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Grigory Perelman

Henri Poincaré

In my last post, on a Wall Street Journal article about Martin Gardner, I noted that it’s “a rare day when any major newspaper has an article with mathematical content.” I didn’t want to let the moment pass, so I wrote about the article. Now the latest New York Review of Books has arrived, with yet more mathematical content, a review by John Allen Paulos of Masha Gessen’s biography of Gregory Perelman, the Russian mathematician who proved the Poincaré Conjecture in 2002.

The conjecture, made by the great French mathematician Henri Poincaré in a 1904 paper, was one of the great unsolved problems in mathematics. Every mathematics graduate student learned about it when studying topology. It’s so easily stated, yet it resisted the efforts of mathematicians for a century, even as analogues were proven in dimensions 5 and higher by Stephen Smale, John Stallings, and E.C. Zeeman in the 1960s and in dimension 4 by Michael Freedman in 1982. Smale and Freedman were awarded Fields Medals for their work.

Perelman posted three papers in 2002 purporting to prove the conjecture, but the papers did not provide complete proofs. This led to a complex sequence of events that I won’t try to recount, with other mathematicians studying the approach he laid out, filling in details, and publishing their own papers verifying that he had indeed proved the conjecture. A 2006 New Yorker article by Sylvia Nasar and David Gruber that received a lot of attention gives an account of some of the controversy that ensued. Perelman was himself awarded a Fields Medal in 2006, but declined it. He was in the news again just two weeks ago when he declined the million-dollar prize awarded him on March 18 by the Clay Mathematics Institute for solving one of their seven Millennium Prize Problems. (Learn more by reading the Clay Institute’s short press release and full press release.

You can learn more about Perelman from the New Yorker article of 2006, Paulos’s review of the Gessen biography, or Jascha Hoffman’s NYT review of the biography last December. Hoffman notes in closing that Gessen “has written something rare: an accessible book about an unreachable man.”

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