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Thanks!

送交者: mathjn 2006年6月06日09:44:40 于 [教育与学术]http://www.bbsland.com
回 答: You also missed the important work of W. Thurston! 由 mathjn 于 2006年6月06日08:35:10:

 

You's better divide Poncare conjecture into 2 parts;
1, high dimension (n>4), proved by S. Smale (and Stalling, but mainly S. Smale);
2, 3-D. In the late 70's, W. Thurston made his famous geometrization conjecture for 3-M compact manifold. He completely classify 3-D manifold, and proven for a large class of manifolds----the so-called Haken manifolds. But not general 3-D. His method is topological and some of the details hadn't been written down until now. (that was why some big guys, such as J.P.Serre, the first winner of the Abel criticize Thurston's Fields medal). Geometrization conjecture is more generel than Poincare conjecture, the xxxxer includes the latter.
3, 4-D. Mike Freedman proved it in the topological category, but not in the smooth category. S. Donaldson used some idea from Yang-Mills gauge theory (your fileds, hehe, you can say more) found astonishing properties of the smooth 4-D manifolds in the early 80's. but 4-D is MUCH HARDER than 3-D case. Because for 3-D, at least we had Thurston's conjecture. But for 4-D, so far, we still have not any even conjecture for the classification of the smooth compact 4-D manifolds. 4-D poincare conjecture is only a special case. BTW, Hamilton's Ricci Flow is also considered as a way to approach 4-D poincare conjecture.

In my personal opinion. You'd better add between youe stage 2 and 3 R. Hamilton's work. in 1982, R. Hamilton INVENTED Ricci flow and proved great theorem in 3-D manifold. His method was geometrical analysis, of course S.T.Yau was very interested in. S.T.Yau realized the potential power of the Ricci Flow so he suggested Hamilton try Poincare conjecture. Please note, S.T. Yau could not do Ricci Flow by himself. Because as a Fields medal, he should not join a field which was created by other people---his friend R. Hamilton. What he could do is to support it and ask his students do it. Yau also suggested Hamilton try to find the Harnack property in the Ricci Flow----the Li-Yau inequality in Ricci Flow. After several hard work, finally, Hamilton found it. In 1995, R. Hamilton propose a great proposal-----in the first time, R. Hamilton raised the "finite time singularities in the Ricci flow". IN that 100 pages long paper, R. Hamilton propose a way to prove 3-D poincare conjecture. That is why 1996-1997, Yau tried to push Chinese mathematision work in Ricci Flow, because R. Hamilton make the dream of proving 3-D poincare conjecture as a reasonably possibility.

I have to say, R. Hamilton himself can not finish his progrom. Just couldn't. Becasue in Perelman's papers, it needed some methods which R. Hamilton did not know. That is why it was G. Perelman win.

I say again, I said it several times: The most important contribution belong to R. Hamilton and G. Perelman, at least 90%!

It is so ridiculous that Yang Le, a person hasn't been doing research more than 10 years, and knows NOTHING about geometry, he claimed Chinese mathematicion make 30% contribution. It is ridiculous, just ridiculous. Cao-Zhu's work is great, but is demaged by Yang Le's stupid words.

Thanks again!



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