MATH NEWS ARCHIVE

LONG before John Forbes Nash, the schizophrenic Nobel laureate fictionalized onscreen in "A Beautiful Mind," mathematics has been infused with the legend of the mad genius cut off from the physical world and dwelling in a separate realm of numbers. In ancient times, there was Pythagoras, guru of a cult of geometers, and Archimedes, so distracted by an equation he was scratching in the sand that he was slain by a Roman soldier. Pascal and Newton in the 17th century, Gödel in the 20th — each reinforced the image of the mathematician as ascetic, forgoing a regular life to pursue truths too rarefied for the rest of us to understand.
Last week, a reclusive Russian topologist named Grigory Perelman seemed to be playing to type, or stereotype, when he refused to accept the highest honor in mathematics, the Fields Medal, for work pointing toward the solution of Poincaré's conjecture, a longstanding hypothesis involving the deep structure of three-dimensional objects. He left open the possibility that he would also spurn a $1 million prize from the Clay Mathematics Institute in Cambridge, Mass.
Unlike Brando turning down an Academy Award or Sartre a Nobel Prize, Dr. Perelman didn't appear to be making a political statement or trying to draw more attention to himself. It was not so much a medal that he was rejecting but the idea that in the search for nature's secrets the discoverer is more important than the discovery.
"I do not think anything that I say can be of the slightest public interest," he told a London newspaper, The Telegraph, instantly making himself more interesting. "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."
Mathematics is supposed to be a Wikipedia-like undertaking, with thousands of self-effacing scriveners quietly laboring over a great self-correcting text. But in any endeavor — literature, art, science, theology — a celebrity system develops and egos get in the way. Newton and Leibniz, not quite content with the thrill of discovering calculus, fought over who found it first.
As the pickings grow sparser and modern proofs sprawl in size and complexity, it becomes that much harder, and more artificial, to separate out a single discoverer. But that is what society with its accolades and heroes demands. The geometry of the universe almost guarantees that a movie treatment heralding Dr. Perelman is already in the works: "Good Will Hunting" set in St. Petersburg, where he lives, unemployed, with his mother, or a Russian rendition of "Proof."
To hear him tell it, he is above such trivialities. What matters are the ideas, not the brains in which they alight. Posted without fear of thievery on the Internet beginning in 2002, his proof, consisting of three dense papers, gives glimpses of a world of pure thought that few will ever know.
Who needs prizes when you are free to wander across a plane so lofty that a soda straw and a teacup blur into the same topological abstraction, and there is nothing that a million dollars can buy? Until his death in 1996, the Hungarian number theorist Paul Erdos was content to live out of a suitcase, traveling from the home of one colleague to another, seeking theorems so sparse and true that they came, he said, "straight from The Book," a platonic text where he envisioned all mathematics was prewritten.
Down here in the sublunar realm, things are messier. Truths that can be grasped in a caffeinated flash become rarer all the time. If Poincaré's conjecture belonged to that category it would have been proved long ago, probably by Henri Poincaré.
It has taken nearly four years for Dr. Perelman's colleagues to unpack the implications of his 68-page exposition, which is so oblique that it doesn't actually mention the conjecture. The Clay Institute Web site carries links to three papers by others — 992 pages in total — either explicating the proof or trying to absorb it as a detail of their own.
Those intent on parceling out credit may have as hard a time with the intellectual forensics: Who got what from whom? Dr. Perelman's papers are almost as studded with names as with numbers. "The Hamilton-Tian conjecture," "Kähler manifolds," "the Bishop-Gromov relative volume comparison theorem," "the Gaussian logarithmic Sobolev inequality, due to L. Gross" — all have left their fingerprints on The Book. Spread among everyone who contributed, the Clay Prize might not go very far.
A purist would say that no one person deserves to stake a claim on a theorem. That seemed to be what Dr. Perelman, who has said he disapproves of politics in mathematics, was implying.
"If anybody is interested in my way of solving the problem, it's all there — let them go and read about it," he told The Telegraph. "I have published all my calculations. This is what I can offer the public."
He sounded a little like J. D. Salinger, hiding away in his New Hampshire hermitage, fending off a pesky reporter: "Read the book again. It's all there."
The Math Was Complex, the Intentions, Strikingly Simple

Illustration by Robert Neubecker.

Posted Friday, Aug. 18, 2006, at 11:59 AM ET

The New York Times recently reported that reclusive Russian geometer Grigory Perelman has apparently proved the century-old Poincaré conjecture. The Times calls Poincaré "a landmark not just of mathematics, but of human thought." But just why it's so significant is left a bit hazy. Big mathematical advances often generate the same kind of lofty but content-free rhetoric found in political speeches about "the family." Like the family, math is a subject everyone agrees is very important without being able to specify exactly why.
I'm here to help. (With the Poincaré conjecture. As for the family, you're on your own.) Poincaré conjectured that three-dimensional shapes that share certain easy-to-check properties with spheres actually are spheres. What are these properties? My fellow geometer Christina Sormani describes the setup as follows:

The Poincaré Conjecture says, Hey, you've got this alien blob that can ooze its way out of the hold of any lasso you tie around it? Then that blob is just an out-of-shape ball. [Grigory] Perelman and [Columbia University's Richard] Hamilton proved this fact by heating the blob up, making it sing, stretching it like hot mozzarella, and chopping it into a million pieces. In short, the alien ain't no bagel you can swing around with a string through his hole.

