**The Principled Genius**Former mathematician Grigori Perelman made the news today by deciding, after considerable delay for pondering, to turn down the million-dollar "Millennium Prize" awarded by the Clay Mathematics Institute, which set up these prizes in 2000 for the solutions to a set of seven basic problems that had been bedeviling the field. Perelman's award, for solving the famous PoincarÃ© Conjecture, was the first to be announced.

Perelman is another of those Russian Jews who so often seem to turn up in stories like this. He studied in the USSR, then held posts in several American universities in the late 1980s and early 1990s. He then turned down positions at Princeton and Stanford to return to Russia. He has since ceased working on mathematics altogether and is unemployed, living with his mother in St. Petersburg.

In turning down the $1 million Millennium Prize, Perelman explained that he did not find the process of awards in his field to be just. He already had been awarded the FIelds Medal for his work in 2006, and declined that one, too. In fact, he consistently turns down prizes, saying things like "[the prize] was completely irrelevant for me. Everybody understood that if the proof is correct, then no other recognition is needed." Or "I'm not interested in money or fame. . . . I don't want to be on display like an animal in a zoo. I'm not a hero of mathematics. I'm not even that successful; that is why I don't want to have everybody looking at me." He has expressed the opinion that prize committees are unqualified to assess his work, even positively.

This quotation sums up his alienation: "[T]here are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest." He has also said, "It is not people who break ethical standards who are regarded as aliens. It is people like me who are isolated."

So what was Perelman's work all about? Wikpedia explains that the PoincarÃ© conjecture claims that if a closed 3-manifold has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. Yeah, that didn't help me either. This is more my speed:

The PoincarÃ© Conjecture says "hey, you've got this alien blob that can ooze its way out of the hold of any lasso you tie around it? Then that blob is just an out-of-shape ball." Perelman and Hamilton proved this fact by heating the blob up, making it sing, stretching it like hot mozzarella and chopping it into a million pieces. In short, the alien ain't no bagel you can swing around with a string through his hole.

The author goes on to explain that mathemeticians classify both 2-dimensional and higher-dimensional shapes. She says that the only 2-dimensional shapes are the surfaces of "doughnuts" with multiple holes:

These surfaces can be classified neatly according to their number of holes. (The picture includes a sphere; does that count as a doughnut with zero holes?) Anyway, it seems the PoincarÃ© Conjecture pertains to the analysis of 3-dimensional shapes, which I guess are like the surfaces of 4-dimensional shapes. Geometer William Thurston (a Fields Medal winner who didn't turn down his prize) "made the daring conjecture that three-dimensional shapes, too, can be classified in a more complicated but equally structured way," which is the kind of thing that makes mathematicians use words like "daring" and really rings their bells. Perelman "proved this conjecture, which has PoincarÃ© as a straightforward corollary" -- once again, using the word "straightforward" in a slightly eccentric sense.

The author of this article readily admits that Perelman's work "won't help anyone build a bridge, aim a rocket, crack a code, or privatize Social Security. " She concludes, a little defensively, that it's nevertheless something worth caring about "if you prefer order to chaos."

This whole thing makes me nostalgic on my father's account. The only time I can remember his expressing an opinion about what I ought to do with my life is when he mused mildly that he'd always thought it would be nice if I studied algebraic topology. That was not, to put in mildly, really in the cards. He must have thought it was awfully boring of me to study law, though he never said so and clearly was pleased that I'd be able to make a living. I still don't know what algebraic topology is. Wikipedia supplies this less-than-helpful definition:

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism. In many situations this is too much to hope for and *it is more prudent to aim for a more modest goal*, classification up to homotopy equivalence.

I'm even more prudent and must aim for goals whose modesty makes that goal look positively overweaning.

Some decades ago Fran Lebowitz wrote a piece about being awakened by something unpleasant, a phone call or alarm clock. She said something like, "This is not my favorite method of being awakened. My favorite method is to have my Swedish lover whisper in my ear that, if I don't want to be late to pick up my Nobel Prize, I'd better ring for breakfast." This is a dream I learned to relinquish quite early on, but I'm still pleased to read about people who might reasonably aspire to these things, even if they're too high-minded to accept.