January 02, 2004
Poincare progress?

Mark Kleiman recently asked about the status of the Poincare conjecture, which a Russian mathematician named Perelman claims to have solved. This Boston Globe article, sent to me by my buddy Matt, would seem to indicate that Perelman's work is gradually gaining acceptance.

"What mathematicians enjoy is the chase of really difficult problems," said Hyam Rubinstein, a mathematician who came from Australia to attend meetings at the Mathematical Sciences Research Institute in Berkeley and the American Institute of Mathematics in Palo Alto, Calif., hoping to better understand Perelman's solution. "This problem is like the Mount Everest of math conjectures, so everyone wants to be the first to climb it."


As the foundation for his proof, Perelman used a method called Ricci flow, invented in the mid-1980s by Columbia University mathematician Richard Hamilton, which breaks a surface into parts and smooths these parts out, making them easier to understand and classify.

Although some mathematicians find it disturbing that Poincare's simple question could have such a complicated answer, Hamilton is not worried. After so many failed proofs, he said, "no one expected it to be easy."

Hamilton calls Perelman's work original and powerful -- and is now running a seminar at Columbia devoted to checking Perelman's proof in all its detail.

A mathematical blogger who goes by the excellent nom de blog Galois has a little more background and sounds a tad more circumspect than the folks quoted here. The article does note that it'll probably be a couple of years before Perelman's proof, if it is valid, is fully accepted.

As is often the case with these famous problems, the journey to the solution offers richer insights than the solution itself.

While working out the Poincare Conjecture, Perelman also seems to have established a much stronger result, one that could change many branches of mathematics. Called the "Geometrization Conjecture," it is a far-reaching claim that joins topology and geometry, by stating that all space-like structures can be divided into parts, each of which can be described by one of three kinds of simple geometric models. Like a similar result for surfaces proved a century ago, this would have profound consequences in almost all areas of mathematics.

Fantastic. Whether Perelman's proof is eventually accepted or not, he's clearly done some great work.

Posted by Charles Kuffner on January 02, 2004 to Technology, science, and math | TrackBack