The shape of things

January 13, 2004

IN THE WORLD of topology, the science of surfaces, the difference between a doughnut and an apple can be explained with a rubber band. The rubber band can be removed from an apple without breaking; that's not the case if it is wrapped in, around and through a doughnut. That's why the surface of an apple is "simply connected" and the doughnut's is not.

Are you with us?

About 100 years ago, the French mathematician Jules Henri PoincarM-i wondered if this same property of connectivity pertained to the geometry of a three-dimensional space, such as the north-south-east-west-up-down area through which an airplane flies. His conjecture spawned a century of mind-bending, head-scratching geometric acrobatics and one doozy of a math problem. It has confounded the best and the brightest for decades - until now. Grigori Perelman, a scientist at the Steklov Institute of Mathematics of the Russian Academy of St. Petersburg, claims he's solved the PoincarM-i Conjecture.

A million dollars is riding on the solution to the century-old problem, prize money offered by the Clay Mathematics Institute of Cambridge, Mass. Solving the PoincarM-i Conjecture is right up there with decoding the structure of the DNA molecule. It's as difficult as, say, putting a man on Mars. If it's taken eight years for Mr. Perelman, working mostly in isolation, to prove PoincarM-i's Conjecture, mathematicians across the country have spent the past two years scrutinizing his proof, which builds on abstruse concepts with obscure names such as "Ricci flow," "modulo diffeomorphism" and "maximal horns."

But besides making one math-manic Russian flush with rubles, why should anyone care about solving a dead Frenchman's query? For one, it's a conquest of monumental proportions in a field where few excel. Think big. Bigger than the Gordian knot, Fermat's Last Theorem, the phone bill.

In a country whose eighth-graders on average perform worse in math than their peers in Latvia, Canada, Malaysia and Bulgaria, Americans should pay attention. By 12th grade, American kids perform even more poorly - the only students they outperform are those from Cyprus and South Africa. Most Americans probably couldn't name a famous contemporary mathematician, even though U.S. scholars have been at the forefront of receiving the Fields Medal, the Nobel equivalent in math.

In 50 years, there may be a half-dozen major advances in math and science, and solving the PoincarM-i Conjecture would be one. If Mr. Perelman's work proves conclusive, researchers say he will have gone further than any other human on the planet in understanding the shape of the universe. And that's one we can't rightly get our hands around.

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