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The Poincare conjecture

Russian may have solved great math mystery

Wednesday, January 7, 2004 Posted: 11:18 PM EST (0418 GMT)

From: http://www.cnn.com/2004/US/West/01/07/math.mystery.ap/index.html

The 100-year-old problem seeks to explain the geometry of three-dimensional space.

SAN FRANCISCO, California (AP) -- A publicity-shy Russian researcher who labors in near-seclusion may have solved one of mathematics' oldest and most abstruse problems, the Poincare Conjecture.

Evidence has been mounting since November 2002 that Grigori "Grisha" Perelman has cracked the 100-year-old problem, which seeks to explain the geometry of three-dimensional space.

If Perelman succeeded, he could be eligible for a $1 million prize offered by the Cambridge, Massachusetts-based Clay Mathematics Institute, formed to identify the world's seven toughest math problems.

Mathematicians around the world have been checking Perelman's work in search of the kind of flaws that have sunk the many other supposed solutions to a problem first presented by the French mathematician Henri Poincare in 1904.

"This is arguably the most famous unsolved problem in math and has been for some time," said Bruce Kleiner, a University of Michigan math professor reviewing Perelman's work.

Perelman's work has advanced the furthest without falling apart, and there is optimism that it will ultimately hold up.

"I don't think there's been a single example of a proof that has gotten this much attention and has withstood the scrutiny as it has so far," Kleiner said.

Not since Princeton University researcher Andrew Wiles cracked the 350-year-old Fermat's Last Theorem a decade ago has the math world been so consumed with one problem.

Perelman is a researcher at St. Peterburg-based Steklov Institute of Mathematics of the Russian Academy. Colleagues describe him as brilliant and say he spent his formative years in the United States, then spent eight years quietly working in Russia without publishing any of his works in science journals.

Whether he attempts to collect the prize money is as much a mystery as the Poincare Conjecture itself. He did not respond to an e-mail query from The Associated Press and has declined interviews with other media in the past.

The institute's rules state that to collect on a proof, winners must publish their work in a science journal and withstand two years of scrutiny afterward.

Though Perelman emerged from relative seclusion last year and gave lectures to math experts at various U.S. colleges, he appears uninterested in submitting his work to a journal and has not openly discussed the prize money. He has instead posted three papers and corresponding data on a Web site.

James Carlson, the institute's president, said that since Perelman's work is undergoing, in effect, a peer review by the world's brightest math minds, he may yet qualify for the prize.

Math experts are confident they will soon be able to decide definitively if Perelman has solved the problem. They are analyzing his use of such esoteric concepts as the "Ricci flow," "modulo diffeomorphism" and "maximal horns."

"They are very complicated papers and there are so many moving parts to them," said Columbia University math professor John Morgan. "It's very easy to slip up a little bit. It's a long process."

The Poincare Conjecture is a highly abstract problem that only the most gifted math wizards love and truly understand.

Poincare made strides in understanding three-dimensional spaces -- the kind, for instance, that an airplane flies through, made up of north-south, east-west and up-down measurements. His question, or conjecture, was whether two-dimensional calculations could be easily modified to answer similar questions about 3-D spaces. He was pretty sure the answer was yes but could not prove it mathematically.

Answering the question may help scientists better understand the shape of the universe. Beyond that, it may have no application to everyday life.

There have been numerous "solutions" to the Poincare Conjecture that have ultimately failed. Two years ago, Martin Dunwoody of Southampton University in England caused a sensation when he posted his six-page proposed solution on a university Web site. Within months, Dunwoody's proposal was shot down.