Chinese Mathematicians Prove Poincare Conjecture

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.'"

Re:Should share credit with Perelman

[gowen]

I can't see why Perelman would share the credit. If his results are right, he proved it first. A second proof is impressive (moreso if it contains anything particularly new), but until shown otherwise, Perelman was the first, so he gets all the marbles.

Re:It's all a conjecture

[Barraketh]

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.

Re:WDWC query

[jopet]

Why do you care about the arts, a clean apartment, love? Well, judging from your question, you probably don't but a lot of people do.
Not everything worthwile doing needs to result in amazing products.

Apart from this, mathematical insights, sometimes of the more dry and abstract sort *have* already resulted in amazing products (take public key encryption, the application of insights gained from number theory).

Re:It's all a conjecture

[Anonymous Coward]

Proving that P==NP wouldn't automatically give us polynomial time algorithms for any NP problem. The proof need not be constructive, and if it's not, it doesn't give algorithms. Granted, it seems easier to prove that P==NP by accidentally finding a polynomial time algorithm for an NP problem than otherwise, but don't assume that the prove would sove anything practical.

Chinese == Good at Math? Wrong!

[Anonymous Coward]

There are so many Chinese, some of them are bound to be good at math.

Re:plain english? Maybe...

[The Mathinator]

Not quite. The fact that the n-sphere is simply connected is pretty easy to prove. Poincare asked whether every closed simply connected 3-manifold is a 3-sphere. A surface is a 2-manifold. The sphere, plane, Mobius strip, Klein bottle, and so on are all 2-manifolds. A 3-manifold is just a natural extension of that idea, except instead of a surface, you have a 3-dimensional object. They're a bit hard to visualize, since most of them don't "fit into" our notion of space, in the same way that a sphere doesn't fit into a plane. Anyway, Poincare's original question In English: if you have a 3-manifold with no holes and no border, is it necessarily the 3-sphere? Translating the more general version into English is a bit more difficult, and I'll leave it to those who actually have experience with the problem. I just read the Wikipedia article. Just a bit more information from there that might be interesting: The problem is actually easier for higher dimensions. It was first shown for dimensions 7 and above, and then worked down to the lower dimensions.

Great. Still waiting for peer review..

[Anonymous Coward]

I'll accept the proof when it's been properly reviewed by peers. Just publishing a proof in a journal doesn't equate to a correct proof, now does it?

not necessarily

[m874t232]

The purpose of a proof is to communicate a sequence of statements such that each and every individual step is easily derivable from axioms or well-known theorems. Let me emphasize this again: a proof is about communication, not merely about making true statements.

Perelman apparently failed to do this: he may have produced a sequence of true statements that could somehow form a subsequence of a complete proof, but he has apparently not supplied enough detail to demonstrate his point to even specialists in his area. The fact that he may have done "the heavy lifting" or that he may have provided the key ideas doesn't change that.

I think it is valid to give all three mathematicians equal credit. And, strictly speaking, the people who actually have done the proof are the ones who "dotted the i's" because that's what ultimately constitutes a proof.

Re:Should share credit with Perelman

[gowen]

You haven't "proved" something until you have written it down in a form in which it convinces at least other specialists in your field.

That assertion is simply untrue.

Suppose (and this a deliberately perverse example), Fermat had secretly developed all the machinery for Wiles' proof of his Last Theorem, and gone on to prove it. None of his contemporaries could possibly understand it. But the theorem would've been proved, even if no-one knew it.

Re:Is there a math geek in the house?

[Wooster_UK]

I'm not too sure what you mean by "broken down into a circular-shaped plane", and I'd much sooner you lost the word "probably". I'll explain the conjecture by means of the two-dimensional version. Before I get there, I've got to explain what I mean by a "sphere", because the mathematical definition is quite specific. A "sphere" is the skin of a ball, okay, so it's all the points lying at a distance r, say, from the origin. Having been so specific about all that, I'm now going to be dreadfully, appallingly loose in the rest of my language. Here we go.

Now, suppose you've got a surface, let's call it S, which is bounded (so it's finite in any direction), closed (so it's not got an edge), and simply-connected (so it's got no holes). Then by twisting, stretching, moving and generally deforming S in any way you like, but without taking scissors to it, you can turn it into a sphere. That's the Generalised Poincaré Conjecture, reduced to 2 dimensions, and it was proved, oh, ages ago. To understand the higher dimension versions, just imagine doing that for an n-sphere, which is the set of all points lying at a distance r from the origin in n-dimensional space.

Re:... not yet. But it may die soon.

[Ithika]

And you apparently have no ability to read what the GP said. Specifically, he suggested that most of Wiles' effort was directed at proving the Taniyama-Shimura conjecture. From that point on, it was a simple step to prove Fermat's Last Theorem (for some extremely esoteric value of 'simple').

Note this line here:

Actually, Wiles proof of FLT is a simplification of the Taniyama-Shimura conjecture which he proved I believe.

Whether the grandparent poster's assertion about this method is accurate or not is neither here nor there. You managed to quote him grossly out of context and completely twisted the original message.

Re:It's all a conjecture

[IWannaBeAnAC]

That isn't quite true: you only need a polynomial time algorithm for a single NP-complete problem, and you can transform that into a polynomial time for all NP-complete problems.

True, that if there was a non-constructive proof that P==NP, it might not be obvious what the polynomial time algorithm actually is. But since such a scenario would be probably the most astounding open problem in the history of mathematics, I don't think it would be an open problem for long ;)

Re:Ok, in plain english

[Pendersempai]

Unfortunately this is wrong. A three-manifold is LIKE a sphere but in four dimensions. A sphere is a three-dimensional object with a two dimensional surface. A three-manifold is a four-dimensional object with a three dimensional surface. I suppose one way to visualize it is by taking our (three-dimensional) universe and imagining that if you traveled far enough in any direction you'd eventually end up back where you started, just like if an ant started walking in a straight line on the (two-dimensional) surface of a sphere he'd eventually end up back where he started.

The Poincare Conjecture says that every three-manifold that meets some conditions (no holes cut from its surface, it's all one object, etc.) can be smoothly distorted (through a process called homeomorphism) into any other three-manifold.

This is NOT true of two-manifolds: while you can smoothly distort (homeomorph) a sphere into a cube, for example, you cannot smoothly distort a sphere into a donut. This is because of the way we define a smooth distortion: at some point in the transformation you'd need to open up a hole in the sphere to make it into a donut, which disrupts the smoothness of the distortion. It's like if the cube were made of flexible rubber, you could bend it into a sphere, but you couldn't turn it into a donut without a pair of scissors and some glue. (This is all very hand-wavy, I know, but it's the best I can do without getting all technical.)

Re:Chinese == Good at Math

[Pinback]

If it follows the recent pattern, there will be a follow up story about this pair of Chinese mathematicians hiring other Chinese mathematicians to pretend the proof is real.