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Science

Poincaré Conjecture May Be Solved 299 299

Flamerule writes "The New York Times is now reporting that Dr. Grigori (Grisha) Perelman, of the Steklov Institute of Mathematics of the Russian Academy of Sciences in St. Petersburg, appears to have solved the famous Poincaré Conjecture, one of the Clay Institute's million-dollar Millennium Prize problems. I first noticed a short blurb about this at the MathWorld homepage last week, but Google searches have revealed almost nothing but the date and times of some of his lectures this month, including a packed session at MIT (photos), in which he reportedly presented material that proves the Conjecture. More specifically, the relevant material comes from a paper ("The entropy formula for the Ricci flow and its geometric applications") from last November, and a follow-up that was just released last month."
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Poincaré Conjecture May Be Solved

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  • Explanation (Score:5, Informative)

    by MaestroSartori (146297) on Tuesday April 15, 2003 @09:52AM (#5735331) Homepage
    Shamelessly stolen from here [claymath.org]:

    If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.


    Now, can someone tell me what practical applications there might be of this? Or is it strictly an abstract concept?
  • Google Partner Link (Score:3, Informative)

    by Anonymous Coward on Tuesday April 15, 2003 @09:52AM (#5735334)
    For the lazy/paranoid [nytimes.com].
  • Explanation (Score:2, Informative)

    by Andy Tanenbaum (655028) on Tuesday April 15, 2003 @09:53AM (#5735337) Journal
    For those who do not know about the Poincare Conjecture, copied from http://www.claymath.org/Millennium_Prize_Problems/ Poincare_Conjecture/ If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.
  • by n3k5 (606163) on Tuesday April 15, 2003 @09:54AM (#5735349) Journal
    for the first time in ages, I'm looking forward to the discussion on this, in the hope that someone explains it in a manner I can understand
    The explanation in the article [nytimes.com] is not too bad; the Wikipedia [wikipedia.org] contains a better explanation [wikipedia.org]:
    [The Poincaré] conjecture is that every simply connected compact 3-manifold without boundary is homeomorphic to a 3-sphere.


    Loosely speaking, this means that every 3-dimensional object that has a set of sphere-like properties can be stretched or squeezed until it is a 3-sphere without breaking it. Note that a 3-sphere consists of all those points in 4-dimensional space R4 that have a distance of 1 from the origin.
  • by Darnit (75420) on Tuesday April 15, 2003 @09:55AM (#5735356)
    Dunwoody [wolfram.com]

    It seems as if he missed a step and couldn't figure it out.
  • What is it ? (Score:2, Informative)

    by Anonymous Coward on Tuesday April 15, 2003 @09:55AM (#5735360)

    Easy, i shall explain

    The conjecture that every simply connected closed 3-manifold is homeomorphic to the 3-sphere. This conjecture was first proposed in 1904 by H. Poincaré (Poincaré 1953, pp. 486 and 498), and subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere iff it is homeomorphic to the n-sphere. The generalized statement reduces to the original conjecture for n = 3.

    The Poincaré conjecture has proved a thorny problem ever since it was first proposed, and its study has led not only to many false proofs, but also to a deepening in the understanding of the topology of manifolds (Milnor). One of the first incorrect proofs was due to Poincaré himself (1953, p. 370), stated four years prior to formulation of his conjecture, and to which Poincaré subsequently found a counterexample. In 1934, Whitehead (1962, pp. 21-50) proposed another theorem which proved to be incorrect, then discovered a counterexample (the Whitehead link) to his own theorem.

    The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical, 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 by Zeeman (1961), n = 6 by Stallings (1962), and by Smale in 1961. Smale subsequently extended his proof to include .

    you see ?, its all quite clear if you think about it
  • by rasafras (637995) <`ude.uhj.ahp' `ta' `samat'> on Tuesday April 15, 2003 @09:57AM (#5735371) Homepage
    It doesn't appear that the paper will survive the two years... [wolfram.com]

    From the site:
    It is unclear as of this writing if Dunwoody's proof will last even a fraction of that duration.

    In fact, it appears that the purported proof has already been found lacking, judging by the facts that (1) the abstract begins, "We give a prospective [italics added] proof of the Poincaré Conjecture" and (2) the revised April 11 version of the preprint contains a small but significant change in title from "A Proof of the Poincaré Conjecture" to "A Proof of the Poincaré Conjecture?" In particular, a critical step in the paper appears to remain unproven, and Dunwoody himself does not see how to fill in the missing proof.
  • by King Babar (19862) on Tuesday April 15, 2003 @09:59AM (#5735386) Homepage
    So, why the excitment about this later Perelman paper? Has the Dunwoody paper been debunked?

