According to this and various other sources. I didn't really know much about this problem, but it is of interest as it is the first of the millenium prizes that have strong claims to it.
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Poincare conjecture solved (1 Viewer)
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hyparzero
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i thought it was proved ages ago
We all thought Perelman had what could have been a proof, but he never submitted it for official verification until last year. Regardless of whether there's any flaws in the proof, it will still be at least two years or so before the prize is actually awarded. Although we have always believed that Poincare is one of the easier Millenium problems.
It is a well known problem in topology that has withstood 100 years of great minds. That and the blokes get $1 000 000. As in most discovories the applications may not be as evident now (or maybe they are but I'm not in the know) nonetheless it is an important advancement for that field.
The Poincare conjecture states that a closed, smooth, simply connected 3-manifold is diffeomorphic to S<sup>3</sup>:
Poincare, H. Cinquieme complementa l’analysis situs, Rend. Circ. Mat. Palermo. 18, 1904, pp. 45-110.
The first person to prove the Poincare conjecture was Grisha Perelman:
Perelman, G., The entropy formula for the Ricci flow and its geometric applications, November, 2002 (http://arxiv.org/pdf/math.DG/0211159)
Perelman, G., Ricci flow with surgery on three-manifolds, March, 2003 (http://arxiv.org/pdf/math.DG/0303109)
Perelman, G., Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, July, 2003 (http://arxiv.org/pdf/math.DG/0307245)
Thereafter it has been a grandiose fill-in-the-blanks exercise. Perelman's proof is sketchy and lacking in detail. The first to fill in the blanks were Bruce Kleiner and John Lott:
Kleiner, B. and Lott, J., Notes on Perelman's papers, May, 2006 (http://arxiv.org/pdf/math.DG/0605667)
The exact reference to the next proof by Cao and Zhu which you mention is
Cao, H-D. and Zhu, X-P., A Complete Proof of the Poincaré and Geometrization Conjectures - Application of the Hamilton-Perelman Theory of the Ricci Flow, Asian Journal of Mathematics 10 (2) pp. 165-492., June 2006. (http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-165-492.pdf)
Poincare, H. Cinquieme complementa l’analysis situs, Rend. Circ. Mat. Palermo. 18, 1904, pp. 45-110.
The first person to prove the Poincare conjecture was Grisha Perelman:
Perelman, G., The entropy formula for the Ricci flow and its geometric applications, November, 2002 (http://arxiv.org/pdf/math.DG/0211159)
Perelman, G., Ricci flow with surgery on three-manifolds, March, 2003 (http://arxiv.org/pdf/math.DG/0303109)
Perelman, G., Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, July, 2003 (http://arxiv.org/pdf/math.DG/0307245)
Thereafter it has been a grandiose fill-in-the-blanks exercise. Perelman's proof is sketchy and lacking in detail. The first to fill in the blanks were Bruce Kleiner and John Lott:
Kleiner, B. and Lott, J., Notes on Perelman's papers, May, 2006 (http://arxiv.org/pdf/math.DG/0605667)
The exact reference to the next proof by Cao and Zhu which you mention is
Cao, H-D. and Zhu, X-P., A Complete Proof of the Poincaré and Geometrization Conjectures - Application of the Hamilton-Perelman Theory of the Ricci Flow, Asian Journal of Mathematics 10 (2) pp. 165-492., June 2006. (http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-165-492.pdf)
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A few days ago a new proof appeared:
Morgan, J.W. and Tian, G., Ricci Flow and the Poincare Conjecture, July, 2006 (http://arxiv.org/pdf/math.DG/0607607)
Morgan, J.W. and Tian, G., Ricci Flow and the Poincare Conjecture, July, 2006 (http://arxiv.org/pdf/math.DG/0607607)
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There's a good article on it with these new developments in the September issue of the AMS Notices at
Jackson, A., Conjectures No More?: Consensus Forming on the Proofs of Poincaré and Geometrization Conjectures, Notices of the AMS, 53 (8), September, 2006, pp. 897-901. (http://www.ams.org/notices/200608/comm-perelman.pdf)
But there is an incorrect statement in it on page 899, about the Cao-Zhu article:
"The issue has not been made available electronically on the journal’s website and is available only as a printed paper publication."
This is incorrect. It is true that for a while, the entire paper was not available, but in fact it now is available. I've emailed the author correcting this.
