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

Random thought...

[HaloZero]

• 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?

Don't you hate that...

[Kjella]

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

Re:A line-by-line proof...

[slubberdegullion]

Why the fuck is this "interesting"?? It's all wrong

A link to the Nature article has been posted, and the linked article includes the supposedly non-existent quote. Furthermore, the quote does turn up on google--try it yourself.

The article is titled "Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background," and the dodecahedral topology they're referring to is Poincare dodecahedral space, so I guess the conjecture has relevance after all.

I think a lot of people have fallen for a troll, one named James A.C. Joyce.

In your face, Clay :-)

[An Anonymous Hero]

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 of snake-oil merchant, who puts money over truth.)