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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.)
Slashdot: when news breaks, we give you the pieces.
welcome our new topological overlord.
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
"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 ... And while the equivalent of the Poincare conjecture has already been proven for dimensions four and up..."
:-)
Being a non-math person, it seems to me if it has been solved for two dimensions (has it?) and four and up, wouldn't three dimensions just be a special case of the many (four and up) dimensions proof? Or is there something special about that proof that limits it to four and up? Or perhaps something in a form like the two dimension proof?
Perhaps my simple understanding of proofs in euclidian geometry doesn't scale up like this
There are 1.1... kinds of people.
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
"(First off, remember that us MATHEMATICANS DO IT SMOOTHLY AND CONTINUOUSLY.)"
They also DO IT with GRATUITOUS USE of CAPITAL LETTERS! Lay off the shift key!
Man, who let Shatner have the keyboard?
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
Here is an article from the current issue of Discover magazine on the state of the Poincare proof, and mathematical proofs in general. Sorry not a full text. Go to your library.
m at hematics/
http://www.discover.com/issues/jan-04/features/
I'm so drunk I can't s up strait and we're asking if some mathematical conjecture has been proved? Is this really the right storey for New Years Eve? Lets go with stories about things that are bright and shiny.
I do security
Last year I assisted with some research involving Poincare along with four other professors. We studied weak wide-angle temperature correlations in the cosmic MICROWAVE background.
There exists a simple geometric model of a NON-INFINITE and NON-NEGATIVE curved space, which we call the POINCARE space.
First, he states that he is either Jean-Pierre Luminet, Alain Riazuelo, Jeffery Weeks, Jean-Philippe Uzan, or Roland Lehoucq, none of whom are Computer Science professors as his sig claims him to be. Second, none of these gentlemen teach at 'slaughter college', which once again does not exist.
Finally, that particular study was interesting, but solving Poincare's theory wouldn't affect it at all. He wrongly used Poincare's significance. The Planck surveryor data should determine Omega0 to within 1%, and from that it will be simple to conclude (as the fine men who studied this did) that if Omega0 is less than 1.01, Poincare's dodecahedron makes a bad model of the universe, and if it's greater then it's a good model. This is not dependant on proving Poincare's theorum.
doggOK, 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.*
Slashdot: when news breaks, we give you the pieces.
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).
It's kinda like Fermat's Last Theorem... when they finally manage to prove it, it's like a "trivial consequence" of some vastly more fundamental and powerful theorem. While it's cool and all that they can solve it now, it's quite frankly fucking annoying to know that this super-duper difficult problem, which you might have tried to bang your head against in the past, is nothing but a mere collorary to something else.
Personally, I got that relevation when I thought I'd "discovered" something real but obscure, only to find out Leonhard Euler had figured out the same 250 years ago. And with some additional stuff I didn't think of either. One moment you feel real smart, the next "that guy with an abacus in the 'stone age' figured it out long long time ago".
It's rarely that you get it so "in your face" as you do it in maths. There's no historical relativity, no real defense. They were smarter than you, plain and simple. If this guy really has figured out something that no other mathematician in all of history has figured out, I applaud him. That is not a small feat in itself.
Kjella
Live today, because you never know what tomorrow brings
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.
Proofs have reached such a level of complexity that I really have my doubts that mathematicians can verify them reliably.
It's rather like writing a 50000 line program from scratch, without ever running it through a compiler, and then having a dozen people look it over for whether it would compile. Do you really believe that a dozen people looking at a 50000 line program would be able to find all the syntax and type errors contained in it just by eye? And, if anything, mathematical proofs are more complex and subtle. With type checking and syntax, there is at least something where people have years of experience with an unforgiving "proof checker", whereas (most) mathematicians have never had to face the rigor of a formal, automated, unforgiving proof checker.
For any proof of this complexity, I think the proof needs to be formalized and the checked by computer. Even then, there is a big risk that there is some bug in the formalization of the proof.
Hats off to Perelman for reminding us that money has never been a mathematician's incentive. The whole Clay thing is a travesty and not the right way to help mathematics.
(Contrast: this sort of snake-oil merchant, who puts money over truth.)