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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'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 :-)
  • by thelizman ( 304517 ) <hammerattack@yah o o .com> on Wednesday December 31, 2003 @11:06PM (#7850706) Homepage
    For those of you wondering "who cares" or "what's the point", well there really is none. Poincare's conjecture has no immediate practical application, or even in the near future. However, proving Poincare's conjecture is a shot in the arm for Special Relativity, which is still a "theory" (much like I have a theory that slashdot exists).

    This is an exciting time to be alive. The Riemann hypothesis has been proven [].The 16th Hilbert problem has been solved [] (by a student no less - proof that important discoveries in science are still an individual sport). After thousands of years, Archimedes Loculus has been solved []. While these are airy egg-head endeavours, so was once the notion of Diracs Quantum Electrodynamics. Today, the antimatter particles predicted by QED are used to image and diagnose diseases of the brain (Positron Emission Tomography), produce light (Light Emitting Diodes), and they make transistors and diodes work. Having a mathematical proof for Poincare's conjecture could lead to new ways of structuring matters behaviors, including time-dependant transformations. For instance, shorter crumple zones which absorb more energy in automotive collisions.
  • Just a thought... (Score:1, Insightful)

    by RedHat_Linux_Man ( 692702 ) on Wednesday December 31, 2003 @11:13PM (#7850743) Journal
    But our universe is not 3-dimensional... it is 4-dimensional(so far as we know):
    Physical dimensions 1.length, 2.width, 3.height
    AND 4.time
    So it would seem to me that Poincare would describe only the physical aspects of our universe, but not the universe as a whole.
    One other thing, we don't know for sure that there is only one way to bend 3 dimensional space into a shape with no holes', the dimension number/num. shapes could be related to a different pattern, such as the fibbonachi sequence (0,1,1,2,3,5,8...)

    I am only a high school math student, so if there are any other mathemeticians out there that can disprove any of my 'conjectures', please post.
  • 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.

  • by jeko ( 179919 ) on Thursday January 01, 2004 @12:45AM (#7851069)
    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 smarter than most."

    "Of course," says the Angel, blithely giving you an IQ of 101. "But wouldn't you like more? I am, after all, an Angel and can grant quite a bit."

    "Well," says your Mother, afraid to push her luck, "let my child be one out of a thousand."

    The Angel smiles as if at a small child and says "Wouldn't one in a million be better?"

    "Yes," says your Mother, scarcely believing her luck, "yes, let me child be one in a million. One in ten million," she blurts out impulsively, and then immediately cows a bit, fearing she's asked too much.

    "Yes," says the Angel, "I think we can do one in ten million," as he ascends to Heaven. Your mother can't believe her fortunes. Her child will be the smartest person in ten million.

    Which means there are about 25 people in the US alone Right Now who can intellectually make you their bitch, another dozen or two in Europe, while India and China have so many they could field a soccer league and not pick you for any teams.

    And throughout all recorded history?

    And suppose the Angel had made you the smartest throughout History? The responsibility would probably have crushed you like a bug.

    At least, thinking of it this way helps me keep my ego in what little tatters are left.

  • 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.
  • by penguin7of9 ( 697383 ) on Thursday January 01, 2004 @06:00PM (#7855003)
    I'm glad you've contributed this gem of insight on such a difficult topic. Just think of all those mathematicians who have been wasting their time!

    They have. Pertti Lounesto, an expert on Clifford Algebras, went through the spinor and Clifford algebra literature with a fine tooth comb and found it to be rife with mistakes. Mathematicians he contacted would generally be unwilling to admit their mistakes even when presented with proofs. And there is no reason to believe that his specialty was any more prone to mistakes than other areas of mathematics--it was just the field he was competent in to find errors by others.

    Mathematics, right now, is a field barely about philosophy in rigor and verifiability. Hopefully, computer science will set mathematics on the right path eventually and give it the tools to verify its results formally.

    By the way, it was probably an oversight, but in your post you forgot to give us the Unified Field Theory and a cure for cancer.

    Well, funny you should mention that. Physics and medicine demand experimental verification. It's only mathematics where people can get away with a bunch of people saying "yep, looks right to me".

"Card readers? We don't need no stinking card readers." -- Peter da Silva (at the National Academy of Sciencies, 1965, in a particularly vivid fantasy)