Poincare update

This, via Chad Orzel, is from last week, but what with all of the CD22 craziness I never got around to posting it. It's an update on the status of eccentric Russian mathematician Grisha Perelman and his reported proof of the Poincare Conjecture (see here for more). Perelman has gone into seclusion in Russia since his proof was published almost three years ago, but the work he did appears to be standing up to scrutiny. Among other things, there's a million bucks riding on this:

Also left hanging, for now, is $1 million offered by the Clay Mathematics Institute in Cambridge, Mass., for the first published proof of the conjecture, one of seven outstanding questions for which they offered a ransom back at the beginning of the millennium.

[...]

In his absence, others have taken the lead in trying to verify and disseminate his work. Dr. Kleiner of Yale and John Lott of the University of Michigan have assembled a monograph annotating and explicating Dr. Perelman's proof of the two conjectures.

Dr. Morgan of Columbia and Gang Tian of Princeton have followed Dr. Perelman's prescription to produce a more detailed 473-page step-by-step proof only of Poincare’s Conjecture. "Perelman did all the work," Dr. Morgan said. "This is just explaining it."

Both works were supported by the Clay institute, which has posted them on its Web site, claymath.org. Meanwhile, Huai-Dong Cao of Lehigh University and Xi-Ping Zhu of Zhongshan University in Guangzhou, China, have published their own 318-page proof of both conjectures in The Asian Journal of Mathematics (www.ims.cuhk.edu.hk/).

Although these works were all hammered out in the midst of discussion and argument by experts, in workshops and lectures, they are about to receive even stricter scrutiny and perhaps crossfire. "Caution is appropriate," said Dr. Kleiner, because the Poincare conjecture is not just famous, but important.

James Carlson, president of the Clay Institute, said the appearance of these papers had started the clock ticking on a two-year waiting period mandated by the rules of the Clay Millennium Prize. After two years, he said, a committee will be appointed to recommend a winner or winners if it decides the proof has stood the test of time.

"There is nothing in the rules to prevent Perelman from receiving all or part of the prize," Dr. Carlson said, saying that Dr. Perelman and Dr. Hamilton had obviously made the main contributions to the proof.

As the article and a commenter at Good Math, Bad Math note, Perelman actually proved a stronger version of Poincare, which is Thurston's Conjecture from 1982. That's really impressive. What's exciting about this for mathematicians is not just that a longstanding historically significant problem has finally been solved, but that the solution has deep connections to other, seemingly unrelated, areas of math. That's sure to generate tons more research opportunities, and who knows where all that will lead.

So in the last decade or so, Fermat and Poincare have fallen. There are still some lucrative problems to solve. Personally, I'm rooting for the Riemann Hypothesis to go next, but to each his own.

Posted by Charles Kuffner on August 22, 2006 to Technology, science, and math | TrackBack