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 2003-12-31, 18:03 #1 Nebob   Sep 2002 11 Posts Poincare conjecture proven?
 2003-12-31, 21:30 #2 GP2     Sep 2003 1010000110012 Posts This is not really news. It's been known for many months now that a proof has been claimed. And it's a very serious proof, it's a Russian guy who pulled an "Andrew Wiles", ie, worked on it in seclusion for many years without publishing anything and finally inventing some new math to come up with a lengthy, complicated proof. He's given seminars and has had a ready answer for any points or issues raised, obviously has thought very deeply about it for a long time. So it will probably be like FLT... there won't be any definite moment in time when people say "aha it's proved". Rather, after enough time without a flaw or counterexample found, acceptance will gradually grow stronger.
 2004-05-17, 14:58 #3 jinydu     Dec 2003 Hopefully Near M48 2×3×293 Posts http://mathworld.wolfram.com/PoincareConjecture.html Anyone heard of updates on the verification of this proof? Its been some time since it was first announced. Unfortunately, there are many times when I hear about new developments in math or science in the news, but no follow-up reports appear for a LONG time, if ever.
2004-05-17, 15:44   #4
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"Luigi"
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Quote:
 Originally Posted by jinydu http://mathworld.wolfram.com/PoincareConjecture.html Anyone heard of updates on the verification of this proof? Its been some time since it was first announced. Unfortunately, there are many times when I hear about new developments in math or science in the news, but no follow-up reports appear for a LONG time, if ever.
AFAIK, the russian discoverer doesn't reply to emails coming from non-math gurus, doesn't like money, is a bit misanthropist...

Luigi

 2004-09-07, 07:10 #5 jinydu     Dec 2003 Hopefully Near M48 2·3·293 Posts A new article on Yahoo News, but the article doesn't seem to have much new in it: http://story.news.yahoo.com/news?tmp...ience_maths_dc By the way, that article doesn't give the links to Perelman's papers, so I give them here: http://www.arxiv.org/abs/math.DG/0211159 http://www.arxiv.org/abs/math.DG/0303109 Last fiddled with by jinydu on 2004-09-07 at 07:11
2004-09-07, 07:26   #6
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"Luigi"
Aug 2002
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Quote:
 Originally Posted by jinydu A new article on Yahoo News, but the article doesn't seem to have much new in it:

