Topology and The Poincare Conjecture
A Russian mathematician Grigori Perelman, wore a rumpled dark suit and sneakers, and paced while he was introduced. Bearded and balding, with thick eyebrows and intense dark eyes, he tested the microphone and started hesitantly: "I'm not good at talking linearly, so I intend to sacrifice clarity for liveliness." Amusement rippled through the audience, and the lecture began. He picked up a huge piece of white chalk, and wrote out a short, twenty-year-old mathematical equation.
The equation, called the Ricci flow equation, treats the curvature of space as if it were an exotic type of heat, akin to molten lava, flowing from more highly curved regions and seeking to spread itself out over regions with lesser curvature. Perelman invited the audience to imagine our universe as an element in the gigantic abstract mathematical set of all possible universes. He reinterpreted the equation as describing these potential universes moving as if they were drops of water running down enormous hills within a giant landscape. As each element moves, the curvature varies within the universe it represents and it approaches fixed values in some regions. In most cases, the universes develop nice geometries, some the standard Euclidean geometry we studied in school, some very different. But certain tracks that lead downhill bring problems the elements moving along them develop mathematically malignant regions that pinch off, or worse. No matter, the speaker asserted, we can divert such tracks; and he sketched how.
The Poincare conjecture provides conceptual and mathematical tools to think about the possible shape of the universe. But let us start with the simpler question of the shape of our Earth. Any schoolchild will say that the Earth is round, shaped like a sphere. And, in these days of airplanes and orbiting spacecraft that can take pictures of our planet from on high, this seems utterly obvious. But, in times past, it was difficult to say with certainty what the shape of the world was.
Belief is one thing, but when did we really know, without any doubt, that the world was shaped like a sphere? We have seen that Columbus began to doubt the sphere theory, thinking of the Earth as pear shaped. And today we know that our planet is not a perfectly round sphere, but is flattened somewhat at the poles.
But, as we shall see presently, there are other more radical possibilities: the question of the Earth's shape is much more than a matter of bumpiness and flattened regions.