主题：Poincaré 猜想被彻底证明

看见一则消息，说Poincaré Conjecture 被彻底证明，见附件。 网址是 http://mathworld.wolfram.com/news/2003-04-15/poincare/

Poincaré Conjecture Proved--This Time for Real April 15--Russian mathematician Dr. Grigori (Grisha) Perelman of the Steklov Institute of Mathematics (part of the Russian Academy of Sciences in St. Petersburg) gave a series of public lectures at the Massachusetts Institute of Technology last week. These lectures, entitled "Ricci Flow and Geometrization of Three-Manifolds," were presented as part of the Simons Lecture Series at the MIT Department of Mathematics on April 7, 9, and 11. The lectures constituted Perelman's first public discussion of the important mathematical results contained in two preprints, one published in November of last year and the other only last month. Perelman, who is a well-respected differential geometer, is regarded in the mathematical community as an expert on Ricci flows, which are a technical mathematical construct related to the curvatures of smooth surfaces. Perelman's results are clothed in the parlance of a professional mathematician, in this case using the mathematical dialect of abstract differential geometry. In an unusally explicit statement, Perleman (2003) actually begins his second preprint with the note, "This is a technical paper, which is the continuation of [Perelman 2002]." As a consequence, Perelman's results are not easily accessible to laypeople. The fact that Perelman's preprints are intended only for professional mathematicians is also underscored by the complete absence of a single reference to Poincaré in either paper and by the presence of only a single reference to Thurston's conjecture. Stripped of their technical detail, Perelman's results appear to prove a very deep theorem in mathematics known as Thurston's geometrization conjecture. Thurston's conjecture has to do with geometric structures on mathematical objects known as manifolds, and is an extension of the famous Poincaré conjecture. Since Poincaré's conjecture is a special case of Thurston's conjecture, a proof of the latter immediately establishes the former. In the form originally proposed by Henri Poincaré in 1904 (Poincaré 1953, pp. 486 and 498), Poincaré's conjecture stated that every closed simply connected three-manifold is homeomorphic to the three-sphere. Here, the three-sphere (in a topologist's sense) is simply a generalization of the familiar two-dimensional sphere (i.e., the sphere embedded in usual three-dimensional space and having a two-dimensional surface) to one dimension higher. More colloquially, Poincaré conjectured that the three-sphere is the only possible type of bounded three-dimensional space that contains no holes. This conjecture was subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere if and only if it is homeomorphic to the n-sphere. The generalized statement is now known as the Poincaré conjecture, and it reduces to the original conjecture for n = 3. The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), n = 3 has remained open up until now, n = 4 was proved by Freedman in 1982 (for which he was awarded the 1986 Fields Medal), n = 5 was proved by Zeeman in 1961, n = 6 was demonstrated by Stallings in 1962, and n >= 7 was established by Smale in 1961 (although Smale subsequently extended his proof to include all n >= 5). Renewed interest in the Poincaré conjecture was kindled among the general public when the Clay Mathematics Institute included the conjecture on its list of million-dollar-prize problems. According to the rules of the Clay Institute, any purported proof must survive two years of academic scrutiny before the prize can be collected. A recent example of a proof that did not survive even this long was a five-page paper presented by M. J. Dunwoody in April 2002 (MathWorld news story, April 18, 2002), which was quickly found to be fundamentally flawed. Almost exactly a year later, Perelman's results appear to be much more robust. While it will be months before mathematicians can digest and verify the details of the proof, mathematicians familiar with Perelman's work describe it as well thought out and expect that it will prove difficult to locate any significant mistakes. References Clay Mathematics Institute. "The Poincaré Conjecture." http://www.claymath.org/Millennium_Prize_Problems/Poincare_Conjecture Johnson, G. "A Mathematician's World of Doughnuts and Spheres." The New York Times, April 20, 2003, p. 5. Perelman, G. "Ricci Flow and Geometrization of Three-Manifolds." Massachusetts Institute of Technology Department of Mathematics Simons Lecture Series. http://www-math.mit.edu/conferences/simons Perelman, G. "The Entropy Formula for the Ricci Flow and Its Geometric Application." November 11, 2002. http://www.arxiv.org/abs/math.DG/0211159 Perelman, G. "Ricci Flow with Surgery on Three-Manifolds." March 10, 2003. http://www.arxiv.org/abs/math.DG/0303109 Poincaré, H. Oeuvres de Henri Poincaré, tome VI. Paris: Gauthier-Villars, pp. 486 and 498, 1953. Robinson, S. "Russian Reports He Has Solved a Celebrated Math Problem." The New York Times, April 15, 2003, p. D3.

