06/06/06 - 21:55Poincaré conjecture is proved!
One of the seven BIG Mathematical Problems, Poincaré conjecture(庞加莱猜想), is finally proved after a century's effort! The final proof is worked out in June 2006, by Chinese mathematicians Xiping Zhu and Huaidong Cao.
Here is the standard form of the conjecture:Every simply connected closed (i.e. compact and without boundary) 3-manifold is homeomorphic to a 3-sphere.
Loosely speaking, this means that if a given 3-manifold is "sufficiently like" a sphere (most importantly, that each loop in the manifold can be shrunk to a point), then it is really just a 3-sphere.
History of attempted solutions:
For a time, this problem seems to have lain dormant, until J. H. C. Whitehead revived interest in the conjecture, when in the 1930s he first claimed a proof, and then retracted it. In the process, he discovered some interesting examples of 3-manifolds, the prototype of which is now called the Whitehead manifold.
In the 1950s and 1960s other famous mathematicians were to claim proofs only to discover a fatal flaw at the last minute. Influential mathematicians such as Bing, Haken, Moise, and Papakyriakopoulos expended great efforts at tackling the conjecture. This period was important to the growth of what would later be called low-dimensional topology.
Over time, the conjecture gained the reputation of being particularly tricky to tackle. Milnor commented that sometimes the errors in false proofs can be "rather subtle and difficult to detect."[1] Overall, work on the conjecture has improved understanding of 3-manifolds. Experts in the field have been most reluctant to announce proofs, and have viewed any such announcement with skepticism. The 1980s and 1990s witnessed some well-publicized fallacious proofs (which were not actually published in peer-reviewed form). Sometimes mathematicians obsessed with this problem are described as suffering from Poincaritis.
In 2000 the Clay Mathematics Institute selected the Poincaré conjecture as one of seven Millennium Prize Problems and offered a $1,000,000 prize for its solution. Undoubtedly its difficulty and the expectation that a significant breakthrough would be needed were important factors in this selection.
The Poincaré conjecture may now attract the first Millennium Prize to be awarded.
In late 2002, Grigori Perelman of the Steklov Institute of Mathematics, Saint Petersburg was rumoured to have found a proof. He claimed to have proven a more general conjecture, Thurston's geometrization conjecture, carrying out a program outlined earlier by Richard Hamilton. In 2003, he posted a second preprint and gave a series of lectures in the United States. After several years of combined efforts of mathematicians from around the globe and intense reworking of Perelman's preprints, several teams of mathematicians have concluded Perelman's work is correct.
In June 2006, the Asian Journal of Mathematics published a paper by Cao Huaidong of Lehigh University in Pennsylvania and Zhu Xiping of Zhongshan University in China, which has filled in the details of Perelman's work, thus "putting the finishing touches to the complete proof of the Poincaré Conjecture", according to the Fields medalist Shing-Tung Yau.
(Last Edited at: 06/06/06 - 21:55)
Leave comments: