26/08/06, Siobhan Roberts, An equation for controversy
Post
With $1-million at stake, the Poincaré conjecture is fuelling competition -- and ethical conflict; Siobhan Roberts reports on the reclusive genius at the centre of it all.
Many a mathematician's richest fantasies were realized last week when:
1.1. The Poincaré conjecture, described as "one of the most burning mathematical questions of all time," was declared almost certainly solved. And:
1.2. This Poincaré plot, full of intrigue, prizes and power plays, transformed the normally anonymous calculations of mathematicians into fodder for headlines worldwide.
The central character in this drama, however, was less than a willing participant. Despite winning the Fields Medal on Tuesday for his work on the Poincaré, reclusive Russian mathematician Grigory Perelman not only declined to attend the International Congress of Mathematicians in Madrid to accept his prize -- from the King of Spain no less -- he refused the medal altogether.
The shy, though obstinate 40-year-old is the first person to reject what is considered the Nobel of math. And, according to an article in The New Yorker, he also resigned from the Steklov Mathematics Institute in St. Petersburg last December, quitting mathematics altogether. His objections: the community's questionable ethics and unseemly politicking around his work.
All of which has raised speculation as to whether Dr. Perelman will also turn down the $1-million (U.S.) Millennium Prize offered by the Clay Mathematics Institute (a think tank in Cambridge, Mass.) for the first person to prove the century-old Poincaré. Or, for that matter, whether any one mathematician in a supremely collaborative field can really claim credit at all.
The Poincaré plot is particularly fraught in this respect, because Dr. Perelman provided only a rough sketch of his strategy to solve the conjecture. He posted two papers on an on-line "preprint" archive, and then ostensibly abandoned his work, deliberately leaving it to stand or fall on its own merit. Three camps of mathematicians rushed in to flesh out the details -- so the question then becomes who really provided a comprehensive and complete proof?
Sir John Ball, the president of the International Mathematical Union, made a not-so-veiled reference to this in his opening address at the ICM. "We freely discuss our work with others without fear of it being stolen, and research is communicated openly prior to formal publication," he said. "Editorial procedures are fair and proper, and work gains its reputation through merit and not by how it is promoted. . . . The exceptions are rare, and they are noticed."
The most notable exception in the Poincaré imbroglio is Chinese mathematician Shing-Tung Yau. Now at Harvard, the 57-year-old is known in the math world for being unpleasantly competitive -- a trait that can be attributed in part to his zeal for building the mathematical enterprise in China.
Dr. Yau is a celebrated mathematician, for a theorem uniting topology and geometry, and seminal work pertaining to string theory. He won a Fields medal in 1982. He has dabbled with the Poincaré since the 1970s, and two of his former students wrote one of the papers explicating Dr. Perelman's results.
But lately, Dr. Yau has been manoeuvring for more photons of the limelight than he deserves, promoting his students' work as an original contribution, rather than a compliment to Dr. Perelman's insights.
In contrast, the purist Dr. Perelman -- who makes quite an impression with nails that are apparently inches long -- is described as living on an "ideal plane," governed, like the discipline of mathematics, by absolute axioms.
Rather than chasing renown, he treasures his privacy and shuns material trappings. In the 1990s, he refused a prize from the European Mathematical Society, and the 6,000 euros that went with it. The current monetary value of the Fields is $13,000 (U.S.).
With such a tempestuous plot, the Poincaré excitement at the ICM was scarcely diminished by Dr. Perelman's repudiation of his accolades. More than 3,000 of the world's mathematicians are congregating and cogitating through the middle of next week, with one main item on the itinerary: a "verdict" on whether Dr. Perelman's proposed solution to the Poincaré is, in fact, correct.
As stated by the ICM, the event "will in all probability go down in history as the occasion when this conjecture became a theorem, a term used by mathematicians to refer to a demonstrated hypothesis."
Frenchman Henri Poincaré -- one of the great mathematicians and an eloquent popularizer of his field -- formulated his namesake conjecture in 1904.
As described by Canadian Donal O'Shea, a math professor and dean at Mount Holyoke College who is finishing a book on the subject, the Poincaré conjecture "is a bold guess about nothing less than the shape of our universe.
"The problem of proving, or disproving, it," he writes, "has exerted a siren-like pull on mathematicians."
The Poincaré conjecture states that the three-dimensional sphere is the only three-dimensional space that does not go on forever, and does not have any walls, and in which you can shrink every loop to a point.
"It's probably the simplest question you could ask about these three-dimensional shapes, but just the fact that it is not known has driven everybody crazy," Dr. O'Shea says, adding that, surprisingly, it has been proved in every higher dimension.
