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.

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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."