In the article about Perelman's putative proof of Bill Thurston's geometrization conjecture that Thane mentioned was in The Princetonian, the writer got things almost exactly backwards when he wrote: << In fact, there are infinitely many distinct two-dimensional surfaces. The ball is different from the donut is different from a two-holed donut is different from a three-holed donut, ad infinitum. But this is not the case in higher dimensions. As was first suggested by former University mathematics professor William Thurston, when dealing with three-dimensional surfaces, there are only eight shapes that are topologically distinct. This statement, known as the Geometrization Conjecture in three dimensions, is currently unproven.

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The latter paragraph is exactly wrong. There are assuredly infinitely many topologically distinct 3-manifolds; in some sense each dimension is more complicated than lower dimensions, but 3 dimensions may be in another sense the most complicated of all. What Bill proved for a large class of (compact) 3-dimensional manifolds and conjectured for all of them is that each such manifold can be decomposed into pieces each of which can be given one of only 8 different types of geometries. For surfaces, the analogue is that each compact surface can be given a metric of constant curvature, either = 1, or = 0, or = -1. Furthermore, surfaces don't need to e decomposed into pieces first. Three of the eight geometries in the conjecture are the 3D analogues of the 3 constant curvature metrics in 2D; the other 5 might be called hybrids. --Dan