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."
MANIFOLD DESTINY

Professor Jon Kleinberg

By Bill Steele
Jon Kleinberg, Cornell professor of computer science, received the International Mathematical Union's 2006 Rolf Nevanlinna Prize at the International Congress of Mathematicians in Madrid, Spain, for his "deep, creative and insightful contributions to the mathematical theory of the global information environment."
The prize, a gold medal, was presented by King Juan Carlos of Spain at the opening session of the conference. On Aug. 25 Kleinberg presented a lecture associated with the award, which has been presented every four years since 1982 in recognition of the most notable advances made in mathematics in the "information society" -- that is, the modern world as influenced by information technology.
In 1996, Kleinberg introduced the idea of "authorities" and "hubs" on the Internet and devised algorithms to analyze their links to one another, which have greatly influenced how today's search engines operate. He has made major contributions to the theory of small-world networks (the origin of the term "six degrees of separation") that have applications in fields ranging from sociology to the design of peer-to-peer file-sharing networks. He developed a mathematical model to recognize "bursts" in data streams, showing what topics are receiving attention at a given time in a large collection of data, from national news to personal e-mail.
Kleinberg received his bachelor's degree from Cornell in 1993, and an S.M. degree (1994) and Ph.D. (1996) from the Massachusetts Institute of Technology.
Among his distinctions are a Sloan Foundation fellowship, a Packard Foundation fellowship and the Initiatives in Research Award of the U.S. National Academy of Sciences. In 2005, Kleinberg received a MacArthur "genius" fellowship from the John D. and Catherine T. MacArthur Foundation.
In addition to the areas cited in the Nevanlinna Prize, Kleinberg works on problems in data mining, computer optimization, computational biology, geometric pattern matching and fault tolerance in distributed computing.
Jon Kleinberg receives international math prize

An artistic-flavored breakthrough in the field of crystal studies made by YU Shuhong from the Hefei National Laboratory for Physical Sciences at Microscale (HFNL), the University of Science and Technology of China (USTC), and his colleagues has recently been published by Chemistry of Materials and highlighted by the August 3 issue of Nature under the title of "Crystal growth: Star Quality."
"Recipe for making geometric 'stars': dissolve copper nitrate in ethylene glycol, add sulphur and bake well. The result, report Shu-Hong Yu of the University of Science and Technology of China in Hefei and his co-workers, is microscopic crystals of copper sulphide that have a beautiful cuboctahedral form, reminiscent of the cages drawn by M. C. Escher in his 1948 engraving Stars. They are composed of four intersecting hexagonal plates and contain 14 concave cavities," Nature indicates.
Before that, in American Chemical Engineering and News published on July 24, a detailed report and comment had already been given on the new outcome in an article entitled "Escher goes chemical," which is quoted as saying:
"In his 1948 wood engraving Stars, Escher celebrated geometric symmetry in a composition of polyhedrons, including a cuboctahedron, in which an octahedron infiltrates a cube. Now chemists from Hefei National Laboratory for Physical Sciences at Microscale, in China, and the Max Planck Institute of Colloids & Interfaces, Germany, have made micrometer-scale cuboctahedral structures.
"Each copper sulfide cuboctahedron consists of four intersecting hexagonal flakes, each about 2 mm across, yielding cuboctahedra with eight tetragonal and six pyramidal cavities.
"To make the Escherian crystals, the researchers prepared an ethylene glycol solution of Cu(NO3)2 and elemental sulfur, which they autoclaved at 140°C for a day. After they collected the resulting black solid by centrifugation, scanning electron microscope imagery gave Shu-Hong Yu and his coworkers a most welcome surprise.
"It is appealing that a synthetic technique as simple as the one presented here can produce such beautiful objects that even a skilled craftsman cannot touch on the microscale level," the researchers noted.
They also suggested "building blocks for larger structures and encapsulating agents for other materials are among the structures' potential uses."
Yu"s research group has been focusing on the studies of multilayer structures of bio-mineralization materials for an effective control over the construction, measurement and performance of special nanostructured materials, and explorations have been made in designing specially-structured and multifunctioned materials under natural circumstances. Their research has been supported by CAS Outstanding Overseas Scholars Recruitment Program, National Outstanding Young Scientists Funds and CAS-MPG partner group in USTC.
Escher was a Dutch graphic artist well-known for his prints and engravings using realistic details to achieve bizarre optical illusions. His works were of interest to mathematicians, cognitive psychologists and the general public, and were widely reproduced especially in the 1960s and 1970s.
http://english.cas.ac.cn
Micro-Crystals Featuring Escherian Architecture

Professor Eric Vigoda

Joy Weaks, College of Computing
Atlanta (August 16, 2006) — College of Computing associate professor Eric Vigoda recently won the 2006 Delbert Ray Fulkerson Prize for his paper titled "A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries," co-authored with Mark Jerrum at the University of Edinburgh and Alistair Sinclair at UC Berkeley.
The "Fulkerson Prize" is a prestigious award given every three years for outstanding papers in the area of Discrete Mathematics, and is sponsored jointly by the Mathematical Programming Society and the American Mathematical Society.
Vigoda is the first from the College of Computing at Georgia Tech to win this celebrated prize, although past Georgia Tech winners include Arkadi Nemirovski (1982) from the School of Industrial and Systems Engineering and Robin Thomas (1994) from the School of Mathematics.
The permanent of a matrix is currently a well-studied combinatorial problem with applications in many fields, as it corresponds to the number of perfect matchings of a bipartite graph. For example in physics, computing the permanent is central to the study of the Dimer and Ising Models, although the exact computation of the permanent is intractable. Mathematicians began studying the permanent about two centuries ago, partly because of its superficial similarity to the determinant, which is a much easier problem.
Vigoda's breakthrough discovery is a randomized algorithm which approximates the permanent to within an arbitrarily close factor in time polynomial and in the size of the input. Therefore, with the use of randomness, arbitrarily good approximations can still be obtained. Vigoda's paper also introduces techniques that have already found several important computing, physics, and mathematical applications. The award was presented at the International Symposium on Mathematical Programming this month in Rio de Janeiro.
For more information about the Fulkerson Prize, visit http://www.ams.org/prizes/fulkerson-prize.html.
To view Vigoda's award-winning paper, visit http://www.cc.gatech.edu/~vigoda/Permanent.pdf.
Computing's Vigoda Wins Renowned Fulkerson Prize

Dr. Simon Chua (right) receives the Paul Erdos Award
from Dr. Petar Kenderov,
president of the World Federation of National Mathematics Competition,
in ceremonies held in Cambridge, England.

