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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.
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Has The Poincare Conjecture Been Solved?

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

    by ameoba ( 173803 ) on Wednesday December 31, 2003 @10:41PM (#7850600)
    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...
  • Finite Universe (Score:2, Interesting)

    by Anonymous Coward on Wednesday December 31, 2003 @10:47PM (#7850624)
    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.

    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 ... which is identical to the last 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? [sethf.com]
    Censorship: The Battle Begins At Home [kuro5hin.org]
  • Random thought... (Score:5, Interesting)

    by HaloZero ( 610207 ) <protodekaNO@SPAMgmail.com> on Wednesday December 31, 2003 @11:02PM (#7850692) Homepage
    • 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:Proof Smoof (Score:1, Interesting)

    by Anonymous Coward on Wednesday December 31, 2003 @11:15PM (#7850750)
    linkified [discover.com]
  • Re:Sphere? (Score:3, Interesting)

    by Saint Stephen ( 19450 ) on Wednesday December 31, 2003 @11:16PM (#7850755) Homepage Journal
    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".

  • by Kjella ( 173770 ) on Thursday January 01, 2004 @12:01AM (#7850935) Homepage
    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.

  • by TedCheshireAcad ( 311748 ) <ted.fc@rit@edu> on Thursday January 01, 2004 @12:16AM (#7850975) Homepage
    Thurston Geometrization conjecture. I knew that guy was onto something. Saw him speak at an MAA meeting a few months ago, his brother is a physics professor at my school. Smart guy, understood the first 10 minutes of his talk though, being a lowly math undergrad and him being a Fields medal winner.
  • by slubberdegullion ( 544119 ) on Thursday January 01, 2004 @01:25AM (#7851192)
    Why the fuck is this "interesting"?? It's all wrong

    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.
  • by harlows_monkeys ( 106428 ) on Thursday January 01, 2004 @02:42AM (#7851406) Homepage
    The "425 2003 593" is simply a US court case reference number

    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.


  • by An Anonymous Hero ( 443895 ) on Thursday January 01, 2004 @09:51AM (#7852297)
    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 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.)

  • by tc ( 93768 ) on Thursday January 01, 2004 @12:48PM (#7852966)
    While I'm sure that professional mathematicians are not influenced by the money, I don't think the Clay Institute prize is by any means a travesty. After all, it raises awareness of mathematics to the general public. Having a big cash prize attached to something makes it more newsworthy (which might be a sad fact, but is hardly the fault of the Clay Institute).

    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.

The only possible interpretation of any research whatever in the `social sciences' is: some do, some don't. -- Ernest Rutherford