Aug 16, 2006

Wanted: The Russian who smashed Poincare

Jules Henri Poincaré was a French mathematician AND physicist AND philosopher who made a lot of important (and beautiful) contributions to all those branches of science. In some sense though, his most famous contribution was an (until now) unproved conjecture which has his name: the Poincare Conjecture. As you can read in the Wikipedia link, the modern version of the conjecture seems very simple:

Every simply connected closed (i.e. compact and without boundary) 3-manifold is homeomorphic to a 3-sphere.

I'll explain it in a moment. First, let me say that it is a conjecture because Poincare asserted it without proving. He was led to think that it could be true by clues in his work, but was unable to prove it in that time. The second thing is that it is so important to mathematics (and probably to physics too) that it is one of the famous Millenium Prize Problems from the Clay Institute, which include among others, the P=NP problem, the solution of Navier-Stokes equation and the Riemann Hypothesis. It means that whoever solves one of them is entitled to receive the US$ 1,000,000.00 prize!

Poincare conjeture talks about topology. A 3-manifold is a mathematical set that looks like a 3-dimensional euclidean space if you look very close (what is called locally in technical language), like the Earth looks like a flat locally, but is really a sphere. The 3-sphere is just the familiar sphere in 3-dimensions, i.e., the whole sphere, including its interior. In mathematics, when you say that two things are homeomorphic, you are saying that, in practice, they are the same. Now, the notion of being simply connected. A set is simply connected if any closed curve (e.g., a circle) in the set can be continuously shrinked to a point without breaking. The classical example is the sphere against the torus (i.e., the doughnut). Inside a sphere, you can shrink any closed curve to a point without breaking, in a torus, if the curve goes around the hole, you cannot shrink it to a point, you are stopped by the hole. The term compact is a little technical and in some sense means that the set is not infinite. The non-boundary condition means that you cannot found the boundary of the set. The circle, seen as a one-dimensional object, has no boundaries. Indeed, it is a one-dimensional manifold which is compact, has no boundary but fails to be simply connected. Actually, you can say that it is a 1-dimensional torus.

As you can see, although it is simply stated, it has a lot of technical subtleties and stood without proof until the russian mathematician Grigory Perelman posted some papers in the arXiv saying that he had found a proof. The problem is that the papers where very compact and took some time to be dissecated. And now, that it seems that they were and the proof seems correct, Dr. Perelman simply disappeared! You can read the detailed story in the NY Times: Elusive Proof, Elusive Prover: A New Mathematical Mystery.

The detailed version of the Poincare Conjecture is in the Clay Mathematics site, as well as the papers explaining his proof. I'm linking them here just in case:

- Ricci Flow and the Poincare Conjecture, John W. Morgan, Gang Tian (math.DG/0607607)

- Notes on Perelman's papers, Bruce Kleiner, John Lott (math.DG/0605667)

- A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow, Huai-Dong Cao, Xi-Ping Zhu

If you find Dr. Perelman, please send an email to the CLay Institute! All mathematicians in the world want to find them. It is very probable, as it is what he likes most to do, that he is looking for mushrooms in some Russian forest.

Picture: The Guy.

No comments: