When I started as a graduate student in mathematics lo these many moons ago, there were maybe four open questions which were then considered to be the pinnacle of mathematics, analogous to scaling Everest or running a sub-four-minute mile. All of them had been unsolved for decades or even centuries, despite the best efforts of the strongest minds. Since then two or the four (the Bieberbach Conjuecture and the Shimura-Taniyama Conjecture (Fermat's Last Theorem)) have been solved by stunning and surprising approaches. Of the two remaining, the Poincare Conjecture and the Riemann Hypothesis, it would appear that the Poincare Conjecure has fallen. It is odd that we are living in the abolute golden age of mathematics and yet almost no one knows it or even has the merest awareness of what that means.
This ignorance occurs for good, rather than nefarious, reasons. Mathematics has basically become too abstract and too sophisticated to be able to be explained any more. It takes years of arduous study just to learn what the terms mean. Yet it has been shown in the last quarter century or so that even the most abstruse and recondite piece of pure mathematics is likely to have important consequences or applications in the physical world, so it seems to me incumbent upon us as citizens of a democracy to try to understand something about the froth of activity going on surreptitiously around us. Continued....
Allow me then to take a crack at explaining the Poincare Conjecture: the only 3-dimensional object that has the shape of a sphere is a sphere. That's deceptively simple, because I'm not talking about the ordinary sphere (called the 2-sphere in mathematics because if you live on one, say the surface of the Earth, your movements are constrained to only 2 dimensions), but rather its analog in 4-space (the set of all points in Euclidean 4-space whose distance is one from the origin is the unit 3-sphere). And by having the "shape" of the sphere I mean that its shape as measured by a very important mathematical yardstick called the "fundamental group" as well as by the "homology groups" is that of a sphere. What this Conjecture does is validate our ordinary intuition in the strongest way. That the Conjecture is true there has been very little doubt for a long time; that we found ourselves unable to prove it has been a huge embarassment to the species. The Conjuecture means that this yardstick really captures the topological essence of a sphere, which is really good news because the fundamental group is far simpler than a sphere and easier to work with.
The proof itself has an interesting human interest story behind it. Like many such intellectual feats (think chessmasters) it emanates not from the Anglo-Saxon world but from Russia, with its Orthodox, Greek-based culture. The author, Perelman, of this alleged proof wrote a couple of short papers claiming to have a proof. He wasn't quite believed because the papers were too short and surprising to be convincing. He went on a whirlwind tour of the US to make his case; American mathematicians have been working hard for three years now to fill in the numerous gaps in Perelman's ideas, and Perelman has apparently completely disappeared. There's a large prize of a million dollars to be had should he reappear, together with (probably) a Fields Medal, the mathematical equivalent of the Nobel Prize. “He came once, he explained things, and that was it,” Dr. Anderson said. “Anything else was superfluous.”
He came, he conquered, he disappeared. Was he an alien?