Has The Poincare Conjecture Been Solved?

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.

Comments

Has the Poincare Conjecture Been Solved?

[James A. C. Joyce]

No.

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

Description of the new shape

[SeanTobin]

Let me guess.. he says that the new topological object is universe-shaped?

Re:In 2002, I researched the COSMIC background

[kurosawdust]

(First off, remember that us MATHEMATICANS DO IT SMOOTHLY AND CONTINUOUSLY.)

Yes but Godel showed that you never do it completely.

Re:I'm confused...

[sam_nead]

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.

Re:Who Cares! or An Exciting Time To Be Alive

[noonien_soong]

You seem to be misinformed. The Riemann hypothesis has not been proven. If it had, we would have heard about it; it is one of the current holy grails of mathematics. The 16th Hilbert problem has not been solved. The student in question only claimed to have solved part of it, and she was dead wrong. Positrons have nothing to do with LEDS, transistors, or diodes, and QED was not relevant to the invention of any of them. "structuring matters behaviors, including time-dependEnt transformations"---what does that even mean? Nothing. You made it up. Having a proof of Poincare's conjecture has absolutely nothing to do with crumple zones, or any engineering problem, for that matter.

I agree that it's an exciting time to be alive, but if you are as ignorant about science as your post would suggest, you would do well to confine your comments to generalities and stop spreading misinformation.