In this sense, the road to Dr. Perelman's proof of the Poincaré has been a cumulative and collaborative venture. Stephen Smale, at the Toyota Technological Institute in Chicago, proved the conjecture held true in five or more dimensions in 1960. Michael Freedman, at Microsoft, proved it was true for four dimensions in 1983. And in the 1970s, William Thurston, now at Cornell, wrote the broader "geometrization conjecture" that encompassed Poincaré.
All three mathematicians subsequently won the Fields -- though none succeeded in proving the elusive Poincaré itself. "The Poincaré conjecture has been a trap, a tar pit, for so many good mathematicians," Dr. Thurston says.
Or, as an ICM press release states, "It has been one of the problems that has consumed the most 'brain power' in mathematics, to such an extent that the issue has been referred to as 'Poincaritis,' a species of contagion affecting those who have spent decades in an attempt to resolve the problem."
Richard Hamilton also has spent decades on the problem. In 1981, the Columbia professor developed a mathematical concept called Ricci flow, treating geometric manifolds in a manner that is analogous to the diffusion of heat. This was the main tool Dr. Perelman used to get his results, and for his crucial contribution to the venture Dr. Hamilton been praised as a "hero."
Dr. Perelman began his work on the Poincaré in the 1990s. He was already well known for his perfect score at the age of 16 at the 1982 International Mathematical Olympiad and for solving a big geometry theorem called the Soul conjecture. After a stint at Berkeley in the early 1990s, he was offered professorships at Princeton and Stanford.
But he rebuffed these opportunities, and in 1995 returned to St. Petersburg, where he lives in an apartment with his mother and has few friends -- the perfect equation for withdrawing into one's head and communing only with mathematics. According to ICM literature, Dr. Perelman "sequestered himself away to tackle the problem, and for the following eight years gave no signs of life."
In November, 2002, he resurfaced and announced his results in a sketchy "preprint" paper posted on the on-line math archive arXiv.org, followed by another in March, 2003.
"There was a buzz all over about the papers," Dr. O'Shea says. "But a ton of caution too. There have been so many wrong proofs of the Poincaré conjecture by very good mathematicians that everybody had been feeling a little burnt."
After Dr. Perelman posted his papers, he was eager to discuss his results and accepted invitations to deliver a series of talks in the U.S. But shortly thereafter he again dropped out of sight. It soon became apparent that he had no plans to publish his results in a refereed journal -- a necessary step to qualify for the Millennium Prize.
As Dr. Perelman bowed out, other mathematicians took up the task of expounding upon his work, transforming his rough papers into comprehensive treatments.
These efforts, which have almost certainly confirmed that yes indeed Dr. Perelman has a complete and correct proof, came to fruition in rapid-fire succession in May, June and July -- just in time for the ICM.
Interestingly, though Dr. Perelman's two papers amount to 61 pages, a book by Columbia's John Morgan and Princeton's Gang Tian (to be published by the Clay Institute) is 474 pages; "Notes on Perelman's Papers," by Bruce Kleiner of Yale and John Lott of the University of Michigan, is 192 pages; and the work of Huai-Dong Cao, at Lehigh in Bethlehem, Pa., and Xi-Ping Zhu, at Zhongshan, in Guangzhou, China, is 328 pages.
"It's a funny role," Dr. Morgan says of the nearly two years he has spent verifying Dr. Perelman's proposal. "One doesn't usually spend so much time writing out somebody else's ideas, for obvious reasons. But I was sufficiently interested . . . And so between that, and doing some public service for the community, those were my main motivations to get involved."
The motivations of others have come to seem less noble. Dr. Morgan's co-author, Dr. Tian, is a former student of Dr. Yau -- and now his rival for status as the pre-eminent mathematician in China.
Dr. Cao and Dr. Zhu, authors of the 328-page paper, touted in its title as "the complete proof," are also former students of Dr. Yau. They executed their Poincaré work under his wing and in the run-up to the ICM, Dr. Yau promoted their work vigorously, at times seeming to suggest they had done more than a mere verification.
Practitioners familiar with Dr. Perelman's proof dispute that the Yau-Cao-Zhu treatment contributed anything of originality. "There have been a lot of major advances from many people [on the Poincaré]. And they kind of tie together. This is nothing unusual. It's how all of science works," Dr. Thurston says.
Dr. Perelman's results, on the other hand, were described in his Fields citation as "revolutionary." And, at a lecture Thursday, Dr. Morgan stated unequivocally that Dr. Perelman had solved the Poincaré.
"Dr. Perelman is after far more, and has seen far deeper, than Dr. Hamilton or Dr. Yau," Dr. O'Shea says.
As for the Millennium Prize, contenders will now undergo a two-year waiting period, to ensure all papers receive intensive scrutiny by the mathematics community.
And although this Poincaré proof was a collective effort that evolved over a century, James Carlson, president of the Clay, comments that "anyone involved in this will regard Hamilton and Perelman as the principal players."
