In honor of Perelman’s recent official rejection of the Clay Institute’s $1 million prize for solving Poincaré’s Conjecture (he’d already declined the Fields’s medal) here’s a re-print from Slate Magazine (of a piece written 4 years ago for the NYTimes).  It bears repeating.

It explores not only the nature of the problem (Poincaré’s conjecture) that Perelman would become famous for solving, but also suggests a possible distinction among mathematical objects fascinating to consider:  that between structured and unstructured objects and the divergent mathematical cultures to which they give rise:

The entities we study in science fall into two categories: those which can be classified in a way a human can understand, and those which are unclassifiably wild. Numbers are in the first class—you would agree that although you cannot list all the whole numbers, you have a good sense of what numbers are out there. Platonic solids are another good example… These mathematical objects behave something like the chemical elements, which are neatly classified by Mendeleev’s periodic table.

In the second class are things like networks (in mathematical lingo, graphs) and beetles. There doesn’t appear to be any nice, orderly structure on the set of all beetles, and we’ve got no way to predict what kinds of novel species will turn up. All we can do is observe some features that most beetles seem to share, most of the time. But there’s no periodic table of beetles, and there probably couldn’t be.

Mathematicians are much happier when a mathematical subject turns out to be of the first, more structured, type. We are much sadder when a subject turns out to be a variegated mass of beetles. (But have a look at Fields Medalist Timothy Gowers’ beautiful essay “The Two Cultures of Mathematics” for a spirited defense of mathematical enterprises of the second sort.)

So, where do three-dimensional shapes, the subject of the Poincaré conjecture, fit in? To simplify, let’s think about two-dimensional shapes first. These fall firmly in the “periodic table” category. The only such shapes are the surfaces of “doughnuts” with multiple holes. The number of holes is called the genus of the surface and plays the role that the atomic number does for chemical elements. (Here is a picture of the surfaces of genus 0, 1, 2, and 3.) Geometer William Thurston (another Fields winner) made the daring conjecture that three-dimensional shapes, too, can be classified in a more complicated but equally structured way. Perelman has proved this conjecture, which has Poincaré as a straightforward corollary. That means, in turn, that we can think about proving general statements about three-dimensional geometry in a way that we can’t hope to about beetles or graphs.

Read the complete article here.

Didn’t give you much insight?  Perhaps a tale of Perelman’s feat in fictional form will help: you can read “Perelman’s Song” here.

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