Saturday, November 07, 2009

Soviet-style mathematics

Via Anand Kulkarni (aka polybot) comes an interesting article in the WSJ by Masha Gessen on Grigori Perelman, Soviet-era mathematics and the question of 'big math'. The premise of the article (Masha Gessen has a book out on Perelman and the Poincare conjecture) is that special environments are needed to prove big results, and the Soviet-era mathematical enclaves fostered this environment both because of, and inspite of the Soviet political system.

It is indeed true that amazing work came out of the isolated confines of Soviet mathematical institutes, often parallel to or well before similar work in the Western world. There's a joke that goes around theoryCS circles that for every theorem proved before the 80s in the west, there's an equivalent result proved 10 years earlier by a Russian mathematician. We need look no further than the Cook-Levin theorem, the Koebe-Andreev-Thurston theorem (on circle packings), Kolmogorov-Chaitin-Solomonoff complexity (and according to some, the Cauchy-BUNYAKOVSKY-Schwarz inequality, though this is disputed).

But in the article is a more thought-provoking claim:
The flow is probably unstoppable by now: A promising graduate student in Moscow or St. Petersburg, unable to find a suitable academic adviser at home, is most likely to follow the trail to the U.S.

But the math culture they find in America, while less back-stabbing than that of the Soviet math establishment, is far from the meritocratic ideal that Russia's unofficial math world had taught them to expect. American math culture has intellectual rigor but also suffers from allegations of favoritism, small-time competitiveness, occasional plagiarism scandals, as well as the usual tenure battles, funding pressures and administrative chores that characterize American academic life. This culture offers the kinds of opportunities for professional communication that a Soviet mathematician could hardly have dreamed of, but it doesn't foster the sort of luxurious, timeless creative work that was typical of the Soviet math counterculture.

For example, the American model may not be able to produce a breakthrough like the proof of the Poincaré Conjecture, carried out by the St. Petersburg mathematician Grigory Perelman.

This is a reflection of one of the enduring myths of mathematical research, "a mathematician would be happy in jail if they had paper and pen", with a bit of the 'a mathematician is a solitary (and slightly crazy) genius'. I can see the allure in the idea: mathematics requires great concentration, and removal of distractions would surely make it easier to focus on a big problem.

But is this really impossible to achieve in the Western model of research ? After all, even Perelman's work built heavily on a program first outlined by Richard Hamilton from Columbia. Andrew Wiles proved Fermat's theorem while at Princeton. Ketan Mulmuley has been banging away at P vs NP while shuttling between Chicago and IIT Bombay (yes, I know it's not a perfect comparison because it hasn't been resolved yet). Stephen Cook proved that SAT is NP-Complete while at Toronto. And so on and so forth.

Possibly one argument in favor of the 'isolation: good' theory is that Perelman didn't need to prove himself for 6-7 years, maintain a steady stream of funding, and teach lots of classes in order to "earn" the right to study such a hard problem. It's hard to imagine a researcher in the US being able to do this before they get some kind of job security (tenure, or otherwise).

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