Monday, July 24, 2006

3D manifolds classified

The WSJ reports:

Major Math Problem Is Believed Solved By Reclusive Russian
July 21, 2006; Page A9

For six years, $7 million in prize money has lay unclaimed at the Clay Mathematics Institute in Cambridge, Mass., waiting for someone to solve any of the seven "millennium prize problems," the oldest of which has been kicking around since 1859. Despite periodic claims, it looked like the institute would hold on to the cash until after the sun burned out.

But the math world is abuzz over the very real possibility that one millennium problem, the Poincaré conjecture, has been proved by a mathematician in Russia. After nearly four years of scrutiny by other mathematicians, the work holds up, even though Grigori Perelman's work is decidedly unusual.

In 2002 and 2003, he posted two papers to an online archive. Usually, a posting serves a flag-planting function -- "I solved this first!" -- until the paper is published in a journal, which can take years. But as the math community waited for him to follow up his postings, a realization set in. Dr. Perelman, long affiliated with the Steklov Institute of Mathematics in St. Petersburg, apparently has no intention of saying more. He probably feels he proved the Poincaré conjecture, mathematicians surmise, and has no interest in the $1 million bounty. (He did not respond to emailed requests for comment.)

Dr. Perelman's style is reminiscent of the Sid Harris cartoon of a board filled with equations and, at a key step, the words, "then a miracle occurs." One mathematician tells the other, "I think you should be more explicit here in step two."

The conjecture Henri Poincaré posited in 1904 is the most famous problem in topology, the branch of math that analyzes the shape of objects and space. He claimed, "if a closed 3-dimensional manifold has trivial fundamental group, [it must be] homeomorphic to the 3-sphere," as John Milnor of Stony Brook University puts it.

Translated, that means that if you wrap one rubber band around the surface of an orange and another around a doughnut, and shrink down both, the rubber bands act differently. The one around the orange keeps shrinking without tearing or leaving the surface. The one around the doughnut can't, without breaking itself or the doughnut. This difference says something profound about the structure of space itself.

Many mathematicians have claimed to prove Poincaré, but the claims flamed out immediately, their fatal flaws obvious. Dr. Perelman's proof has survived. The dilemma for the Clay Institute is that, according to its rules, a proof must be published in a refereed math publication. The archives aren't refereed.

Putting his proof online rather than in a journal is only one example of Dr. Perelman's iconoclasm. He admits that he gives only "a sketch of an eclectic proof of" a more general conjecture from which Poincaré's follows; he never mentions Poincaré. The papers are difficult to understand, and sketchy in the extreme. He asserts that one can prove something by a variation on an earlier argument, but it isn't clear what the variation is. "Perelman's papers are written in a style rather different from what would appear in a journal," says mathematician Bruce Kleiner of Yale University.

The sketchiness may reflect how a genius interacts with mortals. Dr. Perelman may believe some things are so obvious he needn't bother to explain them step by step, say mathematicians. If readers are too dumb to fill in the blanks, he doesn't care. Or, he has better things to do than justify every tortuous step, as proofs must.

Others have taken it upon themselves to explicate his work -- and find no major flaws. Like Torah commentaries, they dwarf the original. Dr. Perelman's 2003 paper is 22 pdf pages; the 2002 paper is 39. But "Notes on Perelman's Papers," in which Prof. Kleiner and John Lott of the University of Michigan explain them almost line-by-line, is 192 pages. A book on the papers is expected to top 300 pages. A "complete proof" of Poincaré, based on Dr. Perelman's breakthrough and published last month in the Asian Journal of Mathematics (which Prof. Milnor describes as throwing "a monkey wrench" into the question of who gets credit), is 328 pages long.

Oddly, either the book or the Kleiner-Lott paper might count as the "refereed" work the Clay Institute demands. If so, we would have the weird situation in which authors of the work that satisfies the prize requirement aren't the people who figured out the proof. But their efforts could win Dr. Perelman $1 million.

"It's definitely an unusual situation, but what's important is that the person who made the breakthrough put it out there so the community could scrutinize and analyze it," says institute president, James Carlson.

Dr. Perelman shuns the limelight, but is known through lectures in the U.S. and for getting a perfect score at the 1982 International Mathematical Olympiad, at age 16. He isn't expected at the quadrennial meeting of the International Congress of Mathematicians, in Madrid. There, the Fields Medal, math's Nobel Prize, will be awarded to the "outstanding" mathematician 40 or under. Dr. Perelman is the odds-on favorite.

And the millennium prizes? "I don't think the other six will be solved in my lifetime," says Dr. Carlson. "But then, I didn't think the Poincaré conjecture would be solved either."
Now that the proof has been peer-reviewed, Perelman should get his million-dollar check in a year.

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