    A gap or three in the proof were found within days, and a mathematician friend of mine reported that it didn't look like solutions to these problems were immediately forthcoming.

    The excitement about this paper comes from the fact that the guy who did the work has come up with impressive results in the past, builds on important and cutting edge work, and seems to have a really thorough command of the potential difficulties. (In other words, when he is asked questions about the tricky points, he immediately responds with what look like strong and well-thought-out answers.) For that matter, his work claims to prove a more general conjecture of which Poincare is a special case, and so this work could have more general significance to many other problems, even if there turns out to be a glitch or two in this iteration of the proof.

    It's a very hard problem, and this answer could be wrong, too. But there's a big difference between tossing a paper up on a preprint server and giving a lecture at MIT where nobody can (yet) touch you. :-)

  • Re:Y'know (Score:2, Informative)

    by robslimo (587196) on Tuesday April 15, 2003 @10:12AM (#5735479) Homepage Journal
    Yah. Looks like NYT got wise to us. Replacing 'www' with 'archive' no longer works. Just redirects to the main page.

    So here is the Google/NYT partner link [nytimes.com]

  • by MarvinMouse (323641) on Tuesday April 15, 2003 @10:18AM (#5735525) Homepage Journal
    translation to make it easier.

    basically all the poincare conjecture says is that if you have a 3 dimensional figure which is closed (therefore, it it bounded (doesn't go off to infinity in either direction), and doesn't have any "holes" in it (like a donut)) then you can take every point and map it to a point in an equivalent sphere without losing continuity (therefore, everypoint will have the same "neighbourhood" of points as it had in the initial shape.)

    ie. You can map a cube into a sphere, or a dodecahedron, or a weird globlike thing that doesn't fold back on itself, or a whole piece of paper (without holes), or a pencil, or a lot of different figures.

    As well, this conjecture also handles figures with holes in them (like donuts), and maps them all to simpler figures.

    It's a very simple concept, but has been incredibly hard to prove, and what makes this conjecture even more frustrating is the fact that 1 and 2-dimensional forms of this conjecture were incredibly easy to prove, as well as 4 and up have been solved, and were reasonably easy as well. Yet for some reason the 3 dimensional version does not lend itself easily to a simple proof.

    Everyone generally believes this is true, but no one has been able to prove or disprove it.

    If proven, this is an important aspect of topology, because then we can map all n-dimensional figures to a simpler form (like a sphere) and know that the continuity and general structure of the figure will remain the same.
  • by Alsee (515537) on Tuesday April 15, 2003 @10:31AM (#5735615) Homepage
    It's so simple when you put it in plain english ...
    [/sarcasm]


    Ok, try this:

    We long ago proved that an ordinary sphere is the only shape in 3 dimentions with no holes in it.

    Note that the "shape" is "made of clay". You are allowed to stretch it, squish it, and bend it all you want. You aren't allowed to cut it or put a hole in it. And you can't "meld" parts togther.

    A coffee cup is the same "shape" as a donut because you can smoothly "flow" the cup part into the handle and you get a donut.

    What they just proved is that a 4-dimentional sphere is the only shape with no holes in it.

    So what? Well if you have some wierd complex 4 dimentional "thing" and you know it doesn't have any holes in it then you now know it has to be equal to a sphere. It SEEMS obvious, but it was still important to prove. It is important for many other proofs.

    Better?

    -
  • by Anonymous Coward on Tuesday April 15, 2003 @10:42AM (#5735718)
    Perelman isn't claiming just to have proved the Poincare Conjecture -- he's claiming to have proved the Thurston's Geometrization Conjecture [http], a much more general (and harder to explain) result. Basically, while the Poincare Conjecture just says things about 3-spheres (namely every "simply-connected" 3-manifold is a 3-sphere), the Geometrization Conjecture says that _any_ compact Riemannian 3-manifold is built in a specific way from a handful of basic building blocks (the important thing here is that you're not just considering the manifold structure, but the metric structure as well).

    Anyway, if true, this is kind of like Wiles proof of Fermat's Last Theorem -- proving an old conjecture by proving a more general (and more modern) one (in Wiles case, it was proving part of Taniyama-Shimura).