HICAP Under his leadership, he catapulted young Filipino mathematicians to world competitions, which produced a string of awards and medals for the country. Now, his time has come to be recognized by the world for his efforts.
Dr. Simon Chua, president of the Mathematics Trainers Guild-Philippines (MTG) and principal of the Zamboanga Chong Hua High School, is the first Filipino to receive the Paul Erdos Award, given by the World Federation of National Mathematics Competitions (WFNMC), an affiliated study group of the International Commission for Mathematical Instruction. Chua received the award in Cambridge, England, during the federation's conference from July 22 to 28.
Chua is proud that through his award, the Philippines was recognized in mathematics.
"It would be an understatement to say that I only felt happy. Indeed, I was on cloud nine. I felt a real sense of achievement for my country and the Filipino people. Since I have been working hard to bring the Philippines into the frontline of international mathematics competition, I feel honored to dedicate this recognition to every Filipino," Chua told The Manila Times.
When he arrived at the Ninoy Aquino International Airport on July 30, he was surprised to find that a welcome party was eagerly waiting for him composed of MTG trainers, parents and students.
The big challenge
For the last 10 years, the MTG, composed of teachers, has been training Filipino students to compete in international competitions. The MTG started in 1995 as a movement called "The Mathematical Challenge for Filipino Kids Training Program" founded by Chua and Rechilda Villame, MTG vice president. Today the MTG has almost 1,000 member-schools and 28 training centers nationwide.
Besides Chua and Villame, other officers of the MTG are Dr. Eduardo de la Cruz Jr., dean of the Institute of Education of Arellano University and MTG vice president for training and development; Robert Degolacion; Lucy Sia; Eugenia Guerra; Sanet Hipolito; Isidro Aguilar; Anthony Ang; Alma Luz Pasiliao and Josephine Tan.
Chua started as a substitute teacher in a Chinese school and spent two years teaching trigonometry, advanced algebra and analytic geometry.
After getting his BS Math degree in college, he quit his job and became the assistant manager to the family's chemical business. After a while of soul-searching, he went back to teaching and became the chairperson of the math department in a Manila university. He taught college students that mathematics was a fun subject.
Establishing the MTG opened doors for young mathematicians to compete and win in international competitions.
This year alone, young Filipino mathematicians won in the 2006 Indonesia Elementary Mathematics International Contest in Bali, Canadian Math Competition, 7th Invitational World Youth Intercity Mathematics Competition in Wenzhou, China, and the 10th Po Leung Kuk Primary World Mathematics Olympiad held in Hong Kong.
Before students can be selected for international math contests, they must undergo rigorous training by the MTG.
Chua said that in the training, MTG emphasizes commitment from every student who want to make it in the international scene.
"Nothing can substitute care and affection for these MTG kids. I always make them feel that they are important in which they are. Any student for that matter would try to achieve excellence if the teacher shows trust on his/her students. As for discipline, well, I only show my commitment and dedication to make them learn and good things follow. Students, in return, give their sense of commitment to excel in international competition. I believe that students can never show their potential and their giftedness in an atmosphere of fear and harsh discipline. What MTG does with these kids is to draw the best from them and for sure the best just come out," he said.
Under Chua's leadership, MTG became a formidable force in mathematics training in the country. Dr. De la Cruz praises Chua for his personal commitment to ensuring the highest level of training.
"[Dr. Chua] is a man of work, a man of service and a man of numbers. He always thinks of how he can help students and teachers improve their math abilities. He is always there to share his expertise and even materials to those who love math," he said.
Degolacion, MTG office director, says Chua is an epitome of someone who "has a great passion for mathematics. Such passion is oftentimes translated to his commitment in making mathematics shine in Philippine education."
He describes Chua as a person with full of humility and a big heart.
"Dr. Chua always makes himself available to everybody. He never makes himself first in anything he does. For him, the welfare of others is his top priority," Degolacion said.
For many students, math is a dreaded subject in school but Chua assures that math is a great subject to learn. He says students should "first open their hearts to mathematics."
"I know there are so many students who have adverse attitude toward this subject. If they can open only their hearts for mathematics, they will see the beauty of the subject. Second, understand its language. Anyone who knows its language that comes in number form learns to appreciate mathematics. Third, students must feel its importance in life. Since mathematics makes the world go round, then, students must consider how math influences our daily life. With these inputs, I can encourage them to give their best foot forward for mathematics and achieve proficiency and excellence," said Chua.
Chua said the Paul Erdos Award has only encouraged him even more to dedicate his life to uplifting math education among students.
"Receiving this award makes feel challenged to do better than my best. It gives me more responsibility to make the mathematics education in the Philippines achieve a greater height so that every student shall be more competitive in any international math challenge," he said.
Dr. Simon Chua: Paul Erdos awardee for math

Associate Professor of Mathematics Sally Cockburn

Associate Professor of Mathematics Sally Cockburn presented a paper at MathFest, the summer meeting of the Mathematical Association of America on Aug. 12 in Knoxville. The paper, titled "Deranged Socks," was joint work with former visiting professor Joshua Lesperance, and grew out of a problem from Cockburn's junior-level graph theory and combinatorics course. Specifically, given n distinct pairs of socks, how many ways are there to distribute 2 socks to each of n people so that no one receives a matching pair? Like many combinatorial problems, it is easy to state, but remarkably difficult to solve.
Cockburn also recently published an article in the online version of the journal Mathematical Programming, titled "On the domino-parity inequalities for the STSP", which was joint work Dr. Sylvia Boyd of the University of Ottawa and Danielle Vella. The paper looks at a new approach for solving the famous Traveling Salesperson Problem, in which a salesman wants to find the shortest route for visiting each of n cities exactly once and returning home.
Also, a paper Cockburn wrote with Dr. R. Bruce Mattingly of SUNY Cortland and Dr. Ben Coleman and Dr. Kay Somers, both of Moravian College, titled "Some Problems are NP-Harder than Others," was published this summer as part of the NSF's DIMACS (Center for Discrete Mathematics and Computer Science) Educational Module Series. The module presents the results of recent results on the computational complexity of a pair of classic optimization problems in graph theory, the dominating set problem and the vertex cover problem. Although both are theoretically intractable in general for large instances, the latter can be solved in a reasonable amount of time for all cases that arise in practical settings.
Cockburn Presents Paper at MathFest Meeting

It would seem, on the surface at least, that art and science have little in common. The first deals with unquantifiable, subjective concepts like beauty and emotion; the latter is absorbed by observable, measurable phenomena. But while they may appear to be opposites, art and science share a fundamental characteristic that binds them inseparably: Both are, at heart, nothing more than a search for truth. The avenues of approach to this truth are necessarily different, of course, but each seeks to express the verities and intangibles of life on this planet and beyond. Whether via a mathematical formula or a painting of exquisite beauty, reality is explored and explained by practitioners who pair empirical observation with imagination to achieve a synthesis that resonates as true.

