Let me try to explain a bit about the mathematics behind the story. One of the most important advances in the field of geometry was the 19th century shift in viewpoint initiated by Gauss and Riemann that lead to thinking of space itself in geometric terms. As you may know from hearing about Einstein's General Relativity, we can think of space itself as being bent...but what shape does it take? Beginning with the 19th century, geometers have studied geometric objects intrinsically, rather than necessarily assuming that they are an object embedded into a larger space that they can move around.

For instance, suppose you found yourself on another world, not the Earth. Just like the Earth, you might find that when you look around you the surface of the planet seems flat. But of course, we know that the Earth is not flat, it curves around and forms a sphere. Similarly, this new planet you are on could curve around. But what shape does it take? Could it be a donut shaped world, one with a hole in it? Could it be shaped like a football?

Now, in order to understand the Poincaré Conjecture, you have to be willing to ignore the question of whether the world could be football shaped. From the point of view of a topologist, a football and a sphere actually look the same. Just squish the football a bit, or tweak the sphere, and one becomes the other. But, the donut is really a different sort of thing all together!

In fact, there is a way to tell for certain if you are on a sphere or not even without flying up into space to look at it from a distance. Take a long rope and fasten one end of it in the spot where you're standing. Then walk a huge distance, stretching the rope out with you as you go. When you eventually get back to the spot where you started, take the two ends of the rope (the one that stayed in one place and the one you brought with you) and tie them into a noose or slip-knot. With such a knot, you can make the loop smaller and smaller...and on a sphere you always can do that while keeping the rope on the surface, but if you were walking on a donut then it might not be possible! Imagine that you were standing on a donut and walked around through the hole and back to where you started. Then once it is tight enough you will not be able to make it smaller without cutting the donut. (You can also walk around the hole in which case the loop cannot get smaller than the hole without leaving the surface.)

The point is this: If you have a surface and you know it is finite in size without any edges, then it must be a sphere if every loop on the surface can be shrunk down to point size (and conversely it cannot be a sphere if there is any loop which cannot be shrunk farther).

Okay, now the Poincaré Conjecture is the statement that the same thing can be said for higher dimensional spaces as well. In particular, Henri Poincaré said that he believed that any compact three dimensional space in which loops can be shrunk in this way (i.e. is simply connected) must be the three dimensional version of a sphere, called S³. That it is called a "conjecture" means that he said this, but had no proof of it, otherwise it would have been known as a theorem. Since then, the statement has been generalized to any number of dimensions. So, when someone talks about the Poincaré Conjecture today they probably mean that a compact, n-dimensional space is homotopy equivalent to the n-sphere, Sⁿ, if and only if it is homeomorphically equivalent.

This was a very famous open problem in mathematics, even being listed as one of the Millenium Problems for which the Clay Institute offered a $1 million prize. However, it now appears to have been resolved. A method of proof was utilized in which any given compact manifold can be squished and squeezed to put it in a form in which it can be recognized whether or not it is an n-sphere. Although this research program was begun by Richard Hamilton, and many other geometers made contributions to the program as well, the final step was completed by Grigori Perelman. Well, perhaps I should be a little bit less definitive, it appears that the last step was completed by Perelman. Certainly, nobody has yet found a flaw in his proof, but since it is rather long and complicated, mathematicians are being cautious about stating explicitly that it is now proved. However, the recent decision to award Perelman a 2006 Fields Medal (which he refused to accept) suggests it is now accepted by the mathematical community that this statement (and, in fact, the more general Thurston Geometrization conjecture) is no longer a conjecture but rather a theorem of geometry.

Of course, I've had to leave out many details in my brief description above. As I said at the beginning, it is hard to explain this explicitly in terms that a non-expert can understand because modern geometry/topology is rather abstract. That's where a short story by an author with both mathematical knowledge and a sense of poetry, like Perelman's Song, can be very useful.