Has The Poincare Conjecture Been Solved? 292
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 thought... (Score:3, Interesting)
I was really hoping that that kind of money would get the P=NP results first...
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 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
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What Happened to the Censorware Project? [sethf.com]
Censorship: The Battle Begins At Home [kuro5hin.org]
Random thought... (Score:5, Interesting)
How do you know that the shape of the universe does not include holes?
Re:Proof Smoof (Score:1, Interesting)
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".
Don't you hate that... (Score:4, Interesting)
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:This Proof Isn't New (Score:3, Interesting)
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 all.
I think a lot of people have fallen for a troll, one named James A.C. Joyce.
Re:A line-by-line proof... (Score:3, Interesting)
That 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.
In your face, Clay :-) (Score:4, Interesting)
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 [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 as a result, but I think in these days when many kids (and adults) are almost proud to be virtually innumerate, anything which brings maths to mainstream attention can't be a bad thing.