Millennium Prize Awarded For Perelman's Poincaré Proof

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...

[MR.Mic]

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

[ThoughtMonster]

For those just in, here's an article [newyorker.com] covering Perelman and his theorem.

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

What does he win?

[name_already_taken]

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

[Frans Faase]

I am very happy that they have awarded the price only to him, although he did meet the requirement that the proof should be published in a peer-reviewed journal. I am very happy that they did not included those two Chinese guys who did write down the proof (about 260 pages) and claimed that they had proven the conjecture. Perelman was very upset by this especially that other mathematics did not raise their voice. I hope that Perelman will accept the price. He said (some years ago) that he would only decide when the offer was made, if he would except the price or not.

So will he accept?

[Puff_Of_Hot_Air]

Perelman has famously turned down the fields medal and shunned the world since the whole Yau political saga. Will he take this prize? I hope that he will. I think that the whole Yau trying to take the credit for the proof issue, sullied the entire world for Perelman. Perhaps now that the honour is being fairly directed at him in response to his work, Perelman will be able to re-enter society and enjoy some of the fruits of his labour.

Re:English Please

[selven]

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

[pomakis]

I think the question is easier to understand if you knock everything down a dimension, because then it can actually be visualized. Take the surface of any three-dimensional object that doesn't contain any holes (e.g., a cup, but NOT a coffee mug with a handle). Can the surface be stretched/distorted to be shaped into a sphere? The answer is fairly obviously yes. But is this also true for four-dimensional objects? Stop trying to visualize it; you can't. You have to rely on the math instead. But that, I believe, is the question.

Re:Who the fuck cares?

[not-my-real-name]

Nerds care.

Re:English Please

[chhamilton]

Not quite true... a 3-sphere is actually the *surface* of a 4-dimensional sphere. So, not exactly something that lives in our world. In topology, the dimensions refer to the dimensionality of the surface, and not the space the surface lives in (ie: a circle drawn on a piece of paper is a 1-sphere, but the surface it was drawn on is 2-dimensional).

Summary of the Poincare conjecture is inaccurate

[TheEmptySet]

As someone who's job involves research into geometry and topology, I would like to point out that the summary is wrong in a couple of places. The Poincare conjecture states (in simple terms) that:

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

[_|()|\|]

Some previous Slashdot coverage, since they don't show up as related stories:

• Poincare Conjecture Proof Completed [slashdot.org]
• 2006 Fields Medalists Announced [slashdot.org]
• New Yorker on Perelman and Poincaré Controversy [slashdot.org]
• Another Millenium Problem May Have Been Solved [slashdot.org]
• Science's Breakthrough of the Year [slashdot.org]
• Grigory Perelman and the Poincare Conjecture [slashdot.org]

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

Re:Who the fuck cares?

[mariuszbi]

The prize is 1 million USD and Perelman is this guy in the picture http://englishrussia.com/index.php/2007/06/15/perelman-in-a-subway/ [englishrussia.com]

Re:So will he accept?

[HaeMaker]

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