Perelman explains proof to famous math mystery

Nobel Laureate John Nash GS '50 sat in the fifth row of Taplin Auditorium yesterday afternoon. Andrew Wiles, the man who proved Fermat's Last Theorem 10 years ago, sat two rows closer. All told, more than 100 mathematicians from three generations gathered to listen to Dr. Grigori Perelman describe his potentially groundbreaking work, which may earn him a share of a $1 million prize from the Clay Mathematics Institute for solving one of the "Millennium Prize Problems," the seven most difficult outstanding problems in mathematics.

For eight years now, The New York Times reported, Perelman has been working alone in Russia developing new techniques to describe geometry in higher dimensions.

While the $1 million prize was offered for the proof of a specific theorem, known as the Poincaré Conjecture, Perelman's work, if verified, is much broader and has far more powerful implications.

"He's not facing Poincaré directly, he's just trying to do this grander scheme," said mathematics professor Peter Sarnak. After creating so much new mathematics, the Poincaré result is just "a million dollar afterthought," he said.

He has essentially pitched a perfect game when the prize was offered for a single strikeout.

While Perelman did not discuss the implications of his results yesterday, he did go into detail about several technical advances he made in understanding complex topological techniques.

Topology is the branch of mathematics that studies properties of surfaces that are independent of stretching and bending without tearing the surface.

For example, inflated and deflated versions of a basketball are topologically similar. However, whether inflated or deflated, the ball is fundamentally different from a donut, because of the donut's central hole. Both of these are examples of what topologists would call two-dimensional surfaces.

In fact, there are infinitely many distinct two-dimensional surfaces. The ball is different from the donut is different from a two-holed donut is different from a three-holed donut, ad infinitum.

But this is not the case in higher dimensions. As was first suggested by former University mathematics professor William Thurston, when dealing with three-dimensional surfaces, there are only eight shapes that are topologically distinct. This statement, known as the Geometrization Conjecture in three dimensions, is currently unproven.

But if Perelman's work survives the peer review process, all that will change.

"He's obviously made a major breakthrough," Sarnak said. But he cautioned that the devil is in the details. While "he hasn't made claims he couldn't substantiate before, mistakes can be in very delicate places," Sarnak said.

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According to Sarnak, Perelman's most important result may be his work with Ricci flow, a technique that essentially "smoothes out" surfaces in such a way that they evolve into their simplest possible shape.

"I do not think the million dollars is the motivation," said mathematics professor Sun-Yung Alice Chang. The Poincaré Conjecture is "in the same scale as Fermat's Last Theorem. [Proving] it puts you in the history of mathematics; the dream of every mathematician."