For his part, Dr. Perelman says he will not decide whether to accept the $1-million Millennium Prize until it is offered.
Siobhan Roberts is the author of the upcoming King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry.
Many a mathematician's richest fantasies were realized last week when:
1.1. The Poincaré conjecture, described as "one of the most burning mathematical questions of all time," was declared almost certainly solved. And:
1.2. This Poincaré plot, full of intrigue, prizes and power plays, transformed the normally anonymous calculations of mathematicians into fodder for headlines worldwide.
The central character in this drama, however, was less than a willing participant. Despite winning the Fields Medal on Tuesday for his work on the Poincaré, reclusive Russian mathematician Grigory Perelman not only declined to attend the International Congress of Mathematicians in Madrid to accept his prize -- from the King of Spain no less -- he refused the medal altogether.
The shy, though obstinate 40-year-old is the first person to reject what is considered the Nobel of math. And, according to an article in The New Yorker, he also resigned from the Steklov Mathematics Institute in St. Petersburg last December, quitting mathematics altogether. His objections: the community's questionable ethics and unseemly politicking around his work.
All of which has raised speculation as to whether Dr. Perelman will also turn down the $1-million (U.S.) Millennium Prize offered by the Clay Mathematics Institute (a think tank in Cambridge, Mass.) for the first person to prove the century-old Poincaré. Or, for that matter, whether any one mathematician in a supremely collaborative field can really claim credit at all.
The Poincaré plot is particularly fraught in this respect, because Dr. Perelman provided only a rough sketch of his strategy to solve the conjecture. He posted two papers on an on-line "preprint" archive, and then ostensibly abandoned his work, deliberately leaving it to stand or fall on its own merit. Three camps of mathematicians rushed in to flesh out the details -- so the question then becomes who really provided a comprehensive and complete proof?
Sir John Ball, the president of the International Mathematical Union, made a not-so-veiled reference to this in his opening address at the ICM. "We freely discuss our work with others without fear of it being stolen, and research is communicated openly prior to formal publication," he said. "Editorial procedures are fair and proper, and work gains its reputation through merit and not by how it is promoted. . . . The exceptions are rare, and they are noticed."
The most notable exception in the Poincaré imbroglio is Chinese mathematician Shing-Tung Yau. Now at Harvard, the 57-year-old is known in the math world for being unpleasantly competitive -- a trait that can be attributed in part to his zeal for building the mathematical enterprise in China.
Dr. Yau is a celebrated mathematician, for a theorem uniting topology and geometry, and seminal work pertaining to string theory. He won a Fields medal in 1982. He has dabbled with the Poincaré since the 1970s, and two of his former students wrote one of the papers explicating Dr. Perelman's results.
But lately, Dr. Yau has been manoeuvring for more photons of the limelight than he deserves, promoting his students' work as an original contribution, rather than a compliment to Dr. Perelman's insights.
In contrast, the purist Dr. Perelman -- who makes quite an impression with nails that are apparently inches long -- is described as living on an "ideal plane," governed, like the discipline of mathematics, by absolute axioms.
Rather than chasing renown, he treasures his privacy and shuns material trappings. In the 1990s, he refused a prize from the European Mathematical Society, and the 6,000 euros that went with it. The current monetary value of the Fields is $13,000 (U.S.).
With such a tempestuous plot, the Poincaré excitement at the ICM was scarcely diminished by Dr. Perelman's repudiation of his accolades. More than 3,000 of the world's mathematicians are congregating and cogitating through the middle of next week, with one main item on the itinerary: a "verdict" on whether Dr. Perelman's proposed solution to the Poincaré is, in fact, correct.
As stated by the ICM, the event "will in all probability go down in history as the occasion when this conjecture became a theorem, a term used by mathematicians to refer to a demonstrated hypothesis."
Frenchman Henri Poincaré -- one of the great mathematicians and an eloquent popularizer of his field -- formulated his namesake conjecture in 1904.
As described by Canadian Donal O'Shea, a math professor and dean at Mount Holyoke College who is finishing a book on the subject, the Poincaré conjecture "is a bold guess about nothing less than the shape of our universe.
"The problem of proving, or disproving, it," he writes, "has exerted a siren-like pull on mathematicians."
The Poincaré conjecture states that the three-dimensional sphere is the only three-dimensional space that does not go on forever, and does not have any walls, and in which you can shrink every loop to a point.
"It's probably the simplest question you could ask about these three-dimensional shapes, but just the fact that it is not known has driven everybody crazy," Dr. O'Shea says, adding that, surprisingly, it has been proved in every higher dimension.
In this sense, the road to Dr. Perelman's proof of the Poincaré has been a cumulative and collaborative venture. Stephen Smale, at the Toyota Technological Institute in Chicago, proved the conjecture held true in five or more dimensions in 1960. Michael Freedman, at Microsoft, proved it was true for four dimensions in 1983. And in the 1970s, William Thurston, now at Cornell, wrote the broader "geometrization conjecture" that encompassed Poincaré.
