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."

Comments

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 covering Perelman and his theorem.

This wikipedia entry covers some controversies following the article.

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.

Re:I'm amazed.

[PopeRatzo]

Yeah, Perelman thinks he's so smart. Feh.

Math ain't rocket science.

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:So will he accept?

[Obyron]

I can't take credit for finding this. Another Slashdotter was kind enough to link it the last time Perelman came up, but I found this to be very enlightening and illustrative of Perelman's personality as well as the whole Yau controversy. It's an article from the New Yorker co-written by Sylvia Nasar, who wrote the biography of John Nash, A Beautiful Mind. It contains what was at the time the only interview with Grigori Perelman, but I'm not sure if that's still true.

Annals of Mathematic: Manifold Destiny

Re:So will he accept?

[Vellmont]

Has anyone had a hard answer as to why he turned down the prizes and medals?

What his friends have said is he believes actually proving it is reward enough. It's like being the first person to land on the moon, and someone gives you a "you landed on the moon" prize.

Still, a million dollars is something that can give you a lot of freedom. Turning it down is something that he might regret later.

Re:So will he accept?

[Puff_Of_Hot_Air]

Perelman is not a normal guy (obvious I realize, but hear me out). People like to subscribe 'normal' motives for behaviour they see as abnormal. I think this is why the idea that the fields medal was rejected as 'beneath' him was put forward. Arrogance is simple to understand. But what did Perelman actually say? "[the prize] was completely irrelevant for me. Everybody understood that if the proof is correct then no other recognition is needed". What Perelman was looking for was recognition for solving the problem. This was more important than the fields medal! What he got instead, was Yau and his cohorts claiming to have "really solved it." In Perelman's mind, political play such as this has no place in mathematics! Worse, his peers were not standing up to a) condemn this behaviour, and b) defend his paper. I think an important missing piece was that Perelman had not been officially recognized as having solved the Poincare conjecture. Now that this had been rectified, perhaps the world will be in enough order for him to rejoin it.

Re:What does he win?

[atomic777]

It's amazing that TFA doesn't mention a thing about whether Perelman will actually accept the prize. What will happen to the prize money if he does not accept? The million dollars disappears into Lichtenstein numbered bank accounts 2718-282 and 3141-519?

English Please

[AP31R0N]

Could someone give us non-math geeks an explaination of this that does not include the following words: manifold homologous homeomorphic?

i'll read the wiki page too, but i'm hoping someone here will take a crack at explaining in it plain English.

Also: What does this mean? What are the applications? Not that it has to have any to be interesting.

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)

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:English Please

[Coryoth]

It's really all about classifying shapes. For two dimensional things this is pretty easy, at least as far as the topology goes: you need to know the curvature and "how many holes does it have" and that's it -- this is the whole topologist not knowing a coffee cup from a donut since they both have one hole and hence can be deformed one into the other (note that this is two dimensional because we are considering the 2-dimensional surface on the donut and coffee cup). In dimensions higher than two things start getting trickier because more bizarre configurations become possible. Perelman's work, which actually goes toward proving the rather more far reaching Geometrization Conjecture (due to Thurston), essentially lays out how you can classify all the different (from a topological point of view) shapes of things in three dimensions and higher.

What are the implications? Well, one reasonable question is: what is the topology of the universe like; what shape is the universe? Since the universe is a three dimensional manifold that turns out to be tricky. Perelman's work lays out the groundwork to be able to answer such a question.

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).

Re:Who the fuck cares?

[not-my-real-name]

Nerds care.

Re:controversial "proof"

[Ukab the Great]

I find the concept of mathematicians having fanboys who flame each other over proofs to be disturbing.

from a mathematician

[l2718]

It's easier to explain the two-dimensional version, that is the version about surfaces. A mathematical surface is a kind of quilt: it's what you get from stitching together patches, each of which looks like a small piece of the plane. Just like with the quilt, if you bend or deform the surface it still is the same surface. Surfaces are completely "floppy".

Now, most real-life quilts are rectangular and have a boundary where they end, but you can also "close" the quilt by stitching the boundary back onto itself -- what you get is a "closed" surface. For example, you can stitch all the boundary together and get a sphere. Or you can stitch opposite sides together and get a "torus" -- the surface of a doughnut. You can also make more complicated quilts, which look like the joining of several doughnuts, i.e. a doughnut with several holes.

Next, one way that the sphere and doughnut-surface differ is that the latter has a hole. The way we express this is by looping a closed piece of string along the surface. With the sphere you can always slide the piece of string off the surface (we say that the sphere is "simply connected"), but with the torus you can run a loop of string along it in such a way that no deformation will allow you to take it off (we say the doughnut is "multiply connected").

Finally, the "2d Poincare conjecture" is the statement that the only simply connected closed 2d surface is the sphere. In other words, if you can't link a loop with your closed quilt then your quilt can be deformed to be a round sphere. A strong version of this was proved by Poincare, among others.

Now for the real "Poincare Conjecture" that was proved by Perelman, replace "2d" by "3d", so the quilt comes from stitching little cubes instead of little squares. The "closed and simply connected" assumptions are the same, and the conclusion is that the quilt is, up to deformation, the 3d sphere. It's much harder to visualize since now the quilt may not fit into regular 3d space. For example, the 3d sphere is what you get by stitching the whole boundary of the 3d cube together into one point (recall how we got a 2d sphere!) -- which is not something that fits into ordinary 3d space.