So, What is Poincare's Conjecture Anyway?

So what is the Poincare' Conjecture anyway.

Since it was proposed by polymath H. Poincare' in 1904, who sought to mathematically define the topology of a sphere, mathematicians had sought to prove the theorem. For more than a century, the best and the brightest fell short of the mark until Russia's Grigory Perelman pulled off the feat in a series of papers he published between 2002 and 2003.

But earlier today, we learned that Perelman had turned down the $1 million in prize money the Clay Mathematics Institute had offered for the first correct solution.

Most laymen - yours truly included - will likely break more than a few teeth trying to understand the topic in all its complexity, so I'll leave it to the folks at the Clay Mathematics Institute, who offered up this instructive - and relatively simple - analogy:

"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. Poincare' 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."