Gothamist Math

So it looks like the Poincare Conjecture may have been proven. While this is not expected to revolutionize the way we here at Gothamist do business, it is interesting nevertheless.

Bear in mind that these things usually take a while to knock out all the kinks. That's what peer review is for, especially in cases like this where some nutcase closets himself for years in order to solve a problem that great minds have broken themselves on for centuries.

On the other hand, if he's got the good stuff, he may be in for a nice cash reward.

Poincare's original conjecture, translated from the French and updated to modern language by this site (pdf), reads:
"Consider a compact 3-dimensional manifold V without boundary. Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?"

What the heck does that mean, you ask?
Lemme break it down like this:
Imagine a two-dimensional object that's not infinitely large. For normal human beings, that means trying to imagine a very thin surface that doesn't fade off into the distance. Now imagine that it doesn't have any loose edges (we don't want you to get a papercut). So now we're thinking of something like a soap bubble shaped like the surface of some everyday object. Now some of these shapes will have holes in them. Not holes like a pinprick hole that would pop a baloon but rather holes in the overall shape, like the hole in a donut. We know (well, they know and you'll accept because I told you so) that any of these bubbles without a hole in it can be smooshed, stretched, or otherwise deformed into a sphere. Those with holes cannot. Poincare conjectured that the same relationship holds for three dimensional surfaces. Now, a word of warning before you think you understand this. When you picture those very thin soap bubble surfaces floating in air you're visualizing a two-dimensional surface embedded in three dimensions. If you want to try to prove Poincare's conjecture you have to be able to discuss three-dimensional surfaces embedded in four dimensions. I won't ask you to do that today.

If that doesn't clear things up for you, try this explaination from Mathworld. Mathworld is an excellent resource for this sort of thing because it's so densely hypertexted. Clicking through the links in the story will take you to better and clearer definitions of terms than I can provide.