The url is http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-165-492.pdf
Jackson, A., Conjectures No More?: Consensus Forming on the Proofs of Poincaré and Geometrization Conjectures, Notices of the AMS, 53 (8), September, 2006, pp. 897-901. (http://www.ams.org/notices/200608/comm-perelman.pdf)
But there is an incorrect statement in it on page 899, about the Cao-Zhu article:
"The issue has not been made available electronically on the journal’s website and is available only as a printed paper publication."
This is incorrect. It is true that for a while, the entire paper was not available, but in fact it now is available. I've emailed the author correcting this.
The url is http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-165-492.pdf
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There's an article on it in today's Australian wherein it says Perelman might get a Fields Medal for it:
http://www.theaustralian.news.com.au/story/0,20867,20155298-30417,00.html
http://www.theaustralian.news.com.au/story/0,20867,20155298-30417,00.html
It was in the SMH 21/8/2006:
http://www.smh.com.au/news/world/maths-genius-living-in-poverty/2006/08/20/1156012411120.html
Perelman won a Fields medal yesterday but didn't go to Spain to get the medal. The Herald article says he can't afford to go to get the Fields medal. But this afternoon's MX says that the real reason is that he's angry at not being re-elected as a member of St Petersburg's Steklov Mathematical Institute:
Major maths event minus a superstar, MX, August, 23, 2006, page 8. (see attachment below)
http://www.icm2006.org/dailynews/nota_prensa_en.doc
http://www.icm2006.org/dailynews/fields_perelman_info_en.pdf
http://www.icm2006.org/dailynews/perelman.jpg
http://icm2006.org/AbsDef/ts/Lottlight-GP.pdf
http://www.smh.com.au/news/world/maths-genius-living-in-poverty/2006/08/20/1156012411120.html
Perelman won a Fields medal yesterday but didn't go to Spain to get the medal. The Herald article says he can't afford to go to get the Fields medal. But this afternoon's MX says that the real reason is that he's angry at not being re-elected as a member of St Petersburg's Steklov Mathematical Institute:
Major maths event minus a superstar, MX, August, 23, 2006, page 8. (see attachment below)
http://www.icm2006.org/dailynews/nota_prensa_en.doc
http://www.icm2006.org/dailynews/fields_perelman_info_en.pdf
http://www.icm2006.org/dailynews/perelman.jpg
http://icm2006.org/AbsDef/ts/Lottlight-GP.pdf
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The most reputable sources are of course the official press releases.
However, I decided to include other sources because they often offer other aspects which are not in the official press releases.
I think the MX article is right.
It says on the AMS website at http://www.ams.org/dynamic_archive/home-news.html#fields-2006 "In announcing the medal awarded to Perelman, John Ball, president of the International Mathematical Union, stated that Perelman had declined to accept the medal. At a later press conference, Ball said that he had spoken personally and at length with Perelman about his decision to decline the medal. While Ball was disappointed that Perelman remained steadfast in refusing the award, he noted that it is Perelman's right to refuse the medal if he wishes and described their discussions as polite and pleasant. Ball also said that Perelman will be recorded as having been awarded a Fields Medal but as having declined to accept it."
Refusal to accept the medal is quite a different proposition than merely not being able to afford to go to Spain to get it.
So I think I'll keep the MX article rather than delete it.
However, I decided to include other sources because they often offer other aspects which are not in the official press releases.
I think the MX article is right.
It says on the AMS website at http://www.ams.org/dynamic_archive/home-news.html#fields-2006 "In announcing the medal awarded to Perelman, John Ball, president of the International Mathematical Union, stated that Perelman had declined to accept the medal. At a later press conference, Ball said that he had spoken personally and at length with Perelman about his decision to decline the medal. While Ball was disappointed that Perelman remained steadfast in refusing the award, he noted that it is Perelman's right to refuse the medal if he wishes and described their discussions as polite and pleasant. Ball also said that Perelman will be recorded as having been awarded a Fields Medal but as having declined to accept it."
Refusal to accept the medal is quite a different proposition than merely not being able to afford to go to Spain to get it.
So I think I'll keep the MX article rather than delete it.
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Interesting article. Ta.buchanan said:
More in today's SMH about Perelman::
http://www.smh.com.au/news/world/wh...p/2006/08/25/1156012739260.html?page=fullpage
http://www.smh.com.au/news/world/wh...p/2006/08/25/1156012739260.html?page=fullpage
Science magazine has announced that Perelman's proof is the "breakthrough of the year" for 2006:
http://www.sciencemag.org/cgi/reprint/314/5807/1848.pdf
http://www.sciencemag.org/cgi/reprint/314/5807/1848.pdf
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