Slashdot Mirror
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.'"
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.
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.
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?
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.
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.
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 ;)