Luigi

 2004-09-07, 08:20 #7 jinydu     Dec 2003 Hopefully Near M48 2×3×293 Posts There has been one more recent paper by Perelman: http://www.arxiv.org/abs/math.DG/0307245 It was published after the Mathworld article, so there was no link to it from the Mathworld article, which is why it didn't appear in my last post. Actually, Poincare's Conjecture isn't mentioned in the abstracts of any of the 3 papers. Thurston's geometrization conjecture is mentioned once in the abstract of the 1st paper: "We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound." [It seems to me he's saying that he's giving just a sketch of a proof, not a fully worked out proof]. The 3rd paper contains the sentence: "Our argument (in conjunction with [P, [some symbol I don't know about] 1-5]) also gives a direct proof of the so called "elliptization conjecture". However, it appears on the 1st page of the paper, not on the absract. This might be the closest that Perelman gets to explicitly claiming that he has solved the Poincare Conjecture. According to: http://mathworld.wolfram.com/Thursto...onjecture.html "Since the trivial group is in particular a finite group, the elliptization conjecture implies the Poincaré conjecture." Last fiddled with by jinydu on 2004-09-07 at 08:28
 2004-09-07, 12:54 #8 Uncwilly 6809 > 6502     """"""""""""""""""" Aug 2003 101×103 Posts 31×317 Posts If he doesn't want the $1,000,000, I will take it. 2006-08-15, 19:56 #9 ewmayer 2ω=0 Sep 2002 República de California 19·613 Posts And the Winner is... There's a long article on this in today's New York Times, to the effect that the consensus is that the PC has been proven by Perelman et al. (the wrangling over the "et al" may take some time), and that Perelman himself has apparently vanished off the face of the earth: http://www.nytimes.com/2006/08/15/science/15math.htm I find the part about the Ricci flow especially interesting, since the mathematics there has deep connections with the maths of general relativity. The resulting problems with singularities remind one of renormalization in quantum mechanics. Fascinating stuff. Article text, in case the above link eventually becomes inactive: Quote:  Elusive Proof, Elusive Prover: A New Mathematical Mystery By DENNIS OVERBYE Published: August 15, 2006 Grisha Perelman, where are you? To a topologist, a rabbit is the same as a sphere. Neither has a hole. Longitude and latitude lines on the rabbit allow mathematicians to map it onto different forms while preserving information. Three years ago, a Russian mathematician by the name of Grigory Perelman, a k a Grisha, in St. Petersburg, announced that he had solved a famous and intractable mathematical problem, known as the Poincaré conjecture, about the nature of space. After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world’s mathematicians to pick up the pieces and decide if he was right. Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose between them. As a result there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought. “It’s really a great moment in mathematics,” said Bruce Kleiner of Yale, who has spent the last three years helping to explicate Dr. Perelman’s work. “It could have happened 100 years from now, or never.” In a speech at a conference in Beijing this summer, Shing-Tung Yau of Harvard said the understanding of three-dimensional space brought about by Poincaré’s conjecture could be one of the major pillars of math in the 21st century. Quoting Poincaré himself, Dr.Yau said, “Thought is only a flash in the middle of a long night, but the flash that means everything.” But at the moment of his putative triumph, Dr. Perelman is nowhere in sight. He is an odds-on favorite to win a Fields Medal, math’s version of the Nobel Prize, when the International Mathematics Union convenes in Madrid next Tuesday. But there is no indication whether he will show up. Also left hanging, for now, is$1 million offered by the Clay Mathematics Institute in Cambridge, Mass., for the first published proof of the conjecture, one of seven outstanding questions for which they offered a ransom back at the beginning of the millennium. “It’s very unusual in math that somebody announces a result this big and leaves it hanging,” said John Morgan of Columbia, one of the scholars who has also been filling in the details of Dr. Perelman’s work. Mathematicians have been waiting for this result for more than 100 years, ever since the French polymath Henri Poincaré posed the problem in 1904. And they acknowledge that it may be another 100 years before its full implications for math and physics are understood. For now, they say, it is just beautiful, like art or a challenging new opera. Dr. Morgan said the excitement came not from the final proof of the conjecture, which everybody felt was true, but the method, “finding deep connections between what were unrelated fields of mathematics.” William Thurston of Cornell, the author of a deeper conjecture that includes Poincaré’s and that is now apparently proved, said, “Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide,” explaining that curiosity is tied in some way with intuition. “You don’t see what you’re seeing until you see it,” Dr. Thurston said, “but when you do see it, it lets you see many other things.” Depending on who is talking, Poincaré’s conjecture can sound either daunting or deceptively simple. It asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere. The conjecture is fundamental to topology, the branch of math that deals with shapes, sometimes described as geometry without the details. To a topologist, a sphere, a cigar and a rabbit’s head are all the same because they can be deformed into one another. Likewise, a coffee mug and a doughnut are also the same because each has one hole, but they are not equivalent to a sphere. In effect, what Poincaré suggested was that anything without holes has to be a sphere. The one qualification was that this “anything” had to be what mathematicians call compact, or closed, meaning that it has a finite extent: no matter how far you strike out in one direction or another, you can get only so far away before you start coming back, the way you can never get more than 12,500 miles from home on the Earth. In the case of two dimensions, like the surface of a sphere or a doughnut, it is easy to see what Poincaré was talking about: imagine a rubber band stretched around an apple or a doughnut; on the apple, the rubber band can be shrunk without limit, but on the doughnut it is stopped by the hole.

Last fiddled with by ewmayer on 2006-08-22 at 19:37

2006-08-15, 19:56   #10
ewmayer
2ω=0

Sep 2002
República de California

19·613 Posts

(continued from above)

Quote:

Last fiddled with by ewmayer on 2006-08-22 at 19:38 Reason: Reduced text size of quoted material

 2006-08-17, 12:24 #11 mfgoode Bronze Medalist     Jan 2004 Mumbai,India 22×33×19 Posts Poincare's conjectured proved ! From Slashdot.org 0]Flamerule writes "A New York Times article has finally provided an update on the status of [1]Grigori Perelman's 2003 rough proof of the [2]PoincarÃ© Conjecture. 3 years ago, Perelman published several papers online explaining his idea for proving the conjecture, but after giving lectures at MIT and several other schools ([3]covered on Slashdot) he returned to Russia, where he's remained silent since. Now, mathematicians in the US and elsewhere have finally finished going over his work and have produced several papers, totaling 1000 pages, that give step-by-step, complete proofs of the conjecture. In addition to winning some or all of the \$1,000,000 [4]Millennium Prize, Perelman now seems to be the favorite to receive a [5]Fields Medal at the International Mathematics Union meeting next week, but it's not clear that he'll even show up!" Discuss this story at: http://science.slashdot.org/comments.../08/16/0144202 Links: 0. http://slashdot.org/~Flamerule 1. http://www.nytimes.com/2006/08/15/science/15math.html 2. http://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture 3. http://science.slashdot.org/article....1337219&tid=14 4. http://en.wikipedia.org/wiki/Clay_Mathematics_Institute 5. http://en.wikipedia.org/wiki/Fields_Medal Mally

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