaux reliant la théorie des probabilités à la physique statistique pour l'examen de phénomènes aléatoires. «Un homme discret et modeste», présente Bourguignon, mais d'une efficacité redoutable, sachant «trouver la faiblesse dans l'armure» qui protège les secrets mathématiques.
(1) Considérons une variété compacte V à 3 dimensions sans frontière. Est-il possible que le groupe fondamental de V soit trivial bien que V ne soit pas homéomorphe à une sphère de dimension 3 ?
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Jeudi 31 Août 2006
L’énigme Perelman
Le gourou russe des équations a estomaqué la communauté scientifique en résolvant l’une des sept énigmes du millénaire. Plus fort : il a refusé la médaille Fields, le « Nobel de mathématiques », et 1 million de dollars de récompense.
Par Fabien Gruhier, Le Nouvel Observateur
Sacré Grisha ! Unanimement considéré comme un génie des maths, ce Russe de tout juste 40 ans – Grigori Perelman pour l’état civil – est encore plus souvent qualifié d’« ermite de Saint-Pétersbourg », voire d’« ours mal léché ». Les rares collègues qui ont pu l’approcher lui trouvaient, voilà dix ans déjà, une tête à la Raspoutine. Avec des ongles presque aussi longs que les cheveux – « pour pouvoir ouvrir un livre exactement à la page voulue », se justifiait-il. Curieux souci pratique de la part d’un spécialiste de l’abstraction qui refuse d’empocher un prix de 1 million de dollars, et qui se réfugia dans sa chambre d’hôtel pour avaler un bocal de bortsch froid lorsqu’on voulut l’entraîner dans un restaurant réputé de Boston. Mais ça, c’était au temps où il acceptait encore de voyager et de se montrer. La dernière fois qu’il a fait des siennes, du 22 au 30 août, c’était à Madrid, au quadriennal et vingt-cinquième CIM (Congrès international de Mathématiques), où il a tenu la vedette… en ne venant pas. Alors qu’il n’y était presque question que de ses travaux. Un de ses collègues se consolait toutefois en déclarant : « Il est toujours plus simple et agréable de discuter de ses démonstrations quand il n’est pas là… » Pourtant, ce reclus qui préfère les balades en forêt aux cérémonies académiques restait à l’écoute du congrès et ne devait pas s’être beaucoup éloigné de son ordinateur. La preuve : il s’est laissé décerner une médaille Fields (la récompense suprême en maths) avant d’annoncer aussitôt par courriel qu’il la refusait. Il semble donc que l’« ours mal léché » possède aussi un caractèrede chien.
Très brillant dès le lycée, « doué dans toutes les matières à l’exception du sport », selon l’un de ses anciens professeurs, Grigori (Grisha) Perelman, dont – c’est tout dire… – les mathématiciens les plus chevronnés jugent les articles « d’une lecture difficile », est une sorte de spécialiste du refus. En 1996 déjà, il avait refusé un prix européen. Motif ? Le jury n’était « pas compétent ». Lors de tournées de conférences aux Etats-Unis, il a repoussé de nombreuses offres de chaires dans les meilleures universités. Il ne répond même pas aux demandes d’interview de la prestigieuse revue « Nature », pour laquelle tout scientifique normal vendrait son âme. Quant à la médaille Fields, il l’avait un jour déclarée « sans intérêt », sans toutefois préciser que le cas échéant il la refuserait. Au célèbre Institut Steklov de Saint-Pétersbourg (dont il a disparu sans s’être donné la peine de démissionner tant il a horreur des formalités), on avait certes noté que « des signes étranges commençaient à apparaître dans son caractère ». Mais pas au point d’imaginer qu’il refuserait avec mépris le million de dollars offert par le Clay Mathematics Institute (CMI) américain, fortune lui revenant de droit pour avoir réussi la démonstration de la conjecture de Poincaré. Démonstration sur laquelle avaient planché en vain depuis un siècle des générations de mathématiciens. Et qui lui a valu par ailleurs sa toute récente médaille Fields, décernée par le roi d’Espagne en personne et aussitôt refusée par retour du courriel.