All three mathematicians subsequently won the Fields -- though none succeeded in proving the elusive Poincaré itself. "The Poincaré conjecture has been a trap, a tar pit, for so many good mathematicians," Dr. Thurston says.
Or, as an ICM press release states, "It has been one of the problems that has consumed the most 'brain power' in mathematics, to such an extent that the issue has been referred to as 'Poincaritis,' a species of contagion affecting those who have spent decades in an attempt to resolve the problem."
Richard Hamilton also has spent decades on the problem. In 1981, the Columbia professor developed a mathematical concept called Ricci flow, treating geometric manifolds in a manner that is analogous to the diffusion of heat. This was the main tool Dr. Perelman used to get his results, and for his crucial contribution to the venture Dr. Hamilton been praised as a "hero."
Dr. Perelman began his work on the Poincaré in the 1990s. He was already well known for his perfect score at the age of 16 at the 1982 International Mathematical Olympiad and for solving a big geometry theorem called the Soul conjecture. After a stint at Berkeley in the early 1990s, he was offered professorships at Princeton and Stanford.
But he rebuffed these opportunities, and in 1995 returned to St. Petersburg, where he lives in an apartment with his mother and has few friends -- the perfect equation for withdrawing into one's head and communing only with mathematics. According to ICM literature, Dr. Perelman "sequestered himself away to tackle the problem, and for the following eight years gave no signs of life."
In November, 2002, he resurfaced and announced his results in a sketchy "preprint" paper posted on the on-line math archive arXiv.org, followed by another in March, 2003.
"There was a buzz all over about the papers," Dr. O'Shea says. "But a ton of caution too. There have been so many wrong proofs of the Poincaré conjecture by very good mathematicians that everybody had been feeling a little burnt."
After Dr. Perelman posted his papers, he was eager to discuss his results and accepted invitations to deliver a series of talks in the U.S. But shortly thereafter he again dropped out of sight. It soon became apparent that he had no plans to publish his results in a refereed journal -- a necessary step to qualify for the Millennium Prize.
As Dr. Perelman bowed out, other mathematicians took up the task of expounding upon his work, transforming his rough papers into comprehensive treatments.
These efforts, which have almost certainly confirmed that yes indeed Dr. Perelman has a complete and correct proof, came to fruition in rapid-fire succession in May, June and July -- just in time for the ICM.
Interestingly, though Dr. Perelman's two papers amount to 61 pages, a book by Columbia's John Morgan and Princeton's Gang Tian (to be published by the Clay Institute) is 474 pages; "Notes on Perelman's Papers," by Bruce Kleiner of Yale and John Lott of the University of Michigan, is 192 pages; and the work of Huai-Dong Cao, at Lehigh in Bethlehem, Pa., and Xi-Ping Zhu, at Zhongshan, in Guangzhou, China, is 328 pages.
"It's a funny role," Dr. Morgan says of the nearly two years he has spent verifying Dr. Perelman's proposal. "One doesn't usually spend so much time writing out somebody else's ideas, for obvious reasons. But I was sufficiently interested . . . And so between that, and doing some public service for the community, those were my main motivations to get involved."
The motivations of others have come to seem less noble. Dr. Morgan's co-author, Dr. Tian, is a former student of Dr. Yau -- and now his rival for status as the pre-eminent mathematician in China.
Dr. Cao and Dr. Zhu, authors of the 328-page paper, touted in its title as "the complete proof," are also former students of Dr. Yau. They executed their Poincaré work under his wing and in the run-up to the ICM, Dr. Yau promoted their work vigorously, at times seeming to suggest they had done more than a mere verification.
Practitioners familiar with Dr. Perelman's proof dispute that the Yau-Cao-Zhu treatment contributed anything of originality. "There have been a lot of major advances from many people [on the Poincaré]. And they kind of tie together. This is nothing unusual. It's how all of science works," Dr. Thurston says.
Dr. Perelman's results, on the other hand, were described in his Fields citation as "revolutionary." And, at a lecture Thursday, Dr. Morgan stated unequivocally that Dr. Perelman had solved the Poincaré.
"Dr. Perelman is after far more, and has seen far deeper, than Dr. Hamilton or Dr. Yau," Dr. O'Shea says.
As for the Millennium Prize, contenders will now undergo a two-year waiting period, to ensure all papers receive intensive scrutiny by the mathematics community.
And although this Poincaré proof was a collective effort that evolved over a century, James Carlson, president of the Clay, comments that "anyone involved in this will regard Hamilton and Perelman as the principal players."
For his part, Dr. Perelman says he will not decide whether to accept the $1-million Millennium Prize until it is offered.
Siobhan Roberts is the author of the upcoming King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry.
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