Jean Constant is a Los Alamos–based artist who works with mathematicians to beta-test software by rendering their formulas as artistic representations. Using a sophisticated computer program, he applies imagination to the computations in such a way as to demonstrate their physical manifestations and, in the process, highlight any errors in logic.

He also creates artwork on his own — in oils, acrylics, digital media — that's inspired by and adheres to the principles of these formulas. "My work is a poetic visualization of mathematical algorithms," the artist says. A mathematician's aim is to understand and define the world as it is. As an artist, I use the tools of mathematics to create new perspectives."

Constant is involved with a number of organizations that promote interaction and collaboration between artists and scientists, and he believes that bringing these groups together is an important next step in our social evolution. "Society has tried to push artists into a corner," he says. "The 'crazy artist' is a convenient image, but it's never really been accurate. These days, science is bringing art back to where it belongs, a partner in the act of discovery of the world around us. Today the computer is as powerful a technology as advances in painting were in the Middle Ages.

Also from nature comes the more recently defined fractal, a geometric shape that has symmetry of scale, such that if you were to zoom in on any part of it at various levels of magnification, it would still look the same, or nearly the same. We find this property in the branches of a tree, rugged coastlines, and planets that orbit stars that, in turn, orbit galaxies — the part is the whole, and the whole is the part.

[Disclaimer: I don't have any relationship with Santa Fe Trend, but Jean Constant is a long time friend. You can find more of his works in his Art Portfolio. And you can even buy some of his paintings or photographs. Even if I don't get a cent on it, I'm sure Jean will buy me a drink the next time he comes to Paris.] Sources: Nancy Zimmerman, Santa Fe Trend, Summer 2006; and Jean Constant web site
When Art meets Maths

Martin Gardner

His mind is highly philosophical, at home with the most abstract concepts, yet his thinking and writing crackle with clarity -- lively, crisp, vivid. He achieved worldwide fame and respect for the three decades of his highly popular mathematical games column for Scientific American, yet he is not a mathematician. He is by every standard an eminent intellectual, yet he has no Ph.D. or academic position. He has a deep love of science and has written memorable science books (The Ambidextrous Universe and The Relativity Explosion, for instance), and yet he has devoted probably more time and effort to -- and has been more effective than any thinker of the twentieth century in -- exposing pseudoscience and bogus science.
He is considered a hard-nosed, blunt-speaking scourge of paranormalists and all who would deceive themselves or the public in the name of science, yet in person he is a gentle, soft-spoken, even shy man who likes nothing better than to stay in his home with his beloved wife Charlotte in Hendersonville, North Carolina, and write on his electric typewriter.
His critics see him as serious, yet he has a playful mind, is often more amused than outraged by nonsense, and believes with Mencken that "one horselaugh is worth ten thousand syllogisms." He is deeply knowledgeable about conjuring and delights in learning new magic tricks. He retired from Scientific American more than fifteen years ago, but his output of books, articles, and reviews has, if anything, accelerated since then. (He's now written more than sixty books, and more are in the works.) His knowledge and interests span the sciences, philosophy, mathematics, and religion, yet he professes no special standing as a Renaissance man. He has received major awards from scientific societies and praise from some of the nation's leading scholars ("One of the great intellects produced in this country in this century," says Douglas Hofstadter), some of whom forthrightly consider him an intellectual hero, yet he remains modest about his contributions.
At eighty-three, Martin Gardner reigns supreme as the leading light of the modern skeptical movement. More than four and a half decades ago, in 1952, he wrote the first classic book on modern pseudoscientists and their views, Fads and Fallacies in the Name of Science, and today it remains in print and widely available as a Dover paperback and is as relevant as ever. It has influenced and inspired generations of scientists, scholars, and nonscientists. He followed that up in 1981 with Science: Good, Bad, and Bogus. In an essay in the New York Review of Books entitled "Quack Detector," Stephen Jay Gould welcomed the book and said Martin Gardner "has become a priceless national resource," a writer "who can combine wit, penetrating analysis, sharp prose, and sweet reason into an expansive view that expunges nonsense without stifling innovation, and that presents the excitement and humanity of science in a positive way." After that, in the same genre, came The New Age: Notes of a Fringe-Watcher (1988), On the Wild Side (1992), and Weird Water and Fuzzy Logic: More Notes of a Fringe-Watcher (1996).
The subtitles refer of course to his column "Notes of a Fringe-Watcher" (broadened from its original title, "Notes of a Psi-Watcher"), which has graced the pages of the Skeptical Inquirer every issue since Summer 1983. His first SI column, "Lessons of a Landmark PK Hoax," dealt with James Randi's then-just-revealed Project Alpha experiment, in which Randi planted two young magicians in a parapsychology laboratory to see if the lead investigator could detect their trickery. The three Gardner anthologies each consist of half SI columns and half reviews and writings for other publications.
When the Committee for the Scientific Investigation of Claims of the Paranormal (CSICOP), publisher of the Skeptical Inquirer, was established in 1976, Martin Gardner was one of its original founding fellows, and he has remained a member of its Executive Council and Editorial Board ever since. When offered the opportunity fifteen years ago to write a regular column for SI, he quickly agreed. He dedicated The New Age anthology to CSICOP's founder and chairman: "To Paul Kurtz, a friend whose vision, courage, and integrity started it all." Although Martin Gardner seldom travels to CSICOP meetings, he remains, through his personal contacts, insights, published writings, and voluminous correspondence, a profound influence on CSICOP, modern skepticism, and intellectual discourse broadly.
He answered questions posed by Skeptical Inquirer Editor Kendrick Frazier.