  • by H*(BZ_2)-Module (536932) on Tuesday April 15, 2003 @11:33AM (#5736161)
    Because if you are standing on a "good-old friendly" sphere then it looks 2-dimensional. This is describing a local property of the object, in other words, if you stand at any point on the sphere what is in your immediate viscinioty looks like a 2-dimentional disc. Simmilarly a circle is called a 1-sphere, because if you pick any point on it it what is around you looks like a 1-dimensional line segment(you could also call it a 1-dimensional disc). You can then carry this up to higher dimentions, so that a 27-dimentional sphere is a sphere that looks like a 26-dimentional disc locally. This way of thinking doesn't really break down until you get to an infinite dimentional sphere.
  • Re:Explanation (Score:4, Informative)

    by Gleef (86) on Tuesday April 15, 2003 @11:38AM (#5736220) Homepage
    Some uses for topology:
    http://www22.pair.com/csdc/car/carhomep.htm [pair.com]

    Granted, none of this is stuff I would expect a gas station attendant to be playing with, but it's apparently increasingly important for researchers and engineers.
  • Re:Explanation (Score:3, Informative)

    by metlin (258108) on Tuesday April 15, 2003 @12:03PM (#5736446) Journal
    Now, can someone tell me what practical applications there might be of this? Or is it strictly an abstract concept?

    Hmmm, a lot of work in mathematics may not have immediate applications or uses. But down the line, they just might get used.

    As many posters have already mentioned, Boolean algebra is one such case, and another example would be the work done by Fourier [mathphysics.com] - particularly his integral transforms and series.

    I mean, today these are used so much in DSP and the like, I doubt Fourier had these in mind when he worked it out in the early 1800s :-)

    Although a lot of pure mathematicians [st-and.ac.uk] may take pride in the fact that their work might just never get used, one can never be so sure ;-)
  • by Anonymous Coward on Tuesday April 15, 2003 @02:33PM (#5737825)
    Well, IAAT so I might as well try to put in a 3-manifold topologist's perspective. First: the Poincare conjecture. Instead of talking about "holes" and whatnot, I would have put it like this:

    First, what is a manifold? Well, take a bunch of tetrahedra, and start gluing their sides together in pairs. Start with finitely many, pair the sides up, and say how the sides are matched. What you have at the end will be a closed 3-manifold, providing every face gets glued to exactly one other face, and providing some number you can calculate called the "Euler characteristic" is equal to zero. The Euler characteristic is just
    the number of vertices, minus the number of edges, plus the number of triangles, minus the number of tetrahedra *after* you have glued it all up. You now have your manifold M.

    Simply connected has "something" to do with holes, but I think it's easy enough to say exactly what it is. Think of the unit circle in the plane, the set of points which are distance 1 from the center of the plane. Then think of a continuous function from the circle to the manifold. That is, for every point in the circle, you get a corresponding point in M such that when you vary the point you choose in the circle continuously, the corresponding point in M moves continuously. Since a circle is 1-dimensional, we can wiggle the image a little bit so that it doesn't cross itself; we call this image a *knot*. "Simply-connected" means that every continuous map from a circle to a knot in M extends to a continuous map of the unit disk. So the knot "bounds" a disk in M (which *is* allowed to intersect itself, and probably has to) which gives a way of shrinking the knot down to a point continuously.

    The Poincare conjecture claims that the only 3-manifold M with this property is the 3-sphere.
    Well, what is the 3-sphere? Take two solid balls and completely glue their boundaries together. What you get is the 3-sphere. Another description is as the set of points in 4-dimensional space which are distance 1 from the origin, just like the circle was the set of points in 2-dimensional space (i.e. the plane) at distance 1 from the origin. How would you prove such a thing? Well, one way is to use a criterion of Bing, who showed that any closed manifold with the following property is S^3: Bing's property says "every knot in M is contained in a solid ball in M". There are other criteria, but part of the problem is that they are very hard to check or verify, and the hypothesis (that S^3 is simply connected) is hard to use.

    So, what does Perelman do? He actually proves not just the Poincare Conjecture but a much stronger conjecture called Thurston's Geometrization Conjecture. Unlike the Poincare Conjecture, which is a conjecture just about S^3, Thurston's conjecture is a conjecture about *every* 3-manifold. It says, roughly speaking, that every closed 3-manifold which is *irreducible* (i.e. every sphere bounds a ball) can be cut up into a finite number of pieces which have a canonical "geometric structure". What is meant by a geometric structure? Well, a football and a soccerball are both *topologically* spheres, but they have different *geometries*; the football is pointy at both ends, but the soccerball is perfectly round everywhere. So the soccerball is "geometric" since it doesn't have odd bumps or lumps, but looks the same everywhere. It turns out in 3 dimensions there are 8 different ways that a small piece of a geometric manifold can look, and one of these is the geometry of S^3. It's not hard to show that any *simply connected* 3-manifold with the geometry of S^3 is actually equal to S^3, so Thurston's conjecture implies the Poincare conjecture, and that's what Perelman proves.

    How does he do it? Well, he starts off with the manifold, and if it's not geometric, he starts to deform it so that it looks more and more geometric. That's the "Ricci flow" bit - it means that if you have some direction that looks pointy, you stretch it out, and if you have some direction which looks more stretched, you squash it do

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