As you may have gathered, the Poincaré Conjecture appears to have been proved by a reclusive Russian mathematician, Grigory “Grisha” Perelman, who failed to turn up to recieve his Fields medal – the Nobel prize of mathematics – on tuesday. He is also expected to be turning down the $1 million Clay Institute prize for the accomplishment. Perelman has said that he is not interested in prizes and awards, he is repulsed by media attention, and that acknowledgement of his proof is recognition enough. Sounds like my kind of guy.

But what exactly is it that he has achieved?

The Poincaré conjecture was formulated in 1904, and states simply that the only compact, three dimensional, simply connected manifold is a three dimensional sphere. Half an hour’s introduction to basic topology will lead most people to see the intuitive truth of this statement, but actually proving it is a complete pig. Attempted proofs of the conjecture have appeared almost every year since the conjecture was published, but none of them stood up, until now.

This article in the Slate goes some way to explaining why this proof is important.

Mathematicians, no dummies, like to point out that, in some unspecified future, Perelman’s theorem might pitch in to help with these problems in ways that aren’t obvious now. But its real significance is like that of the fact that a times b is equal to b times a; it’s a basic structural statement about how the world is organized.

There’s a good round-up of articles on the Perelman-Hamilton proof at Christina Sormani’s page at CUNY, in which she refers to the New Yorker piece as “sensationalist”. Tee-hee.

The linkage spreads in all sorts of peculiar directions, perhaps none more peculiar than this piece of short fiction by Tina Chang, Perelman’s Song – due to be published in February 2008 issue of Mathematical Fiction (must. renew. subscription…) but available right now at her blog.

BTW, I’ve been noticing that the examples of topological forms used in some of these articles aren’t particularly well chosen: the New Yorker article gives “the surface of a football” as an example of a topographic sphere, when they really should have specified the outer surface; footballs have an inner surface, too, no less topoligically significant for being not generally seen. (If, however, we consider the outer and inner surfaces to be continuous, through the valve hole, the football becomes, topologically speaking, a sphere again. Hooray!)
Similarly, the New York Times article offers a rabbit as being topographically similar to a sphere, when biology teaches us that the inside of the gut of an animal is, strictly speaking, outside the body of the animal, and so the rabbit’s body, like the human body, is topologically a toroid (a bagel or ring doughnut is the most common exemplar). None of this, of course, affects the proof of the conjecture at all: I’m just indulging my Inner Smart-Aleck.

(via Robot Wisdom)