## Has The Poincare Conjecture Been Solved? 292

Posted
by
simoniker

from the let's-conject-together dept.

from the let's-conject-together dept.

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.
## I proved it (Score:2, Funny)

## Has the Poincare Conjecture Been Solved? (Score:5, Informative)

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

## Re:Has the Poincare Conjecture Been Solved? (Score:2)

Article also generously says

"This problem is like the Mount Everest of math conjectures, so everyone wants to be the first to climb it."I'd say Riemann-zeta holds that title; at least now that FLT is done.

## Re:Has the Poincare Conjecture Been Solved? (Score:2)

A torus also meets the same requirements. Plus makes mapping trig function like TAN() work.

Think four origins pairs... (0,0) (0,%) (%,0) and (%,%) Where % is infinity (sideways 8).

## Re:Has the Poincare Conjecture Been Solved? (Score:2)

## Re:Has the Poincare Conjecture Been Solved? (Score:2)

Actually, it has been solved. I've talked with some people in the know. It's the real deal this time.Until it has been peer reviewed and published, isn't that just Poincare Conjecture conjecture?

## I, for one, (Score:4, Funny)

## I thought... (Score:3, Interesting)

I was really hoping that that kind of money would get the P=NP results first...

## Re:I thought... (Score:3, Funny)

You might say that it's an NP hard problem. Hahah. *crickets* Oh well.

## Re:I thought... (Score:2)

Well, sure. If this vast class of problems thought to be intractable is found to be tractable, that would sure be nice. That doesn't mean it will ever happen. Realistically the only oddity is that nobody can pr

## Re:I thought... (Score:2)

Well, sure. If this vast class of problems thought to be intractable is found to be tractable, that would sure be nice.Defining all problems in P to be "tractable" turns out to be a pretty useless definition.

Realistically the only oddity is that nobody can prove P != NP, as is thought to be the case.It is bad to be prejudiced about such things.

## Re:I thought... (Score:2)

If P were equal to NP, then you could have a nice contructive proof in the form of a polynomial-time algorithm for an NP-complete problem. That's a pretty straigh

## Re:I thought... (Score:2, Informative)

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 po

## Re:I thought... (Score:2)

"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).P=NP where N=1. Now, where's that cheque?

## Re:I thought... (Score:2)

## Description of the new shape (Score:5, Funny)

## Fuck! (Score:1, Funny)

Now, what am I supposed to talk about ?

## mirror (Score:1, Troll)

BERKELEY, Calif. -- A reclusive Russian mathematician appears to have answered a question that has stumped mathematicians for more than a century.

After a decade of isolation in St. Petersburg, over the last year Grigory Perelman posted a few papers to an online archive. Although he has no known plans to publish them, his work has sent shock waves through what is usually a quiet field.

At two conferences held during the last two weeks in California, a range of specialists scrutini

## TROLL (Score:1, Informative)

"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 grinLast line, devious bugger

## Finite Universe (Score:2, Interesting)

Now we take the two circular edges and we glue them together, giving a donut (a torus). Now if you go in [what you see as] a straight line in any direction, you'll never reach an

## Re:Finite Universe (Score:2)

## Re:Finite Universe (Score:5, Informative)

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!

## Re:Finite Universe (Score:2, Funny)

a donut has a hole you dork!!!!Just what is this "dorking," and why do it to a donut?

But seriously. You're not following along. When you loop back the two openings to touch each other, you get a tube, just like a donut.

## Sphere? (Score:2)

## Re:Sphere? (Score:3, Interesting)

Even though they're topologically equivalent, I would have expected them to call the "obloids" or "closed simply connected two dimensional surfaces", instead of spheres. In linear algebra or measure theory its usually called a "ball".

## Is Collatz Next? (Score:1)

People have been going at Collatz Conjecture For Years, and maybe this recluse is giving that a swing next time.

For Information regarding Collatz Conjecture seek The Collatz Conjecture [216.239.39.104]

## Re:Is Collatz Next? (Score:2)

ah, yes, the red-headed stepchild of the conjecture family - Collatz!The answer to the collatz problem is yes, it does. Thank you very much.

## Re:Is Collatz Next? (Score:2)

"Mathematics is not yet ready for such problems."

And I agree with you 100%... thats what makes it so interesting... there are more interesting open problems than one could even read about in a lifetime... let alone solve.

## This Proof Isn't New (Score:5, Informative)

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).

## Re:This Proof Isn't New (Score:3, Funny)

What's really important is that this proof was put out by a reclusive Russian mathematician. That pretty much clinches it.

## Don't you hate that... (Score:4, Interesting)

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

## Re:Don't you hate that... (Score:2, Insightful)

It's rarely (sic) 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.Let's suppose that an angel appeared to your mother before you were born and asked her what gifts God should give to her child.

She, like all mothers, responds, "Please just let my child be healthy."

"Done," says the Angel, "but come on, surely you would like more for your child than that."