SI: In your book of essays The Night Is Large: Collected Essays 1938-1995, you organized your lifelong intellectual interests into seven categories: physical science, social science, pseudoscience, mathematics, the arts, philosophy, and religion. Do they have equal importance to you? How do you rank them in importance or interest -- to you? to others? Do you see them as complementary aspects of one coherent worldview, or are some separate?
Gardner: My main interests are philosophy and religion, with special emphasis on the philosophy of science. I majored in philosophy at the University of Chicago (class of 1936), having entered the freshman class as a Protestant fundamentalist from Tulsa. I quickly lost my entire faith in Christianity. It was a painful transition that I tried to cover in my semi-autobiographical novel The Flight of Peter Fromm (now a Prometheus Books paperback). I actually doubted the theory of evolution, having been influenced by George McCready Price, a Seventh-day Adventist creationist. A course in geology convinced me that Price was a crackpot. However, his flood theory of fossils is ingenious enough so that one has to know some elementary geology in order to see where it is wrong. Perhaps this aroused my interest in debunking pseudoscience.
After I returned from four years in the Navy as a yeoman, I returned to Chicago and would have gone back to my former job in the university's press relations office had I not sold a humorous short story to Esquire. This was my first payment for anything I'd written. It persuaded me to see if I could survive as a freelancer, and for the next year or two I lived on income from sales of fiction to Esquire. My second sale was a story based on topology titled "The No-Sided Professor." It was my first effort at science fiction.
While freelancing, I took a seminar (using GI bill funds) from the famous Viennese philosopher of science Rudolf Carnap. It was the most exciting course I ever took. Years later I persuaded Carnap to have the course tape-recorded by his wife and to let me shape the recording into a book. Basic Books issued it under the title Philosophical Foundations of Physics. The title was later changed to Introduction to the Philosophy of Science. All the ideas in the book are Carnap's, all the wording mine. Dover recently reprinted it in paperback with an afterword about how the book came about and my memories of Carnap. During the writing of this book, I exchanged many pleasant letters with Mrs. Carnap, but before the book was published, for reasons unknown to me, she killed herself by hanging.
Carnap had a major influence on me. He persuaded me that all metaphysical questions are "meaningless" in the sense that they cannot be answered empirically or by reason. They can be defended only on emotive grounds. Carnap was an atheist, but I managed to retain my youthful theism in the form of what is called "fideism." I like to call it "theological positivism," a play on Carnap's "logical positivism."
Shortly before he died, Carl Sagan wrote to say he had reread my Whys of a Philosophical Scrivener and was it fair to say that I believed in God solely because it made me "feel good." I replied that this was exactly right, though the emotion was deeper than the way one feels good after three drinks. It is a way of escaping from a deep-seated despair. William James's essay "The Will to Believe" is the classic defense of the right to make such an emotional "leap of faith." My theism is independent of any religious movement, and in the tradition that starts with Plato and includes Kant, and a raft of later philosophers, down to Charles Peirce, William James, and Miguel de Unamuno. I defend it ad nauseam in my Whys.
SI: How have you managed to retain such a phenomenal breadth of interest and knowledge?
Gardner: Philosophy gives one an excuse to dabble in everything. Although my interests are broad, they seldom get beyond elementary levels. I give the impression of knowing far more than I do because I work hard on research, write glibly, and keep extensive files of clippings on everything that interests me.
There are big gaps in my knowledge, one of the largest of which is classical music. I have a poor ear. My tastes run to Dixieland jazz and melodies I am able to hum and play on a musical saw (one of my minor self-amusements). I know nothing about sports other than baseball. I have never played a game of golf or seen a horse race. I never watch football or basketball. I think boxing should be outlawed as too primitive and cruel. Ditto for Spanish bullfighting.
In high school I was on the gymnastic team (specializing in the horizontal bar), and I played lots of tennis. I would enjoy tennis today except that I had cataract surgery early in life. Without eye lenses, one cannot continually alter one's focus, so there is no way to anticipate exactly where the ball is as it comes toward you.
SI: Do you wish you had pursued one field more, to the exclusion of the others?
Gardner: I'm glad I majored in philosophy, though had I known I would be writing some day a column on math, I would have taken some math courses. As it was, I took not a single math course. If you look over my Scientific American columns you will see that they get progressively more sophisticated as I began reading math books and learning more about the subject. There is no better way to learn anything than to write about it!
SI: You probably could have been either a philosopher or a mathematician -- which a lot of fans of your Scientific American recreational mathematics columns probably thought you were. Did you ever think about going into academia?
Gardner: Early on in college I decided I wanted to be a writer, not a teacher, and I have never regretted this decision. It is the reason I took only one year of graduate work, and never cared to go for a master's.
SI: Given your breadth and variety of interests, how would you describe yourself -- your professional field?
Gardner: I think of myself as a journalist who writes mainly about math and science, and a few other fields of interest.
SI: I appreciate the becoming modesty, but I think you may be too self-effacing. Douglas Hofstadter has said, "Martin Gardner is one of the greatest intellects produced in this country in this century." Stephen Jay Gould has said you have been "the single brightest beacon defending rationality and good science against the mysticism and anti-intellectualism that surround us." Certainly you must be pleased to be so highly regarded.
Gardner: Yes, I am pleased, though Hofstadter, a good friend, surely exaggerates, and ditto for Gould, a marvelous writer I hope to meet some day.
SI: What do you consider to be the relationship between your interests in writing about science and in debunking pseudoscience? Which has more appeal to you?
Gardner: In a way, I regret spending so much time debunking bad science. A lot of it is a waste of time. I much more enjoyed writing the book with Carnap, or The Ambidextrous Universe, and other books about math and science.
SI: What motivates you? You have been writing on pseudoscience and fringe-science since at least 1950. The Washington Post reviewer of The Night Is Large described you -- correctly, I think -- as "almost certainly the most eminent debunker of pseudoscience since World War II." Do you find pseudoscience and paranormal claims inherently fascinating -- you seem both wryly amused and deeply concerned -- or do you consider critiquing them more a task that has to be done? If the latter, what keeps you going at it?
Gardner: I'm not sure why I enjoy debunking. Part of it surely is amusement over the follies of true believers, and partly because attacking bogus science is a painless way to learn good science. You have to know something about relativity theory, for example, to know where opponents of Einstein go wrong. You have to know something about probability and statistics to recognize Michael Drosnin's The Bible Code as hogwash. You have to know the power of the placebo and faith to see why Mary Baker Eddy is the very model of a quack.
Another reason for debunking is that bad science contributes to the steady dumbing down of our nation. Crude beliefs get transmitted to political leaders and the result is considerable damage to society. We see this happening now in the rapid rise of the religious right and how it has taken over large segments of the Republican Party. I think fundamentalist and Pentecostalist Pat Robertson is a far greater menace to America than, say, Jesse Helms who will soon be gone and forgotten.
