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.

Has the Poincare Conjecture Been Solved?

[James A. C. Joyce]

No.

(It even says in the freaking article stub that the proof is merely alluded to, for crying out loud.)

This Proof Isn't New

[muon1183]

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).

TROLL

[Anonymous Coward]

"It's interesting how a really good felching can sometimes be much better than a really good man-on-man blowjob," Rubinstein said with a grin

Last line, devious bugger ;)

Re:Finite Universe

[Bombcar]

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! [slashdot.org]

With italics and everything, including the link!

Google!

Proof Smoof

[Anonymous Coward]

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.

http://www.discover.com/issues/jan-04/features/m at hematics/

Re:In 2002, I researched the COSMIC background

[Smitedogg]

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.

dogg

A line-by-line proof...

[James A. C. Joyce]

...of why this guy is a troll and all who modded him up must be smoking the $2 crack.

"(First off, remember that us MATHEMATICANS DO IT SMOOTHLY AND CONTINUOUSLY.) Hehehe, wow, too many New Year's drinks. Anyway, on to the story."

OK, a fairly unfunny introduction. Fair enough.

"Last year I assisted with some research involving Poincare along with four other professors."

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.

"We studied weak wide-angle temperature correlations in the cosmic MICROWAVE background."

This has nothing to do with the Poincare Conjecture at all. Nor mathematics in general. This makes little sense, and is totally offtopic.

"There exists a simple geometric model of a NON-INFINITE and NON-NEGATIVE curved space, which we call the POINCARE space."

This is the only ontopic sentence here, and it's just been copy-and-pasted from the article and capitalised strangely.

"This may sound foreign to you, and I'd probably be worried if it didn't, but this POINCARE space can account for these observations with no fine-tuning."

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.

"From our "Nature" (425 2003 593) article: "If confirmed, the model will answer the ancient question of whether space is finite or infinite, while retaining the standard Friedmann-Lemaitre foundation for local physics.""

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.

"So, yes, Poincare is VERY important"

Sweeping into the conclusion in response to a nonexistent question ("Is Poincare important?")

"and this postulate"

Why does he refer to it as a postulate and not 'Conjecture' all of a sudden?

"as well as the query as to whether it's been appropriately solved has a HUGE impact on all kinds of other research (math, physics, computer science, etc.) such as this very research that I participated in."

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.*

Re:Who Cares! or An Exciting Time To Be Alive

[noonien_soong]

You seem to be misinformed. The Riemann hypothesis has not been proven. If it had, we would have heard about it; it is one of the current holy grails of mathematics. The 16th Hilbert problem has not been solved. The student in question only claimed to have solved part of it, and she was dead wrong. Positrons have nothing to do with LEDS, transistors, or diodes, and QED was not relevant to the invention of any of them. "structuring matters behaviors, including time-dependEnt transformations"---what does that even mean? Nothing. You made it up. Having a proof of Poincare's conjecture has absolutely nothing to do with crumple zones, or any engineering problem, for that matter.

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.

Re:I'm confused...

[Ibag]

From Mathworld [wolfram.com]

The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known to 19th century mathematicians), n = 3 (the original conjecture) remains open, n = 4 was proved by Freedman (1982) (for which he was awarded the 1986 Fields medal), n = 5 was demonstrated by Zeeman (1961), n = 6 was established by Stallings (1962), and n>=7 was shown by Smale in 1961 (although Smale subsequently extended his proof to include all n>=5).


So, to answer your question, the proof for higher dimensions doesn't hold if n10 or something (where 10 is a random number depending on the proof). Sometimes, the argument in one case relies on properties that just aren't present for smaller n. It just means you have to go hunting for a more elegent proof!

Re:I thought...

[Anonymous Coward]

the speaker said that "if P=NP is proven, then all the others are going to fall in short time, making that solution worth $8M" (or $1M * the number of problems).

That's something of an exaggeration. What the speaker was probably referring to was that a non-deterministic Turing machine can easily find any mathematical proof (of a given length) once it is equipped with a formal proof verifier.

Therefore if P=NP we need only set up a sufficiently expressive verifier and then solve the Riemann hypothesis in polynomial time by searching the space of all potential proofs of less than, say, 10,000 pages of AutomatedTheoremProverSpeak. And if it came up empty then we'd know that it's false/true but unprovable/provable but the proof is ridiculously long.

But just because something is polynomial time doesn't mean it's practical to implement. Take the AKS primality test, for example, which has far greater value to number theorists than to cryptographers, since its O(n^6) running time is still too slow for primes of more than a few dozen digits. And if the P=NP algorithm was fast enough to be practical, why bother with only $1 million (or even $8 million) when the world's bank accounts are yours for the taking?

Nah, actually I'd be more in it for the mathematical fame than the money, so I'd want to publish it rather than going underground. But by then the U.S. would probably extradite me and have me executed under the terms of the super-DMCA or something.

Re:Has the Poincare Conjecture Been Solved?

[twistedcubic]

Actually, it has been solved. I've talked with some people in the know. It's the real deal this time.

Article by Milnor

[Tityrus]

Over at the site of the AMS, there is an interesting overview article by J. Milnor on the ideas behind the Poincare hypothesis and Perelman's proof. You don't have to be an expert in low dimensional topology to read this...
Milnor's article [ams.org]

Wrongly Stated

[Anonymous Coward]

The conjecture is state wrongly. Whoever wrote it should be banned to hell, or SCO.
First of all, you will NEVER bend a plane into a sphere. If you do so, you have solved our problems with mapping OUR planet, :o). You state that a 2D simply connected closed surface is always homotopic to a S^2(regular sphere).
The right conjecture is : any simply connected(no holes) 3 dimensional closed surface is homotopic to S^3. Simple hum?

But it seems the Russian professor did it, so I heard in the halls :o) .

Re:Has the Poincare Conjecture Been Solved?

[Anonymous Coward]

"into a shape without holes."

Dorkwad.

Poincare_Conjecture(n=3) := smooth Ricci Flow

[stock]

The Ricci spacetime curvature tensor is a contraction of the general Riemann spacetime curvature tensor. A contraction here just means a special case of Riemann. Basicly one has :

Ricci (Rij) = Riemann (Riajb) with "slots" 1 and 3 "contracted".

Perelman and Hamilton (correct me if mistaken) tried to do a opposite contraction of the Ricci spacetime curvature by making either "slot 1" or "slot 3" variable again. And of course also prove that Ricci Flow is Homeomorphic. Hamilton proved it for some relaxed Ricci Flow conditions, Pavelman took the full scale curvature to the test and apparently succeeded.

For some details read page 218 onto 224 and page 289,290 in the black book called "Gravitation". Those last 2 pages show how by applying the simplification of Riemann to a Ricci spacetime curvature in the case of a Euclidian/Newtonian metric (no special relativity) F = m.a = m.d2x/dt2, which is our daytime geodesic path on earth, the Newton law of gravitation shows up:

Fgrav = G.(m1.m2)/r^2

Searching for "Gravitation" on www.bn.com/ will show that book. The papers of Perelman can be found like this:

checkout http://eprints.lanl.gov/lanl/ and fillout "Perelman" in the Author Field and "Ricci Flow" in the Title/Subject/Abstract field

Robert