Slashdot Mirror
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.
But I've been too busy trying to get first post to tell someone. I wonder how many other huge discoveries are stopped by the same problem. It's a good thing Einstein didn't have Slashdot.
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.
I remember seeing a (webcasted) talk given by the Clay Institute about their $1M math prizes, in particular, the one about P=NP. In it, 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).
I was really hoping that that kind of money would get the P=NP results first...
my sig's at the bottom of the page.
Let me guess.. he says that the new topological object is universe-shaped?
Karma: SELECT `karma` FROM `users` WHERE `userid`=138474;
Imagine a square sheet of rubber (so we can stretch, bend as we like). It has a finite area, and four edges. We choose one edge and glue it to its opposite edge. Now if you start from one point and draw a line in the right direction, you'll get back to where you started. Otherwise you'll just spiral around until you hit an edge.
... which is identical to the last one.
...
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 edge. The surface of the donut doesn't have any sides in the way the original sheet of rubber did, but it still covers a finite area.
N.b. The problem with this example is that it's difficult to think of just the surface of the donut, without imagining it being 'in' some larger space such as the 3D world.
Now if you want a headache, try to imagine doing this starting not with a square, but rather a cube, and joining opposing faces together. The first pair is easy - you get a sort of square donut shape. The second pair gives you a donut with an inner donut removed - something like the inner tube in a tyre.
The third one is the real bugger - you have to imagine joining the inner surface of the tube to the outer one, without going through the tube. I've seen a video [uiuc.edu] that included a representation of what a similar manouvre (sp?) would look like in the 3D world that the cube started in, and I still can't fully get my head around it.
No matter what direction you moved in this weird twisted-cube-thingy, you'd never see an edge. It would give you the same effect as if there were an infinite array of cubes , with the exact same thing happening in each one. When you reach the edge of one cube, you ust move into the next one
This article says that the Universe is doing the same sort of thing, only starting with a dodecahedron instead of a cube (i.e. 6 pairs of faces instead of 3). Don't seriously try to picture this, or your head'll explode
-----
What Happened to the Censorware Project?
Censorship: The Battle Begins At Home
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?
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/
Oh, I knew that. I got a BS in math, but I never took any topology classes. I know a bit informally though, via my bro. in law who got a masters.
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".
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.
In his case, it's all imaginary.
One line blog. I hear that they're called Twitters now.
I heard Al Gore solved this years ago.
John Kerry is a Joke!
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
Come on, what's all this science crap? Let's get back to rumor and innuendo.
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 have a proof of Poincare's Conjecture, but it is too big to fit in the margins of this Slashdot post.
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.
Did you hear about the constipated mathematician?
He worked it out with a pencil.
When you look at the state of the world, how can you not become a radical, liberal anarchist?
From the article:
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.
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.
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
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.)
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