Poincaré Conjecture
If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.
Ah - from Clay Mathematics Institute comes this more accurate statement of this conjecture:
Poincaré Conjecture
= = = = = = = = = = = = = = = = = = = = = = = = = =
Yes, one of the implications for the author, if this proof holds up and is published in a vetted math journal, is a Clay prize of $1 million. The Poincare Conjecture is one of their "Millenium Problems". For the other six, with formal statements of all seven, see the Clay Institute site (Google it!).
I hate apples and donuts.