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Chinese Mathematicians Prove Poincare Conjecture

Posted by ryuzaki0 on from the late-night-math dept.

Joe Lau writes to mention a story running on the Xinhua News Agency site, reporting a proof for the Poincare Conjecture in an upcoming edition of the Asian Journal of Mathematics. From the article: "A Columbia professor Richard Hamilton and a Russian mathematician Grigori Perelman have laid foundation on the latest endeavors made by the two Chinese. Prof. Hamilton completed the majority of the program and the geometrization conjecture. Yang, member of the Chinese Academy of Sciences, said in an interview with Xinhua, 'All the American, Russian and Chinese mathematicians have made indispensable contribution to the complete proof.'"

9 of 288 comments (clear)

  1. It's all a conjecture by The+Bungi · · Score: 5, Funny
    I looked at TFA, and I was kind of lost after reading this:

    In its original form, the Poincaré conjecture states that every simply connected closed three-manifold is homeomorphic to the three-sphere (in a topologist's sense) S^3, where a three-sphere is simply a generalization of the usual sphere to one dimension higher.

    Homeomorphic. Thank god, they dumb it down a bit later:

    More colloquially, the conjecture says that the three-sphere is the only type of bounded three-dimensional space possible that contains no holes. This conjecture was first proposed in 1904 by H. Poincaré Eric Weisstein's World of Biography (Poincaré 1953, pp. 486 and 498), and subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere if it is homeomorphic to the n-sphere. The generalized statement reduces to the original conjecture for n==3.

    More colloquially, it's homotopy-equivalent to the n-sphere! Of course!

    Slow news day?

    1. Re:It's all a conjecture by Barraketh · · Score: 5, Insightful

      Technically, it's more like proving P != NP, since that's the current accepted belief. Proving P=NP would be huge - this would give polynomial time algorithms for Travelling Salesman Problem, Boolean Satisiability Problem, and a slew of others (that all reduce to each other in polynomial time). Proving P != NP pretty much confirms what everyone believes to be true, similar to how the Poincaire conjecture was generally accepted to be true. Still, this is a major result, and clearly falls under the "News for nerds, stuff that matters" heading.

  2. This is... by mlow82 · · Score: 5, Interesting

    This is one of the Millennium Prize problems! One down, seven more to go!

    1. Re:This is... by nacturation · · Score: 5, Funny

      One down, seven more to go!

      Given that there are seven questions total, maybe you know the mystery surrounding the elusive eighth question: "What is seven minus one?"

      --
      Want to improve your Karma? Instead of "Post Anonymously", try the "Post Humously" option.
  3. Should share credit with Perelman by Stalyn · · Score: 5, Informative

    I thought the general consesus was that Perelman had proved Thurston's geometrization conjecture. If this proof by Zhu Xiping and Cao Huaidong is correct it must be a rephrasing of Perelman's work. Perelman is credited with making the major theoretical advances in order for any such proof. Basically he did most of the heavy lifting while these Chinese mathematicians basically dotted the i's and crossed the t's.

    The proof is 300 pages but I would guess the majority of it is an overview of Perelman's extension of Hamiltion's Ricci Flow.

    --
    The best education consists in immunizing people against systematic attempts at education. - Paul Feyerabend
  4. Math isn't dead by colin353 · · Score: 5, Interesting

    This is another reason why math isn't dead. The world's problems aren't solved, and they aren't impossible, either.

    I was just having a conversation about this yesterday with my math teacher.

    Lots of people think that high level math is just advanced adding and subtracting.

    This is good stuff. Props to Zhu Xiping and Cao Huaidong- this shows people that a career in studying mathematics is actually an interesting and rewarding career.

    --
    -- If unsure, say "Why?"
  5. Re:Ok, in plain english by Stalyn · · Score: 5, Informative

    In topology spheres are identical to cubes and pyramids. However spheres are not identical to doughnuts. What PC says is that spheres are the only class of objects that are not doughnut-like (has holes). This seems trivial and obvious to most of us however to prove it is really hard. What it shows is that there is something fundamental and important about the sphere-like class of objects. It also says something important about space itself.

    --
    The best education consists in immunizing people against systematic attempts at education. - Paul Feyerabend
  6. A translation... by FhnuZoag · · Score: 5, Informative

    First, think in Four Dimensions. Not in terms of time, or something, but as a fourth spacial dimension - like in terms of up down, left right, in out, and foo bar. A 3 sphere is a sphere in that sort of space. For example, in three dimensions, a 2-sphere is just a normal sphere - a group of points that are all the same distance from a certain centre point. A 3-sphere in 4 dimensions is just the set of points in four dimensional space that are the same 'distance' from a point in 4D. (We define distance using the pythagorus formula sqrt(x^2 + y^2 + z^2 + k^2).)

    A 3-manifold is another four dimensional object - in fact, a class of objects. They are the analogies of surfaces in 3D space, only again we have it in 4D space. The 3-sphere, for example is an example of a 3-manifold. Simple, connected and closed are two topological properties describing what a surface is like. In layman's terms, simple connected and close means that the surface is well... just an obvious surface. The simple-connected-closed-3-manifold taken together essentially rule out the bizzare sorts of objects that mathematicians come up with. There won't be any 'holes' in the object, and there won't be any non-solid boundaries, the object can't go through itself, and you can't take two seperate objects and pretend the pair is a single one.

    So what does the conjecture say? It says that if we have any 3-manifold satisfying certain properties, there is way of distorting it (that's basically what homeomorphism means. Like you take the object as a piece of putty and stretch and pull it, or fold it, or whatever without cutting or gluing bits together) to make it into a 3-sphere.

    It's a sort of bubblegum theorem. You can chew up the manifold and blow it into a bubble. (Okay, it's not really like that, topologists.... But it's close enough)

    1. Re:A translation... by sammy+baby · · Score: 5, Informative

      so you should be able to distort a shape in 4D to a 4D sphere! and it looks like it ought to apply to any number of dimensions as well - was proof required?

      There are some things which "seem" obvious to us which aren't necessarily so. In math classes that discuss Cantor's theorem, there are always a few holdouts that refuse to believe that one infinite set can be bigger than another infinite set. After all, they're both infinite. How could one be bigger than the other? And yet it's true, and Cantor demonstrated it in a way that's so cool that you can literally explain it on the back of a napkin.

      Likewise, there are certain things that are accepted as a given, until someone discovers/proves something that causes the known world to fall around your ears, mathematically speaking. Kurt Godel pulled the rug out from a whole slew of logicians by demonstrating that not everything that's true can be proven. Up until that time, the "completeness" of mathematics had been considered a given by some people.

      So yeah - on a naive level, it may seem like "making things all bendy" is obvious, but that doesn't mean it wasn't in need of a proof.