I am happy to see the job of debunking bad science being taken over by others, especially by scientists like the late Carl Sagan who came to realize the importance of speaking up. I am delighted that Philip Klass is doing such a good job on UFO nonsense because it allows me to avoid this dismal topic. I was tempted to wade into The Bible Code. Now I don't need to after reading Dave Thomas's definitive blast in the Skeptical Inquirer [November/December 1997].
SI: You are generally considered one of the harshest critics of the paranormal and its proponents. How would you characterize your position?
Gardner: I like to think I am unduly harsh and dogmatic only when writing about a pseudoscience that is far out on the continuum that runs from good science to bad, and when I am expressing the views of all the experts in the relevant field. Where there are areas on the fringes of orthodoxy, supported by respected scientists, I try to be more agnostic. I am certain, for example, that astrology and homeopathy are totally worthless, but I have no strong opinions about, say, superstring theory. Superstrings are totally lacking in empirical support, yet they offer an elegant theory with great explanatory power. I wish I could be around fifty years from now to know whether superstrings turn out to be a fruitful theory or whether they are just another blind alley in the search for a "theory of everything."
There are dozens of monumental questions about which I have to say "I don't know." I don't know whether there is intelligent life elsewhere in the universe, or whether life is so improbable that we are truly alone in the cosmos. I don't know whether there is just one universe, or a multiverse in which an infinite number of universes explode into existence, live and die, each with its own set of laws and physical constants. I don't know if quantum mechanics will someday give way to a deeper theory. I don't know whether there is a finite set of basic laws of physics or whether there are infinite depths of structure like an infinite set of Chinese boxes. Will the electron turn out to have an interior structure? I wish I knew!
I can say this. I believe that the human mind, or even the mind of a cat, is more interesting in its complexity than an entire galaxy if it is devoid of life. I belong to a group of thinkers known as the "mysterians." It includes Roger Penrose, Thomas Nagel, John Searle, Noam Chomsky, Colin McGinn, and many others who believe that no computer, of the kind we know how to build, will ever become self-aware and acquire the creative powers of the human mind. I believe there is a deep mystery about how consciousness emerged as brains became more complex, and that neuroscientists are a long long way from understanding how they work.
SI: What trends have you seen in popular belief in pseudoscience and the paranormal in the past half century? Has it gotten better or worse? What are your greatest concerns?
Gardner: I think popular belief in bogus sciences is steadily increasing. When I was a boy, there were only one or two astrologers who wrote newspaper columns. Now almost every paper except the New York Times, not to mention dozens of magazines, features a horoscope column. Professional astrologers now outnumber astronomers. For Pete's sake, a president of the United States and his first lady were astrology buffs! This would have seemed unthinkable a hundred years ago.
Alternative medical views are growing rapidly, especially on college campuses where more students are relying on homeopathic remedies than ever in history. Real tragedies occur when persons avoid sound medical help and rely on worthless claims.
SI: What do you see for the future in that regard?
Gardner: I see the immediate future as having a steady increase in superstitions. Fundamentalism, especially the Pentecostal variety, is growing rapidly, not only here but in other nations, notably in South America. And not only among Protestants but also among Catholics and Jews. The Catholic Church is on the brink of its greatest blunder since it condemned Galileo. It is close to declaring that Mary is a "co-redeemer" with Christ! (Mother Teresa was a strong supporter of this.) Of course, if the pope declares infallibly that this doctrine is true, it will kill the ecumenical movement.
Did you know that Dr. Raymond Damadian, the distinguished inventor of magnetic resonance imaging (the MRI test), has declared himself a creationist and a young-Earther?
SI: Apart from popular belief in pseudoscience, how about what we might call experimental parapsychology -- work done by Ph.D.'s in the laboratory that some keep pointing to as evidence of ESP or PK -- going back to J. B. Rhine's experiments in the 1930s and 1940s and most recently the ganzfeld experiments, the persistent claims about remote viewing, and Robert Jahn's random-number-generator work? Where does all that stand in your view?
Gardner: I'm all in favor of parapsychologists continuing to look for evidence of psi, and their experiments certainly are more carefully controlled than in the days of Rhine. It has often been pointed out that as Rhine slowly learned how to tighten his controls, his evidence of psi became weaker and weaker. However, the evidence will not become convincing to other psychologists until an experiment is made that is repeatable by skeptics. So far, no such experiment has been made. Jahn's evidence for psi is statistical, and there are many ways his statistics, which favor psi to a very slight degree, can be unconsciously biased. As far as I know, no one else has been able to duplicate his computer-generated results.
SI: Are you discouraged by the rejection of science in certain parts of academia heavily influenced by the postmodernist antipathy toward science and reason? The Sokal hoax, which you wrote about so amusingly, certainly exposed that movement's scientific vacuousness.
Gardner: Yes, I am dismayed by the increasing effort of the postmoderns to view science as a solely cultural phenomenon rather than as a highly successful and ongoing search for objective truths about the universe. No one wants to deny that science is corrigible, but it is a wonderfully successful self-correcting process that gets ever closer to objective truth. Postmodern nonsense has even invaded mathematics, as witnessed by Reuben Hersh's just-published book What Is Mathematics, Really? I have a lengthy critical review of this book in the Los Angeles Times Book Review (October 12, 1997), defending the opinion of almost all mathematicians today or in the past that mathematics has a curious kind of reality independent of human minds. The universe is made of particles and fields about which nothing can be said except to describe their mathematical structures. In a sense, the entire universe is made of mathematics. If the particles and fields are not made of mathematical structure, then please tell me what you think they are made of!
SI: When you wrote the book Fads and Fallacies in the Name of Science, did you expect that it would become the classic it has become?
Gardner: No, I never expected Fads and Fallacies would long remain in print. The first edition, titled In the Name of Science, sold so poorly that Putnam quickly remaindered it. Not until Dover picked it up did its sales take off, thanks in large degree to Long John Nebel, then a popular all-night radio talk-show host. For many months, he had guests on almost every night to attack the book. I remember one night, when I had gotten out of bed to change a diaper on our first born, I turned on the radio and heard John Campbell, then editor of Astounding Science Fiction, say "Mr. Gardner is a liar." I had a chapter about his role in introducing L. Ron Hubbard's dianetics. Campbell claimed it had cured his sinusitis. I never dreamed that Scientology would last more than a few years, because its claims were so preposterous. It maintained, for example (and still does), that immediately after conception, long before the embryo develops ears, it makes recordings (called engrams) of all the words spoken by or to the mother! I would never have dreamed that UFOs, to which I also devoted a chapter, would become a mania that would increase steadily over the next half century. I expected Wilhelm Reich's orgone therapy to be short lived, yet it is still going strong. Come to think of it, phrenology is the only major pseudoscience I know about that once flourished around the world and has since faded away.
SI: Which of your own books are your favorites? Which have been most popular? Which are most important?
Gardner: Of my books, the one that I am most pleased to have written is my confessional, The Whys of a Philosophical Scrivener, with my novel about Peter Fromm running second.
SI: And which of your books have been the most popular, have sold best? Which do you think have been the most influential?
