## Millennium Prize Awarded For Perelman's Poincaré Proof 117 117

epee1221 writes

*"The Clay Mathematics Institute has announced its acceptance of Dr. Grigori Perelman's proof of the Poincaré conjecture and awarded the first Millennium Prize. Poincaré questioned whether there exists a method for determining whether a three-dimensional manifold is a spherical: is there a 3-manifold not homologous to the 3-sphere in which any loop can be gradually shrunk to a single point? The Poincaré conjecture is that there is no such 3-manifold, i.e. any boundless 3-manifold in which the condition holds is homeomorphic to the 3-sphere. A sketch of the proof using language intended for the lay reader is available at Wikipedia."*
## Re:what about... (Score:5, Informative)

You can easily determine the cubic volume of a spherical cavity by using the formula: V = 4/3 PI R^3.

However, in the case of your image, the volume would probably be better matched by a cylindrical volume: V = PI R^2 H

On second thought, a one-sheet hyperboloid would probably be the best match.

## Some background (Score:5, Informative)

This wikipedia entry [wikipedia.org] covers some controversies following the article.

## What does he win? (Score:3, Informative)

Since neither the summary nor either article tell you what the guy wins, (almost like it's a secret), here's a wikipedia entry [wikipedia.org] that does.

It's a million dollars.

## Great news (Score:5, Informative)

## So will he accept? (Score:5, Informative)

## Re:English Please (Score:4, Informative)

Manifold = a surface created by taking pieces of paper and warping them. For example, cylinder is a manifold since it can be formed by attaching the two opposite sides of the paper to each other. If you then attach the two circles at the ends of the cylinder, you get a torus (ie. donut).

Homeomorphic = there's a continuous function mapping points from one object to the other. This means that if two points are close to each other in the first object, they will be close together when the homeomorphism (the function) is used to map the points onto the second object. A square and the surface of a sphere, for example, are not homeomorphic since the square has edges and the sphere doesn't, so the mapping function has to jump somewhere, making it not continuous. Generally, two shapes are homeomorphic if you can deform one into the other (see animation here [wikipedia.org])

Homologous = I don't know how that word got in there. It's not in the Wikipedia article.

Simply connected = Any line drawn on the manifold that starts and ends at the same point can be slowly shrunk down to one point without taking any part of it off the manifold. A torus is not simply connected, since you can draw a line going around the cylinder and there's no way to take it off.

As for implications, as far as I can see, it just tells us that lots of things can be deformed into spheres and gives us a simple test for determining if something can.

## Re:English Please (Score:5, Informative)

## Re:Who the fuck cares? (Score:3, Informative)

Nerds care.

## Re:English Please (Score:4, Informative)

## Summary of the Poincare conjecture is inaccurate (Score:3, Informative)

Any closed smooth three dimensional space ('manifold') without boundary where all loops can be contracted to a point is 'homeomorphic' (essentially the same as) the three dimensional sphere (that is, the unit sphere in 4 dimensions).

The words "homologous" and "boundless" have little/nothing to do with it.

## Re:Some background (Score:3, Informative)

I also see a headline from June 2006, "Chinese Mathematicians Prove Poincare Conjecture," but the link is broken.

## Re:Who the fuck cares? (Score:2, Informative)

## Re:So will he accept? (Score:3, Informative)

He is not a native speaker of English. He might have mistranslated his thoughts.