In the early 20th Century, Henri Poincare conjectured that the three-dimensional sphere is the only compact three-dimensional manifold which is simply connected. This is now called the Poincare Conjecture.

This conjecture is the dimension three case of the Generalized Poincare Conjecture, which makes a similar assertion for manifolds in an arbitrary dimension. Amazingly, the dimension three case (which the original case conjectured by Poincare) is the only one which has not yet been solved.

This conjecture is one of the seven Millenium Prize Problems proposed by the Clay Mathematics Institute.

It appears that topologist Martin Dunwoody has proposed a general proof for the Poincare Conjecture. As of April 7, 2002, this proof is in preprint, available for now at http://www.maths.soton.ac.uk/~mjd/Poin.pdf.
As I am not a mathematician, I will not attempt to provide a layman's explanation of the conjecture. It is worth noting that if the proof stands, Dr. Dunwoody may be a very wealthy man - the Clay Institute prize is one million dollars!

The Poincaré Conjecture is one of the seven Clay Mathematics Institute's Millennium Prize Problems. It is about one hundred years old. Roughly speaking, the conjecture pins down the exact requirements a region of space (a three-manifold) must have in order for it to look like the three-sphere, the surface of a hypersphere. Alternatively, one could say that the three-sphere is the only "important" (from a topological standpoint) three-dimensional manifold that doesn't have any holes. Such a manifold must be both compact, without a boundary, and simply connected. These topological properties aren't exactly intuitive, but both are explained better in their respective nodes.

What makes this conjecture so amazing is that there is a whole plethora of very complicated, weird spaces that fit the bill of being compact and simply connected. The three-sphere is simple in comparison to these weird manifolds. But the other thing that makes this conjecture famous is that it was one of the first conjectures in the field of topology. It also proved to be a difficult one; many powerful mathematicians have attempted to put forth proofs of it, only to fail at some crucial juncture. But this recent proof looks to be legitimate, and Fields medals have been distributed accordingly.

The new proof comes out of a series of three preprints posted to the internet preprint archive arXiv by Russian mathematician Dr. Grisha Perelman, the first being posted in November 2002 [1, 2, 3]. The preprints do not actually claim to prove the Poincaré Conjecture; much as Dr. Andrew Wiles worked with the Taniyama-Shimura conjecture and popped Fermat's Last Theorem out like a past-due infant, Dr. Perelman worked with Thurston's geometrization conjecture. But if this proof is correct, the Poincaré Conjecture is also true as a pleasant side-effect.

The tool that made this proof possible was developed by Dr. Richard Hamilton about 25 years earlier. The Ricci flow is a differential equation in terms of the Ricci tensor; that is, it describes a change in the curvature of the manifold over time in a manner very similar to the way the heat equation changes temperature. In simpler terms, the Ricci flow acts like an iron, smoothing out the wrinkles in a complicated manifold until what is left resembles the surface of a hypersphere, which has constant curvature. Unfortunately, differential equations are hard, and there were many loose ends that needed to be tidied up before one could justify using the Ricci flow to solve the Poincaré Conjecture. This is the gap that Dr. Perelman's three papers fills.

Or does it? Dr. Perelman's papers don't actually contain a proof of the Poincaré Conjecture in the sort of way you and I would think of a proof. Taken separately, one might even think they were about different things. In 2006, two Chinese mathematicians, Dr. Cao Huaidong and Dr. Zhu Xiping [4], under the then widely held opinion that the papers constituted work on the conjecture and not a complete proof, brought together the work done by Dr. Perelman and Dr. Hamilton, along with a good bit of original research, and wrote the proof of the Poincaré Conjecture down in a monumental 300-page paper. Others have also tried to write papers to clear up how Dr. Perelman's work directly relates to the conjecture.

So we have ourselves a minor bit of mathematical controversy. Is Dr. Cao and Dr. Zhu's paper the proof of the Poincaré Conjecture, or is the work done by Dr. Perelman the proof? The Fields Medal judges and the Clay Institute both chose Dr. Perelman (even though he didn't choose them). There is a very small bit of talk about possible discrimination, but not as much as one would expect. The mathematics community seems to be happy enough to have another difficult puzzle mastered and brought under its wing.

1. Perelman, Grisha. "The entropy formula for the Ricci flow and its geometric applications", http://www.arxiv.org/abs/math/0211159.
2. — "Ricci flow with surgery on three-manifolds", http://www.arxiv.org/abs/math/0303109.
3. — "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds", http://www.arxiv.org/abs/math/0307245.
4. "Chinese mathematicians solve global puzzle", Xinhua News. June 3, 2006, http://news.xinhuanet.com/english/2006-06/03/content_4642313.htm. I'll be the first to admit that a Chinese state-run newspaper isn't the best source for information about Chinese mathematicians.