Gardner: The one book of mine that has sold the most copies is far and away my Annotated Alice. It has never been out of print since it was published in 1960, and has now sold over a million copies in hard- and soft-cover editions here and in England. Of my books about pseudoscience, I suppose the first one, now titled Fads and Fallacies in the Name of Science, has been the most influential on later writing about similar topics.
SI: Whatever you write about, you seem always to call on great storehouses of specific information -- journal papers, magazine articles, newspaper clippings, etc., going back decades. I've heard Randi describe with some awe your filing system. Can you tell us about it?
Gardner: Yes, my files are my number one trade secret. It began in college with 3 3 5 file cards that I kept in ladies shoe boxes. I had a habit then (this was before copy machines) of destroying books by slicing out paragraphs and pasting them on cards. A friend once looked through my cards on American literature and was horrified to discover I had destroyed several rare first editions of books by Scott Fitzgerald.
When I began to earn some money I moved the cards into metal file cabinets, and started to preserve complete articles and large clippings and correspondence in manila folders. These folders are now in some twenty cabinets of four or five drawers each. And I have a large library of reference books that save me trips to the library. I have not yet worked up enough courage to go on line for fear I would waste too much time surfing the Internet.
SI: How do you manage to keep up with everything?
Gardner: I keep up my interests by taking scores of periodicals that deal with topics I may write about, especially science and math journals. I have been a lifelong subscriber to Science News, which you once edited. I could never have written my Scientific American columns without access to math magazines that ran articles and problems that could be considered recreational in nature.
SI: For as world-famous and respected as you are -- your writings have been inspirational to two generations of prominent scientists and scholars -- you usually have worked alone. You seldom, if ever, go to conferences or meetings. Only a few of your many fans and readers have ever seen or heard you in person.
Why? Has this been an advantage to you -- no distractions, for instance -- more time for writing? Have there been drawbacks to this solitary work style?
Gardner: I have often been called shy, and with justification. I prefer one-to-one relationships to crowds. I hate going to parties or giving speeches. I love monotony. Nothing pleases me more than to be alone in a room, reading a book or hitting typewriter keys. I consider myself lucky in being able to earn a living by doing what I like best. As my wife long ago realized, I really don't do any work. I just play all the time, and am fortunate enough to get paid for it.
SI: You seem to be curious about everything. What most delights you? Scientifically? Professionally? Personally?
Gardner: I am most delighted by learning something new and significant. (I leave aside the delights of relationships with my wife, with relatives, and with friends). This year I had the pleasure of updating and expanding a 1910 book by Sylvanus Thompson called Calculus Made Easy. It was a great pleasure to learn, for the first time, some basic calculus, and to appreciate fully its enormous elegance and power.
Next to learning something about science or math that I didn't know before, my next greatest pleasure is learning a newly invented magic trick. Conjuring has been a hobby since I was a boy. Some of the best magic tricks operate on scientific or mathematical principles. One of my earliest books, Mathematics, Magic and Mystery (still in print as a Dover paperback) deals with this overlap of magic and math.
Let me give one example. Arrange the cards in a deck so they alternate blacks and reds. Cut the deck in half, making sure the bottom cards of each half are of opposite color. Riffle shuffle the halves together once, making the shuffle as careless or thorough as you please. Now remove cards from the top of the deck in pairs. Each pair will contain a red and a black card! Dozens of clever card tricks have been based on this curious principle. To prove that it must work leads straight into nontrivial combinatorial theory.
SI: Many prominent skeptics are likewise knowledgeable about magic. How important is such an understanding in evaluating paranormal claims?
Gardner: I don't think a knowledge of magic is important in countering paranormal claims, except in connection with self-styled psychics who claim extraordinary paranormal powers. Such psychics use methods which have in common the methods of magicians. A man can be a great scientist, or a greater writer, and be so easily fooled by simple methods of deception that his opinions about extraordinary claims of psi powers are utterly worthless. Conan Doyle, for example, would never have believed in the genuineness of spirit mediums who levitate tables and themselves, float trumpets, produce visible spirits of the dead, exude ectoplasm through their noses, and so on, if he had had even the most superficial training in the methods of conjuring. The parapsychologists who once took Ted Serios and others like him seriously would have been spared their embarrassments had they known anything about magic. A knowledgeable magician, watching these "psychics" perform on stage, knows at once how they obtain their wonders. It is a scandal that even today so few parapsychologists think it worthwhile to study the methods of magicians before they test a psychic who performs incredible feats, then publish papers testifying to the genuineness of the psychic's powers.
An outstanding instance of this failure is John Beloff's unwillingness to learn anything about magic. Not many years ago, he wrote that the card tricks of a certain magician represented one of the strongest recent proofs of paranormal powers! When Persi Diaconis watched this magician do his simple card magic, it was perfectly obvious how he was obtaining his effects by methods well known to card magicians.
SI: You were a founding fellow of CSICOP and have been a member of the Executive Council since the beginning. How have you seen our role? What advice do you have for us for the future?
Gardner: CSICOP is obviously doing a much-needed job in combating America's dumbing down, especially in providing a source to which editors of magazines and newspapers, and the makers of TV shows can turn to get information about bogus claims. It is a role that will be increasingly important in the years ahead.
SI: It's hard to believe you have been writing your "Notes of a Fringe-Watcher" column in SI for almost fifteen years -- especially since you didn't start it until retiring from your long-running Scientific American column. Do you miss doing the latter?
Gardner: I do indeed miss writing the Scientific American column. I had reached a point where I could no longer keep up the column and write the books I hoped to write as long as I had my wits about me. Also, I felt it was time for younger writers to take over the column.
One of the lasting benefits of having done the column was getting to know, as personal friends, so many mathematicians, real mathematicians, far more knowledgeable than I, and whose work I could only dimly appreciate. It would take a page just to list their names. Another continuing pleasure is getting letters from mathematicians telling me it was my column that aroused their interest in math when they were in high school and led them to decide on math as a career.
SI: Well, we all hope you will continue writing your "Notes of a Fringe-Watcher" column in SI for a long time to come. It clearly continues to be provocative.
Gardner: Thanks!
SI: Your readers worldwide have been blessed by your thinking and writing over your long and prolific career, well into a time most people have retired. We all hope you can continue for a long time to come. How is your health?
Gardner: At eighty-three, I tell people I don't feel a day over seventy-five. Seriously, I have few complaints except an enlarging prostate that occasionally bothers me at night, and mild high blood pressure, which I control with Hytrin. Short-term memory is not what it used to be. My wife and I frequently spend twenty minutes at the dinner table trying to recall the name of someone we both know well until suddenly one of us shouts it out. I am fortunate in having parents who each lived into their nineties. I hope my dear wife Charlotte outlives me, although we both look forward to celebrating the arrival of the year 2000, and seeing our grandchildren become adults.
A Mind at Play