"Well," says your mother, "let my child be sm

## Re:Don't you hate that... (Score:2)

## Re:Don't you hate that... (Score:2)

## Re:Don't you hate that... (Score:2)

## Re:This Proof Isn't New (Score:3, Interesting)

## Re:This Proof Isn't New (Score:2)

## Re:This Proof Isn't New (Score:2)

I am not a math guru by any stretch of the imagination. I kicked butt in high-school geometry, but that was a long time ago.

If I have a one dimensional line and want it to bend it so it has no holes (or gaps I guess), it must be promoted to at least two dimensions. It becomes a circle in two and could be a knot (like a piece of string) in three.

If I have a two dimensional plane and want to bend it so it has no holes, it must be promoted to at least three dimensions. It becomes a sphere in

## Re:This Proof Isn't New (Score:2)

If I have a one dimensional line and want it to bend it so it has no holes (or gaps I guess), it must be promoted to at least two dimensions.Not true. The outline of a circle is one-dimensional because you can describe any point on it with a single Cartesian coordinate.

## Re:This Proof Isn't New (Score:2)

## I'm confused... (Score:3, Insightful)

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

## Re:I'm confused... (Score:5, Insightful)

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.

## can it be used in User Interfaces Re:I'm confused. (Score:2)

I was wondering if the concepts in the Proof can be used to User Interface (UI) Design because the User Interface is really a

## Re:I'm confused... (Score:2)

## Re:I'm confused... (Score:3, Informative)

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, t

## Pure science is pure science... (Score:1)

## Random thought... (Score:5, Interesting)

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.How do you know that the shape of the universe does not include holes?

## Re:Random thought... (Score:2)

And if you can find its significance, you can determine whether it's true.

## Re:Random thought... (Score:2)

We don't know that. There's nothing in physics to rule out topological holes, but the solutions to GR are so messy that no one really wants to go there (and there's no compelling reason to expect the univers has a a topologocal hole).

-JS (yes, IAAP)

## Re:Random thought... (Score:2)

## Re:Random thought... (Score:2)

## Re:Random thought... (Score:2)

## My fucking hero. (Score:2)

Colleagues say Perelman, who did not attend the California conferences and did not respond to a request for comment, couldn't care less about the money, and doesn't want the attention. Known for his single-minded devotion to research, he seldom appears in public; he answers e-mails from mathematicians, but no one else.My fucking hero. Thank you Russia for consistently producing maniacs like this man.

## Proof Smoof (Score:3, Informative)

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

## Usefulness? (Score:2)

Grand Unified Theory? Time Travel? Big Crunch?

Dan East

## Not the time... (Score:4, Funny)

## where are the documents? (Score:2)

Robert

## Re:where are the documents? (Score:2)

checkout http://eprints.lanl.gov/lanl/

and fillout "Perelman" in the Author Field and "Ricci Flow" in the Title/Subject/Abstract field

Robert

## Old News... (Score:2, Funny)

## I Fail to See the Relevance to SCO (Score:3, Funny)

## Mike's Last Theorem (Score:2, Funny)

## My uncle's joke (Score:2, Funny)

He worked it out with a pencil.

## Oh man... (Score:3, Funny)

Even in mathematical circles, surprisingly little is known about him, and those who know him often don't want to speak publicly about his work.Oh boy. People who know him won't talk about his work. That means bad news, I'm sure. Like... the proof solves the Poincare Conjecture, but as a byproduct it also proves that Cthulhu's going to wake up in 2005, and that he's really pissed.

## formalize the proof (Score:4, Insightful)

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.

## Re:formalize the proof (Score:2)

Even then, there is a big risk that there is some bug in the formalization of the proof.That doesn't matter so much. Think about it. Let's leave aside the possibility of mis-formalizing the axioms or the conclusion, since those are highly unlikely to be mis-formalized without being spotted as such pretty quickly.

Under that assumption, either you find out the proof can't be verified, and so you work out

why- which means you analyse it carefully and you eventually find an error either in the proof, the## Re:formalize the proof (Score:2)

That doesn't matter so much. [...] Under that assumption, either you find out the proof can't be verified,I think many incorrect formalizations of a proof will give rise to proofs that pass the verifier but don't actually prove what you thought they proved.

It's the proof verifier which really needs to be gone over with a fine tooth-comb - which is why I'd advocate that proof verifying software should be itself proven correct, and checked by a different piece of proof verifying software.I do not believ

## Re:formalize the proof Godel completeness issue (Score:2)

I think there is actually a famous problem about this having to do with orders of logic and completeness, maybe Godel's problem.No, this has nothing to do with Godel, or with anything complicated logical puzzle. It's a simple software engineering and risk management question.

It's probably best to just prove the proof verifier manually.As opposed to what? Proof verifiers take as input manually constructed proofs. So, a proof of the proof verifier's correctness is, of course, a manual proof. And you

## Re:formalize the proof Godel completeness issue (Score:2)

As opposed to verifying the proof verifier with another proof verifier.You are making a type error here. A proof verifier does not "check other proof verifiers", as you state, a proof verifier verifies manually constructed proofs.

