“…there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought.” Reports Dennis Overbye.
If solved, Overbye continues, the implications of Dr. Grigory Perelman’s discovery will unfold for decades, but “the excitement came not from the final proof of the conjecture, which everybody felt was true, but [from] the method, ‘finding deep connections between what were unrelated fields of mathematics’” (quoting Dr. John Morgan from Columbia, emphasis added).
This matters, because “’Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide,’ explaining that curiosity is tied in some way with intuition.
The problem has to do with the nature of space: “In effect, what Poincaré suggested was that anything without holes has to be a sphere. The one qualification was that this “anything” had to be what mathematicians call compact, or closed, meaning that it has a finite extent: no matter how far you strike out in one direction or another, you can get only so far away before you start coming back, the way you can never get more than 12,500 miles from home on the Earth.”
Are there truly limited shapes of space?
“In the late 1970’s, Dr. Thurston extended Poincaré’s conjecture, showing that it was only a special case of a more powerful and general conjecture about three-dimensional geometry, namely that any space can be decomposed into a few basic shapes. Mathematicians had known since the time of Georg Friedrich Bernhard Riemann, in the 19th century, that in two dimensions there are only three possible shapes: flat like a sheet of paper, closed like a sphere, or curved uniformly in two opposite directions like a saddle or the flare of a trumpet. Dr. Thurston suggested that eight different shapes could be used to make up any three-dimensional space.”
The Ricci flow doesn’t sound fun (no thanks, I’d rather my head didn’t morph into a sphere!) But Perelman’s discovery is that apparently the singularities (densities) that occur in the morphing process are all “friendly” – apparently meaning they don’t interfere with the eventual revelation (exposure) of the space’s essential shape.
Perelman himself appears to be satisfied with having produced a monumental work and shuns the spotlight. I like the characterization by Dr. Anderson: “He came once, he explained things, and that was it…Anything else was superfluous.”