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Has The Poincare Conjecture Been Solved?
Zack Coburn writes "An article in the Boston Globe alludes to the Poincare Conjecture being solved, possibly. For those who are unfamiliar with the conjecture, the article gives a brief description: "To solve it, one would have to prove something that no one seriously doubts: that, just as there is only one way to bend a two-dimensional plane into a shape without holes -- the sphere -- there is likewise only one way to bend three-dimensional space into a shape that has no holes. Though abstract, the conjecture has powerful practical implications: Solve it and you may be able to describe the shape of the universe." Apparently Grigory Perelman may have proved it, which would mean a $1 million award from the Clay Mathematics Institute." We've previously discussed other possible Poincare proofs.
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BERKELEY, Calif. -- A reclusive Russian mathematician appears to have answered a question that has stumped mathematicians for more than a century.
After a decade of isolation in St. Petersburg, over the last year Grigory Perelman posted a few papers to an online archive. Although he has no known plans to publish them, his work has sent shock waves through what is usually a quiet field.
At two conferences held during the last two weeks in California, a range of specialists scrutinized Perelman's work, trying to grasp all the details and look for potential flaws.
If Perelman really has proved the so-called Poincare Conjecture, as many believe he has, he will become known as one of the great mathematicians of the 21st century and will be first in line for a $1 million prize offered by the Clay Mathematics Institute in Cambridge.
Colleagues say Perelman, who did not attend the California conferences and did not respond to a request for comment, couldn't care less about the money, and doesn't want the attention. Known for his single-minded devotion to research, he seldom appears in public; he answers e-mails from mathematicians, but no one else.
"What mathematicians enjoy is the chase of really difficult problems," said Hyam Rubinstein, a mathematician who came from Australia to attend meetings at the Mathematical Sciences Research Institute in Berkeley and the American Institute of Mathematics in Palo Alto, Calif., hoping to better understand Perelman's solution. "This problem is like the Mount Everest of math conjectures, so everyone wants to be the first to climb it."
The Poincare Conjecture, named after the Frenchman who proposed it in 1904, is the question that essentially founded the field of topology, the "rubber-sheet geometry" that looks at the properties of surfaces that don't change no matter how much you stretch or bend them.
To solve it, one would have to prove something that no one seriously doubts: that, just as there is only one way to bend a two-dimensional plane into a shape without holes -- the sphere -- there is likewise only one way to bend three-dimensional space into a shape that has no holes. Though abstract, the conjecture has powerful practical implications: Solve it and you may be able to describe the shape of the universe.
Dozens of the best mathematicians of the last century tried with all kinds of approaches to solve the conjecture. Some thought they had it for months, even years, but counter-examples and flaws just kept springing up. Simply-stated but elusive to prove -- like Fermat's Last Theorem -- this conjecture has spurred the development of whole branches of mathematics.
A decade ago, after some work in the United States that colleagues described as "brilliant," Perelman gave up a promising career to work in seclusion in St. Petersburg. Although he appears occasionally, most recently for lectures at the Massachusetts Institute of Technology and several other US schools last spring, he keeps a very low profile.
Even in mathematical circles, surprisingly little is known about him, and those who know him often don't want to speak publicly about his work.
At any rate, he seems to have used his time alone wisely. While working out the Poincare Conjecture, Perelman also seems to have established a much stronger result, one that could change many branches of mathematics. Called the "Geometrization Conjecture," it is a far-reaching claim that joins topology and geometry, by stating that all space-like structures can be divided into parts, each of which can be described by one of three kinds of simple geometric models. Like a similar result for surfaces proved a century ago, this would have profound consequences in almost all areas of mathematics.
As the foundation for his proof, Perelman used a method called Ricci flow, invented in the mid-1980s by Columbia University mathematician Richard Hamilton, which breaks a surface into parts and smooths these parts out, making
There is more coverage at the Smegma Research Institute's information page