Grisha Perelman recently crawled out the woods with the solution to the Poincare conjecture, and snuck back. There was talk of his love for mushrooms. ...er... Well, it turns out he was correct, so we probably shouldn't question his inspiration.
Since I'm about to hit the road, I can't get into as much detail as I'd like. I've quoted Henri Poincare a number of times (here and here) and reffered to him as a grandfather to chaos theory. (I think... if I haven't, I should have.. he is.) His conjecture, which Perelman managed to prove, was that any object, in any dimension, could either be reduced to a sphere or a tube. Sound weird? It is, but accurate, thanks to the Ricci flow:
In the early 1980's Richard Hamilton of Columbia suggested a new technique, called the Ricci flow, borrowed from the kind of mathematics that underlies Einstein's general theory of relativity and string theory, to investigate the shapes of spaces.
Dr. Hamilton's technique makes use of the fact that for any kind of geometric space there is a formula called the metric, which determines the distance between any pair of nearby points. Applied mathematically to this metric, the Ricci flow acts like heat, flowing through the space in question, smoothing and straightening all its bumps and curves to reveal its essential shape, the way a hair dryer shrink-wraps plastic.
Dr. Hamilton succeeded in showing that certain generally round objects, like a head, would evolve into spheres under this process, but the fates of more complicated objects were problematic. As the Ricci flow progressed, kinks and neck pinches, places of infinite density known as singularities, could appear, pinch off and even shrink away. Topologists could cut them away, but there was no guarantee that new ones would not keep popping up forever.
"All sorts of things can potentially happen in the Ricci flow," said Robert Greene, a mathematician at the University of California, Los Angeles. Nobody knew what to do with these things, so the result was a logjam.
It was Dr. Perelman who broke the logjam. He was able to show that the singularities were all friendly. They turned into spheres or tubes. Moreover, they did it in a finite time once the Ricci flow started. That meant topologists could, in their fashion, cut them off, and allow the Ricci process to continue to its end, revealing the topologically spherical essence of the space in question, and thus proving the conjectures of both Poincare and Thurston.
From today's New York Times.
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Consider a spherical cow....
Despite the recent rise of anti-scientific attitudes, Americans strongly contributed to the final solution. Thurston formulated the framework for the problem and Richard Hamilton started the Ricci Flow approach. The final work, after Perelman's 3 papers, took three and half years and about 1000 pages of hard mathematics most of which was done by Americans.
Topologists can do whatever they like to. I think people should just stop listen to them, if it comes to unique shapes.
Thank you for sharing this story with me !
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