140 // Hippopotamian Logic

The main complication in three dimensions is that `singula- rities' can develop, where the manifold pinches together and the flow breaks down. Perelman's new idea was to cut the surface apart near such a singularity, cap off the resulting holes, and then allow the flow to continue. If the manifold manages to simplify itself completely after only finitely many singularities

´ have arisen, then not only is the Poincare Conjecture true, but a more far-reaching result, the Thurston Geometrisation Conjecture, is also true. And that tells us about all possible three- dimensional manifolds.

Now the story takes a curious turn. It is generally accepted that Perelman's work is correct, although his arXiv papers leave a lot of gaps that have to be filled in correctly, and that has turned out to be quite difficult. Perelman had his own reasons for not wanting the prize ­ indeed, any reward save the solution itself ­ and decided not to expand his papers into something suitable for publication, although he was generally willing to explain how to fill in various details if anyone asked him. Experts in the area were forced to develop their own versions of his ideas.

Perelman was also awarded a Fields Medal at the Madrid International Congress of Mathematicians in 2006, the top prize

........................................... in mathematics. He turned that down, too.

Hippopotamian Logic

I won't eat my hat.

If hippos don't eat acorns, then oak trees will grow in

Africa.

If oak trees don't grow in Africa, then squirrels hibernate

in winter.

If hippos eat acorns and squirrels hibernate in winter,

then I'll eat my hat.

Therefore ­ what?

...........................................

Answer on page 280

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