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Has The Poincare Conjecture Been Solved?
Zack Coburn writes "An article in the Boston Globe alludes to the Poincare Conjecture being solved, possibly. For those who are unfamiliar with the conjecture, the article gives a brief description: "To solve it, one would have to prove something that no one seriously doubts: that, just as there is only one way to bend a two-dimensional plane into a shape without holes -- the sphere -- there is likewise only one way to bend three-dimensional space into a shape that has no holes. Though abstract, the conjecture has powerful practical implications: Solve it and you may be able to describe the shape of the universe." Apparently Grigory Perelman may have proved it, which would mean a $1 million award from the Clay Mathematics Institute." We've previously discussed other possible Poincare proofs.
No.
(It even says in the freaking article stub that the proof is merely alluded to, for crying out loud.)
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Let me guess.. he says that the new topological object is universe-shaped?
Karma: SELECT `karma` FROM `users` WHERE `userid`=138474;
This proof has been out for about 9 months, and so far has stood up to intense scrutiny. Perelman is considered one of the top mathematicians in his field, and other mathematicians believe his proof is likely correct, although it is still being scrutinized. I recently attended a lecture by Richard Hamilton, who has been leading a team going through the proof, and he showed the method used and which sections of the proof had already been verified. It appears that the Poincare Conjecture finally has been solved.
If you are interested in the method of proof, Perelman used the Ricci Flow, blow-up arguments, and surgery to prove the Thurston Geometrization conjecture (a theorem far more powerful than the Poincare Conjecture alone).
There's no sig like SIGSEG
Yes but Godel showed that you never do it completely.
> In 2002, I researched the COSMIC background
Yeah, lots of people do that in college... Usually with the help of LSD and stuff.
Sheesh, evil *and* a jerk. -- Jade
How do you know that the shape of the universe does not include holes?
Informatus Technologicus
I've seen a video [uiuc.edu]
Just a little note for moderators: If you see something like that, it means the post was cut 'n' pasted from another slashdot post!
Here!
With italics and everything, including the link!
Google!
Fellowship 9/11
Indeed, the Poincare Conjecture (that every n-manifold with the homotopy groups of an n-sphere is homeomorphic to an n-sphere) has been solved in dimensions n = 1, 2, 4, 5, 6, ... The only missing case is n = 3, which is the case originally conjectured (well, really "asked about") by Poincare.
The cases n = 1, 2 are not so hard and may be explained to undergraduates. n = 5 and above are not easy but not impossible to explain, either -- Smale got a Fields medal for his work in this area. It can now be covered in a single graduate level mathematics course. The idea (if I remember correctly) basically boils down to "in high enough dimensions, there is enough elbow room". To give a better analogy, generically straight lines in two dimensions meet but in three dimensions they do not. (And to really say what is going on "Two-dimensional surfaces generically do not meet each other if embedded in a five-dimensional space")
The case n = 4 was handled by Michael Freedman using very subtle techniques (at least to me!) but again relying on "having enough space to move around in".
I don't understand the n = 3 case at all, really -- no one has given a simple "These techniques should work because x, y, znd z" sort of explaination, yet. The closest they come is to mutter uncomprehensible things about the heat equation... Suffice to say -- in dimension three there is not enough room to move around in. So it is not a complete surprise that the proof for n = 3 is rather different from higher n.
OK, a fairly unfunny introduction. Fair enough.
There's no evidence of this; we don't even know who this person is. There's very little research done merely 'involving' Poincare, and this claim is just so nonspecific it could mean anything. 'Poincare' could mean anything of his, not necessarily his infamous Conjecture.
This has nothing to do with the Poincare Conjecture at all. Nor mathematics in general. This makes little sense, and is totally offtopic.
This is the only ontopic sentence here, and it's just been copy-and-pasted from the article and capitalised strangely.
The reason it sounds foreign is because it makes no sense. "I'd probably be worried if you didn't" is just message padding, and the final clause of the sentence refers to 'observations' which no one, not even the poster himself, mentioned. "no fine-tuning" is just more message padding.
I can't find any such quote on Google. The "425 2003 593" is simply a US court case reference number. Friedmann-Lemaitre is just two random names stuck together. "foundation for local physics" means nothing.
Sweeping into the conclusion in response to a nonexistent question ("Is Poincare important?")
Why does he refer to it as a postulate and not 'Conjecture' all of a sudden?
This very research which you just made up out of thin air, yes. And while Poincare's Conjecture is quite important in number theory, topology and consequently numerical cryptography, it has little relevance to physics or other sciences. He's just listed these to sound credible.
And there you have it. One of the most effective trolls today, and you all fell for it. *Sigh.*
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Pure mathematicians don't do it, they leave it as an exercise to the reader.
Applied mathematicians do it with a real-world model.
I agree that it's an exciting time to be alive, but if you are as ignorant about science as your post would suggest, you would do well to confine your comments to generalities and stop spreading misinformation.