A fake medieval painting added by a forger in the 20th Century hides the Archimedes text. (Credit: Archimedes Palimpsest Project)

A series of hidden texts written by the ancient Greek mathematician Archimedes are being revealed by US scientists.
Until now, the pages have remained obscured by paintings and texts laid down on top of the original writings.
Using a non-destructive technique known as X-ray fluorescence, the researchers are able to peer through these later additions to read the underlying text.
The goatskin parchment records key details of Archimedes' work, considered the foundation of modern mathematics.
The writings include the only Greek version of On Floating Bodies known to exist, and the only surviving ancient copies of The Method of Mechanical Theorems and the Stomachion.
In the treatises, the 3rd Century BC mathematician develops numerical descriptions of the real world.
"Archimedes was like no-one before him," says Will Noel, curator of manuscripts and rare books at the Walters Art Museum in Baltimore, Maryland and director of the imaging project.
"It just doesn't get any better than re-reading the mind of one of the greatest figures of Western civilisation."
'Eighth wonder'
Revealing Archimedes' writings presents a huge challenge to the imaging team.
The original texts were transcribed in the 10th Century by an anonymous scribe on to parchment.
Three centuries later a monk in Jerusalem called Johannes Myronas recycled the manuscript to create a palimpsest.
Palimpsesting involves scraping away the original text so the parchments can be used again. To create a book, the monk cut the pages in half and turned them sideways.
To create a book Myronas also used recycled pages from works by the 4th Century Orator Hyperides and other philosophical texts.
Mr Noel describes the palimpsest as "the eighth wonder of the world".
"You never get three unique palimpsested texts from the ancient world together in one book," he told the BBC News website. "That's just completely unheard of."
The monks filled the recycled pages with Greek Orthodox prayers.
Later, forgers in the 20th Century added gold paintings of religious imagery to try to boost the value of the tome.
The result was the near total obliteration of the original texts apart from faint traces of the ink used by the 10th Century Scribe.
Bright light
Previously the privately-owned palimpsest has been investigated using various optical and digital imaging techniques. However, much of the text remained hidden behind paint and stains.
The researchers have now turned to a technique known as X-ray fluorescence to tease out the final details of the writings.
The method is used in may branches of science including geology and biology. It has previously been used by other researchers to decode ancient texts.
In August 2005 a team from Cornell University successfully deciphered a series of 2,000-year-old worn down stone inscriptions.
The X-rays are formed in a synchrotron - a particle accelerator that uses electrons travelling at close to the speed of light to generate powerful "synchrotron" light. The light covers a wide range of the electromagnetic spectrum, including powerful X-rays, a million times more intense than a transmission X-ray used in medical imaging.
"In fluorescence it's like looking at the stars at night whereas in transmission it's like looking during the day," explains Dr Uwe Bergmann of the Stanford Synchrotron Radiation Lab in the US, where the work is being done.
The light enables scientists to look inside matter at the molecular and atomic scale.
Glowing words
The technique is particularly useful for probing the palimpsest because the ink used by the scribe to record Archimedes' work contains iron.
"When the X-rays hit an iron atom it emits a characteristic radiation, it glows," says Dr Bergmann. "When you record the glow you can reconstruct an image of all of the iron in the book."
The glowing words are displayed on a computer screen, giving the researchers the first glimpse of the text in nearly 800 years.
"It's like receiving a fax from the 3rd Century BC," said Mr Noel. "It's the most sensational feeling."
Each page takes 12 hours to reconstruct as the highly focused beam of X-rays, the width of a human hair, sweeps across the page.
The team have until the 7 August this year to scrutinise the palimpsest, before the synchrotron is switched off for maintenance.
During that time they hope to scan between 12 and 14 pages, paying particular attention to the areas covered with the forged paintings.
The public can watch the researchers as they reveal the glowing ancient text during a live webcast at 2300 GMT on 4 August.

Story from BBC NEWS:
http://news.bbc.co.uk/go/pr/fr/-/2/hi/science/nature/5235894.stm
Published: 2006/08/02 15:01:16 GMT X-rays reveal Archimedes secrets