What if the proof verifier is faulty? If it finds bugs, they may not be bugs at all and vice versa.Then you do what you always do when a proof verifier finds bugs in your manual proof: you fix your manual proof. That's the whole purpose of having a proof verifier: they help

## Re:formalize the proof (Score:2)

## Re:formalize the proof (Score:3, Insightful)

I'm glad you've contributed this gem of insight on such a difficult topic. Just think of all those mathematicians who have been wasting their time!They have. Pertti Lounesto, an expert on Clifford Algebras, went through the spinor and Clifford algebra literature with a fine tooth comb and found it to be rife with mistakes. Mathematicians he contacted would generally be unwilling to admit their mistakes even when presented with proofs. And there is no reason to believe that his specialty was any more pr

## Article by Milnor (Score:2, Informative)

Milnor's article [ams.org]

## In your face, Clay :-) (Score:4, Interesting)

If the proof is vetted, the Clay Mathematics Institute may face a difficult choice. Its rules state that any solution must be published two years before being considered for the $1 million prize. Perelman's work remains unpublished and he appears indifferent to the money.Hats off to Perelman for reminding us that money has

neverbeen a mathematician's incentive. The whole Clay thing is a travesty and not the right way to help mathematics.(Contrast: this sort [salon.com] of snake-oil merchant, who puts money over truth.)

## Re:In your face, Clay :-) (Score:3, Interesting)

Now, I'm sure it's a stretch to imagine that many kids are going to see coverage of the Poincare Conjecture and be sparked to become mathematicians

## IANAM - but... (Score:2)

## Poincare_Conjecture(n=3) := smooth Ricci Flow (Score:3, Informative)

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

## Re:In 2002, I researched the COSMIC background (Score:5, Funny)

(First off, remember that us MATHEMATICANS DO IT SMOOTHLY AND CONTINUOUSLY.)Yes but Godel showed that you never do it completely.

## Re:In 2002, I researched the COSMIC background (Score:2)

## Re:In 2002, I researched the COSMIC background (Score:2, Funny)

## Re:In 2002, I researched the COSMIC background (Score:5, Funny)

Applied mathematicians do it with a real-world model.

## Re: In 2002, I researched the COSMIC background (Score:4, Funny)

>

In 2002, I researched the COSMIC backgroundYeah, lots of people do that in college... Usually with the help of LSD and stuff.

## Re:In 2002, I researched the COSMIC background (Score:4, Funny)

"(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?

## -1 Troll (Score:2)

## Re:In 2002, I researched the COSMIC background (Score:2)

## Re:In 2002, I researched the COSMIC background (Score:2)

[...]this article has no mention of Poincare spaceUhm, it's mentioned 20 times, including a mention right in the abstract. Download the TeX source and look at lines 76, 144, 147, 158, 164, 175, 176, 180, 197, 215, 223, 226, 340, 342, 345, 386, 390, 399, 400, and 405.

## Re:In 2002, I researched the COSMIC background (Score:2)

## A line-by-line proof... (Score:4, Informative)

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.*

## Re:A line-by-line proof... (Score:3, Interesting)

A link to the Nature article has been posted, and the linked article includes the supposedly non-existent quote. Furthermore, the quote does turn up on google--try it yourself.

The article is titled "Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background," and the dodecahedral topology they're referring to is Poincare dodecahedral space, so I guess the conjecture has relevance after al

## Re:A line-by-line proof... (Score:2)

## Re:A line-by-line proof... (Score:3, Interesting)

The "425 2003 593" is simply a US court case reference numberThat doesn't look anything like a court case reference. However, it does look like a journal reference with the parens misplaced...and gosh, what do we find at Nature 425 (1993) 593?

Why, the article he cites, with the quote you claim is made up.

Idiot.

## Re:A line-by-line proof... (Score:2)

## Re:A line-by-line proof... (Score:2)

Did anyone else read this as PornCare?

## Re:A line-by-line proof... (Score:2)

One of the founding assumptions in Physics is that the laws of Physics are the same everywhere. We have no proof that that is the case, though, so sometimes it is necessary to specifically talk about Physics in our domain, ie locally.

## Re:In 2002, I researched the COSMIC background (Score:3, Informative)

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,

## LEDs? transistors? Er, no. (Score:2)

I don't think CPUs emitting hundreds of rads of gamma rays would go over very well anyway

## Re:Who Cares! or An Exciting Time To Be Alive (Score:5, Informative)

Ent 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:Who Cares! or An Exciting Time To Be Alive (Score:2)

Mathematically speaking, that's a pretty small world of difference you're talking about.

## Re:I've found a remarkable proof of this fact... (Score:2)

Work it out with a pencil.

## Re:Followup (Score:2)

As the old saying goes, work it out with a pencil.

## Re:Just a thought... (Score:2)

Lots of people think twelve, right now. Look into branes, superstrings, collapsed dimensions, rotational dimensions, the gyroverse, and so on.

http://arxiv.org/html/gr-qc/9912073

http://www

## Re:Just a thought... (Score:2)