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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."
Look, if you're going to use Ricci Flow to complete the proof, we all might as well pack up and go home. It's like the cheat code for all these manifold questions.
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.
For those just in, here's an article covering Perelman and his theorem.
This wikipedia entry covers some controversies following the article.
I hope Perelman will be able to afford better food than bread and cheese now.
Since neither the summary nor either article tell you what the guy wins, (almost like it's a secret), here's a wikipedia entry that does.
It's a million dollars.
Putting moderation advice in your
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.
Yeah, Perelman thinks he's so smart. Feh.
Math ain't rocket science.
You are welcome on my lawn.
It's not like wants the money or anything. He should at least take it and form a scholarship in his name. Jeez, the man is like a ./er, he lives with his mother.
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.
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.
Utilizing the synergization of benchmark e-solutions to pre-workaround action items!
Most rocket science isn't rocket science.
The /. eds could make this article 10x more relevant to most people by titling it 'Man wins million dollar mental masturbation prize' or by explaining the practical applications of this discovery. Instead the summary is a list of techno jargon that'd put Star Trek to shame with no mention of the $$ prize nor details of the winner. Who is this guy? Why did someone give him so much money for solving for x? Can I too win cash money for balls? If not, can I out source next year's winner to india and take a cut of the prize?
Anyway, this article's a lot better:http://www.newscientist.com/blogs/culturelab/2009/11/grigori-perelman-the-genius-in-hiding.php
I find the concept of mathematicians having fanboys who flame each other over proofs to be disturbing.
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.
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.
Well, I think the real question in this case should be what is the topology of the shape in question (the human body)? Isn't the so-called "cavity" really just a long tube connecting two openings to the outer surface? If that be the only set of connected openings, then the body would be homeomorphic to a torus.
However, there's a complex set of connected openings in the head: 2 nostrils, 2 tear ducts, and the mouth all connect to each other inside. I don't know what this is referred to as, topologically. Perhaps someone can help me out here. I'm guessing it's a quad-torus, and combined with the hole above makes the total a quintuple-torus?
We do, of course, assume that no other piercings have been made.