Retenue par le CMI comme l’un des sept principaux et plus difficiles problèmes mathématiques à résoudre au cours du millénaire (voir encadré), la conjecture de Poincaré peut, selon le mathématicien Jean-Paul Delahaye, professeur à l’université de Lille (1), s’illustrer avec un œuf, un beignet percé d’un trou (genre doughnut, un tore) et… un bout de ficelle. « Entourez l’œuf avec la ficelle : en serrant la boucle, vous pouvez la réduire à un seul point, qui restera en contact avec la surface de l’œuf sans le casser ; faites la même chose avec le beignet, en l’entourant de la ficelle et en passant par le trou : pour rétracter la boucle en un seul point, vous devrez rompre le beignet. » Ce critère définit deux (et seulement deux) catégories, car une série de déformations continues peut ramener tout objet soit à la forme ovoïde, soit à la forme torique. C’est évident ? Encore fallait-il le dé-montrer ! Tel est l’exploit de Grisha, au terme d’un raisonnement ardu qui couvre des centaines de pages illisibles par le commun des mortels, mais dûment validées par la communauté mathématique. Oui, mais à quoi bon ? Et surtout, à quoi ça sert ? « Cette démonstration de la conjecture de Poincaré, étendue à toutes les dimensions, donne une structure à l’espace, répond Jean-Paul Delahaye. Elle va au moins aider les cosmologistes, qui s’interrogent sur la géométrie de l’Univers, à exclure certaines hypothèses. »
Elle pourrait avoir d’autres applications, imprévisibles, comme c’est la règle. Voilà près de deux cents ans, le Russe Nikolaï Lobatchevski prophétisait : « Il n’existe aucune branche des mathématiques, si abstraite soit-elle, qui ne doive un jour trouver à s’appliquer aux phénomènes réels. » L’avenir ne cesse de lui donner raison, car l’univers technologique dans lequel nous baignons doit énormément, en pratique, à des travaux abstraits de matheux du passé qui œuvraient sans souci d’utilité, juste pour la beauté et la cohérence de leurs théories. Or les maths, sans l’avoir voulu, sont devenues une formidable boîte à outils au service de toutes les activités humaines. Ainsi les courbes de Peano, conçues il y a un siècle comme de purs jeux esthétiques, permettent de dessiner des antennes radio. La quête frénétique des nombres premiers les plus gigantesques, ou de l’infinité des décimales du nombre pi, ont fait faire d’énormes progrès à la cryptographie. Les prodigieux développements de l’arithmétique, de l’analyse statistique, de la logique – conduits au nom de la recherche fondamentale – se retrouvent dans les codages qui ont permis les cédéroms et les DVD, la compression des données informatiques, le téléphone portable, le wi-fi, les télé-paiements sécurisés, etc. Les abstractions matheuses sont entrées au service de l’imagerie médicale, de l’architecture des microprocesseurs, de la prévision météo, des modèles climatiques, de l’analyse boursière ou de l’évaluation du risque d’extinction de telle ou telle espèce vivante… Bref, elles sont partout, et leur utilité pratique n’est plus à démontrer. C’est si vrai qu’en France un bon tiers des 6 000 mathématiciens en activité – soit environ 2 000 – travaillent dans l’industrie au sens large, au lieu d’une petite centaine il y a vingt ans. La Chine, pays au développement accéléré, investit aujourd’hui massivement dans les maths, au point de proposer des salaires élevés et des conditions de travail hors pair pour faire revenir ses meilleurs spécialistes émigrés aux Etats-Unis. C’est ainsi que le monde moderne, pour devenir toujours davantage performant, technologique et rationnel, a paradoxalement besoin de ces gourous des équations, de ces purs esprits qui fuient souvent le monde réel et s’évadent volontiers dans une sorte de mysticisme contestataire.
Reste à savoir si, en Chine par exemple, dans le cadre d’un régime totalitaire, un mathématicien peut vraiment s’épanouir. Certes, ils ne sont pas tous aussi « atteints » que Grisha, mais un ancien collègue russe de Perelman à l’Institut Steklov estime qu’« un certain degré de folie douce se retrouve chez tout bon mathématicien ». La France possède elle aussi quelques spécimens du genre.
Le plus emblématique étant Alexander Grothendieck, immense spécialiste de l’analyse fonctionnelle et de la géométrie algé-
brique. Ce lauréat de la médaille Fields – qu’il refusa… – a subitement disparu au cours des années 1990 pour se réfugier en ermite quelque part dans les Pyrénées, en un lieu connu seulement de quelques intimes, où il se consacre à… l’élevage des chèvres.
(1) Dernier ouvrage paru : « les Inattendus mathématiques », Belin, « Pour la science ».
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April 17, 2003
Perelman explains proof to famous math mystery
By Josh Brodie
Nobel Laureate John Nash GS '50 sat in the fifth row of Taplin Auditorium yesterday afternoon. Andrew Wiles, the man who proved Fermat's Last Theorem 10 years ago, sat two rows closer. All told, more than 100 mathematicians from three generations gathered to listen to Dr. Grigori Perelman describe his potentially groundbreaking work, which may earn him a share of a $1 million prize from the Clay Mathematics Institute for solving one of the "Millennium Prize Problems," the seven most difficult outstanding problems in mathematics.
For eight years now, The New York Times reported, Perelman has been working alone in Russia developing new techniques to describe geometry in higher dimensions.
While the $1 million prize was offered for the proof of a specific theorem, known as the Poincaré Conjecture, Perelman's work, if verified, is much broader and has far more powerful implications.
"He's not facing Poincaré directly, he's just trying to do this grander scheme," said mathematics professor Peter Sarnak. After creating so much new mathematics, the Poincaré result is just "a million dollar afterthought," he said.
He has essentially pitched a perfect game when the prize was offered for a single strikeout.
While Perelman did not discuss the implications of his results yesterday, he did go into detail about several technical advances he made in understanding complex topological techniques.
Topology is the branch of mathematics that studies properties of surfaces that are independent of stretching and bending without tearing the surface.
For example, inflated and deflated versions of a basketball are topologically similar. However, whether inflated or deflated, the ball is fundamentally different from a donut, because of the donut's central hole. Both of these are examples of what topologists would call two-dimensional surfaces.
In fact, there are infinitely many distinct two-dimensional surfaces. The ball is different from the donut is different from a two-holed donut is different from a three-holed donut, ad infinitum.
But this is not the case in higher dimensions. As was first suggested by former University mathematics professor William Thurston, when dealing with three-dimensional surfaces, there are only eight shapes that are topologically distinct. This statement, known as the Geometrization Conjecture in three dimensions, is currently unproven.
But if Perelman's work survives the peer review process, all that will change.
"He's obviously made a major breakthrough," Sarnak said. But he cautioned that the devil is in the details. While "he hasn't made claims he couldn't substantiate before, mistakes can be in very delicate places," Sarnak said.
According to Sarnak, Perelman's most important result may be his work with Ricci flow, a technique that essentially "smoothes out" surfaces in such a way that they evolve into their simplest possible shape.
"I do not think the million dollars is the motivation," said mathematics professor Sun-Yung Alice Chang. The Poincaré Conjecture is "in the same scale as Fermat's Last Theorem. [Proving] it puts you in the history of mathematics; the dream of every mathematician."
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