Math takes Science’s spotlight in 2006

Alan Boyle Science editor • Profile • E-mail

A controversial proof of a 102-year-old mathematical puzzler has taken the top spot on the journal Science's annual list of scientific breakthroughs.

Other entries on the top 10 list ranged from the deciphering of Neanderthal DNA to studies of the world's shrinking ice sheets. But it was Russian mathematician Grigory Perelman's proof of the Poincare Conjecture, literally one of the field's million-dollar challenges, that most impressed America's premier peer-reviewed science publication.

For mathematicians, Perelman's achievement would qualify "at least as the Breakthrough of the Decade," Science's editors said.

Science's editor-in-chief, Donald Kennedy, admitted that mathematical papers don't often get the star treatment, "partly because higher mathematics is a subject that’s technically difficult to explain."

"It's difficult even for this poor biologist to understand fully," Kennedy told MSNBC.com.

However, the Russian mathematician's work on the Poincare Conjecture stands out for several reasons. First of all, Kennedy noted that the conjecture has "defeated many brilliant minds" since it was first proposed in 1904 by Henri Poincare, who is generally regarded as the founder of topology.

Rubber-sheet geometry
Topology, also known as "rubber-sheet geometry," focuses on the properties of surfaces that are preserved despite any amount of stretching or poking. To a topologist, there's no difference between a doughnut and a coffee cup, because they both have exactly one hole poking through their surfaces. But there's a big difference between a beach ball (no holes) and an inner tube (one hole).

In simple terms, Poincare proposed that there was no way to transform a no-hole surface into a one-hole surface without somehow tearing it, and that any no-hole surface can be stretched into a sphere. By the early 1980s, mathematicians were able to prove that the conjecture held true for any dimensional space — except for three-dimensional space.

To get to that last step, Perelman created a fresh mathematical vocabulary to describe the problem, then broke it down into steps that could be addressed one by one.

Science's editors said Perelman's hundreds of pages of analysis addressed the "indigestible seed at the core of topology," and could open the way to much broader breakthroughs, such as a "periodic table" of three-dimensional spaces that could do for mathematicians what the periodic table of the elements has done for chemists.

Adding to the buzz
The math alone made Perelman's achievement notable, and the circumstances surrounding his proof only added to the buzz: Starting in 2002, Perelman released the proof as a series of three papers, posted freely on the Internet for review. Other mathematicians helped fill in gaps in the step-by-step argument — and this year, the International Mathematical Union finally decided to give Perelman its highest honor, the Fields Medal.

Perelman created a stir when he refused the medal, reportedly saying that he felt isolated from the rest of the mathematical community and did not want to be seen as its "figurehead." An article about the controversy in The New Yorker added to the mystery — with Perelman complaining about lapses in the "ethical standards" of his colleagues, and other mathematicians squabbling over credit for the proof.

If the proof withstands scrutiny for another couple of years, Perelman could become eligible for a $1 million prize from the Clay Mathematics Institute — though it's not clear whether he would accept that prize, either. Meanwhile, the controversy continues to simmer. "It's clear that mathematicians regard claims as something to attack," Science's Kennedy said.

As for the practical applications of the Poincare proof, Kennedy acknowledged that there weren't any in sight. "I'm tempted to say the same thing about a lot of the exciting cosmology that has been on the menu on the last five years," he said. "We don’t have a practical application for dark energy either. But it's important for our constant probing of our understanding of the world."

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