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Has The Poincare Conjecture Been Solved? 292

Posted by simoniker 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.
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Has The Poincare Conjecture Been Solved? More

Has The Poincare Conjecture Been Solved?

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  • I proved it (Score:2, Funny)

    by Anonymous Coward
    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.

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

    by James A. C. Joyce ( 733782 ) on Wednesday December 31, 2003 @10:38PM (#7850579) Homepage Journal
    No.

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

  • I, for one, (Score:4, Funny)

    by bersl2 ( 689221 ) on Wednesday December 31, 2003 @10:39PM (#7850584) Journal
    welcome our new topological overlord.

  • 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...
    • Well, it's a hard problem. If it was an easy problem, then it would have been solved.

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

    • 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).

      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

      • 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.
        • While it's bad to be prejudiced about such things, there is a huge amount of 'circumstantial' evidence that P != NP, and so it really would be a huge surprise if they turned out to be equal. (Similarly, most mathematicians believe that the Riemann Hypothesis and Poincare Conjecture are both true, and would be extremely surprised if they were not.)

          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)

      by 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 po
    • "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? :-P

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

    by SeanTobin ( 138474 ) * <byrdhuntr AT hotmail DOT com> on Wednesday December 31, 2003 @10:41PM (#7850602)
    Let me guess.. he says that the new topological object is universe-shaped?

  • Fuck! (Score:1, Funny)

    by Anonymous Coward
    I was planning on proving it for my Phd Thesis...
    Now, what am I supposed to talk about ? :/

  • mirror (Score:1, Troll)

    by Anonymous Coward
    in case of slashdotting...

    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)

      by 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 ;)

  • Finite Universe (Score:2, Interesting)

    by Anonymous Coward
    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
    • It helps, for the 3-D version, to visualize a square sheet of rubber with some thickness, rather than a cube. My visualizing intuition didn't want to swallow the idea of a rubber cube that was stretchy enough so that one face could be pulled around to touch the other.

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

      by Bombcar ( 16057 ) <racbmob@@@bombcar...com> on Wednesday December 31, 2003 @11:08PM (#7850716) Homepage Journal
      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!
  • Surely they mean obloid? An egg doesn't have holes. Can anybody provide a better description?
  • ah, yes, the red-headed stepchild of the conjecture family - Collatz!

    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]
    • 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.

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

    by muon1183 ( 587316 ) <(muon1183) (at) (gmail.com)> on Wednesday December 31, 2003 @10:49PM (#7850635) Homepage
    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).

    • 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)

      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.

      Kjella
      • 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

    • 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.
    • Question:

      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
      • 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.

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

    by Magic5Ball ( 188725 ) on Wednesday December 31, 2003 @10:49PM (#7850637)
    "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 :-)

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

      by sam_nead ( 607057 ) on Wednesday December 31, 2003 @11:17PM (#7850762)

      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.

      • Called the "Geometrization Conjecture," it is a far-reaching claim that joins topology and geometry, by stating that all space-like structures can be divided into parts, each of which can be described by one of three kinds of simple geometric models. Like a similar result for surfaces proved a century ago, this would have profound consequences in almost all areas of mathematics.

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

    • No. Dimensions don't scale like that, either up or down. The concept of a knot is nonsense in 2D, as an example. Making the case for up is more difficult, as it's hard to convince someone that something done in two dimensions in a 3D system isn't therefore significant in 3D. A good case of difficulty in explaining is that of complimentary angles, which can be displayed with two sticks and a hinge, but which may never have an effect outside that of an arbitrary plane cast in a 3D space. It's harder stil

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

      by Ibag ( 101144 )
      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, t
  • Pure science is pure science. All great discoveries that have ever existed have been because of small, previously unrelated pure mathematical works (or other pure true sciences), which when done on their own seem to have no superficial meaning to someone such as an engineer or common layman, but pure mathematics is akin to pieces of a grand puzzle. Each piece is intrinsically linked to the whole picture. Looking at each piece will not reveal the puzzle, although solving each piece on its own will. This proo

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

    by HaloZero ( 610207 ) <protodeka@@@gmail...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?
    • All easy jokes about New Jersey aside, that's pretty interesting. Presuming that the universe doesn't contain a mechanism for violating its own shape, I wonder if there's any significance to the shape of the universe. The bits about its growth and contraction seem likely targets, though honestly their mechanisms are above my head.

      And if you can find its significance, you can determine whether it's true. ;)
    • How do you know that the shape of the universe does not include holes?

      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)

    • What about sqaures, other 3D regular polygons or ellipsoids?
  • Comment removed based on user account deletion

  • Proof Smoof (Score:3, Informative)

    by Anonymous Coward on Wednesday December 31, 2003 @11:12PM (#7850737)
    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/
  • In what way would this proof be applied outside the realm of the mathematica and theory?

    Grand Unified Theory? Time Travel? Big Crunch?

    Dan East

  • Not the time... (Score:4, Funny)

    by Gyorg_Lavode ( 520114 ) on Wednesday December 31, 2003 @11:26PM (#7850803)
    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.
  • So where are the documents/papers by Dr. Perelman describing his proof of the Poincare Conjecture? Or are they on purpose not being put available for the grand public?

    Robert
  • I heard Al Gore solved this years ago.
  • Come on, what's all this science crap? Let's get back to rumor and innuendo.
  • I have a proof of Poincare's Conjecture, but it is too big to fit in the margins of this Slashdot post.
  • Did you hear about the constipated mathematician?

    He worked it out with a pencil.

  • Oh man... (Score:3, Funny)

    by robson ( 60067 ) on Thursday January 01, 2004 @01:51AM (#7851274)
    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.

  • formalize the proof (Score:4, Insightful)

    by penguin7of9 ( 697383 ) on Thursday January 01, 2004 @05:49AM (#7851831)
    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.
    • 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

      • 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
    • No, but they might find one such error, and that's all they'd need, right?

  • Article by Milnor (Score:2, Informative)

    by Tityrus ( 547161 )
    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]

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

    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.)

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

      by tc ( 93768 )
      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
  • I am not a mathematician, but would not either a mobius strip or a tauroid be without holes, smooth and continuous?

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

    by stock ( 129999 ) <stock@stokkie.net> on Friday January 02, 2004 @02:38AM (#7858187) Homepage
    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

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