Now that it is offered I would like to know now if he decided to accept it.
Now, we'll have to see whether Dr. Perelman will accept this honor. He astonished people worldwide when he declined the Field Medal: http://www.nytimes.com/2006/08/22/science/22cnd-math.html
"I can't say I'm outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest."
( from "Manifold Destiny", New Yorker, Aug 28, 2006: http://www.newyorker.com/archive/2006/08/28/060828fa_fact2?p... )
Imagine a sphere (in our 3 dimensions). If you drip a bit of water on it, the water will run down to a particular point. It goes down to the same point no matter where you drip the water. If you rotate it, the water will still all drip down to a particular point (just not the same one as before).
The above is true for some shapes (eggs, lima beans) and not true for others (Lego bricks)
So the question is: what about a 4 dimensional sphere? If you could somehow drip water on it... would the drips all converge to a particular point?
The answer is "yes"
The Poincare Conjecture postulates that if any loop on a "nice" surface can be shrunk to a point, it's topologically equivalent to a sphere. ("Nice" here means connected, finite, and without a boundary -- like a sphere or pyramid, but not a disk or infinite plane.) For instance, if the conjecture is true, a cube is topologically equivalent to a sphere, because if you draw a loop on it you can always shrink it down to a point; but a torus (donut) isn't, because a loop around a vertical cross-section can't be shrunk.
Perelman proved the conjecture for three-dimensional surfaces (which are the boundaries of four-dimensional objects).
No, the cube and sphere are homeomorphic regardless. (Pf: Points on S^3 are unit 4-vectors. Projecting onto the unit cube is continuous and invertible)
If the conjecture is true, and if you find yourself on a nice 3-surface, then you can conclude you're on the 3 sphere.
Unfortunately, HN will not allow me to delete or even edit my previous post.
:-(