抽空翻译了一部分，希望抛砖引玉，能有其他朋友把其余部分翻译出来，方便大家阅读。毕竟这些词汇对于一二年级的本科生还是有点不熟悉。 4月15日消息，俄罗斯科学院在圣匹兹堡的斯杰克洛夫（Steklov）数学研究所的俄罗斯数学家Grisha Perelman博士上周在麻省理工作了系列公开演讲。 这些演讲（题目为“3维流形的几何化与Ricci流”），作为麻省理工数学系的Simons系列讲座的一部分，是在4月7，9，11日举行的。这是Perelman的两篇预印本中的重要的数学结果的第一次公开讨论。这两篇预印本一篇是去年11月发表的，而另外一篇是上个月刚刚完成的。 Perelman是一个令人肃然起敬的微分几何学家，在数学界被认为是Ricci流方面的专家。所谓Ricci流是和光滑曲面的曲率有关的数学结构。Perelman的结果是用数学家的专业语言称述的，即用抽象的微分几何的语言陈述的。

Perelman在他的第二篇预印本（2003年）的开篇以极不寻常的明白的语言告诉大家：“这是一篇技术报告，是文献[Perelman2002]的延续”。因此，Perelman的结果对于外行的人来说是不容易搞懂的。Perelman在两篇文章中都没有一个地方提到这是poincare猜想，且仅在一个地方提到Thurston猜想，这更进一步说明Perelman的文章本来就是打算让专业数学家看的这个事实。

撇开技术细节，Perelman的结果证明了数学中一个非常深刻的定理，即所谓的Thurston几何化猜想。Thurston猜想是

Thurston猜想是处理流形等数学结构，是著名的Poincare猜想的推广。因为Poincare猜想是Thurston猜想的一个特殊情形，因此，只要证明了Thurston猜想，Poincare猜想自然就成立了。

用Poincare在1904年提出的最初形式的语言来说，Poincare猜想是说：任意闭单连通三维流形是和（四维欧几里得空间中的）三维球面同胚的。这里拓扑学家所指的三维球面其实就是通常的三维欧几里得空间中二维球面往更高一维的推广。用更口语话的语言来说，Poincare猜想是说：没有洞的有界三维空间一定与三维球面同伦。这个猜想推广到n维情形即是说：紧的n维流形与n维球面同伦的充要条件是它与n维球面同胚，这即是现在所指的Poincare猜想，在n＝3的时候就是Poincare猜想最初的形式。

n = 1 时猜想是显然的, n = 2 的情形是经典的结论 (早在19世纪的数学家就已经知道了), n = 3 的情形一直到现在才证明, n = 4的情形由 Freedman在1982年证明 (他也因此被授予 1986年的菲尔兹（Fields）奖), Zeeman 在1961年证明了n = 5的情形,Stallings 在1962年证明了n = 6的情形， Smale在 1961年证明了 n >= 7的结论（随后Smale把他的方法推广到n>=5的情形）。

Clay数学研究所把Poincare猜想列入其“百万美元大奖征集解法的问题”目录中，这在公众中引起了新一轮的兴趣。根据Clay所的规则，任何宣称的证明必须经过学术界两年的仔细审查后才能颁奖。最近的一个例子是：M. J. Dunwoody 在2002年4月提交的一份5页的证明还没有到两年就被否决了，他的证明中有一个根本性的漏洞。.

差不多一年以后，Perelman的结果看起来更具说服力。尽管数学家们得花数月得时间来消化和验证Perelman的证明的细节，不过熟悉Perelman的研究工作的数学家称他的工作是经过仔细思考过的，并且认为很难找到一个严重的错误。