Wednesday, September 08, 2004

Poincaré Conjecture Proven

Grigori Perelman, a Russian mathematician, has apparently solved the Poincaré conjecture, which states that if a closed 3-dimensional manifold has the homology of the sphere S3, them it is necessarily homeomorphic to S3. Say what? The Clay Mathematics Institute has a great layperson description -- it boils down to this: a sphere has a continuous, uninterrupted surface. If you were to take a sphere and slowly shrink it, it would eventually become a point, without rupturing. That's because the sphere is 'simply connected.' If you were to do the same with a donut, you could never get to a point without breaking the donut, because it doesn't have 'simple connectivity.' This 'simply connected' property is arises from the fact that a 2-dimensional sphere has a continuous surface in 3-dimensional space. What Poincaré asked over 100 years ago was, does this 'simple connectivity' property exist for a 3-dimensional sphere in a 4-dimension space? This question turned out to be a rather difficult question for mathematicians. So difficult in fact, that the Clay Mathematics Institute offered a $1million prize to anyone who can solve it. Yes kiddies, math can actually make money! Then along came Grigori Perelman, and apparently solved it -- 'apparently' because other mathematicians now need to dissect his math to find any mistake. This in itself would make an interesting story, but the story gets more interesting. Perelman apparently, has just presented his math, in fairly technical detail, with little or no explanatory notes. He's put it out there, and is not entertaining much discussion about the topic -- instead, he's just leaving it up to his peers to correct him or vindicate him. He doesn't even appear to have an interest in the $1million prize he may have won. In fact, he's known for being 'very unmaterialistic' and has previously turned down a prize from the European Mathematical Society.


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