2006年国际数学家大会上会对庞加莱猜想的解决做出结论吗？

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2006年国际数学家大会上会对庞加莱猜想的解决做出结论吗？
（转自2006年国际数学家大会官方网站）

Experts on Perelman’s Solution Explain Their Work
A Verdict on Poincaré’s Conjecture at the ICM2006?

The verification of a mathematical solution can take years, but this task is indispensable in a science developed on solid but increasingly complex foundations. The leading experts on one of the most burning mathematical questions of the time, Poincaré’s Conjecture, will meet at the ICM2006 International Congress of Mathematicians to be held this August in Madrid. It would be no exaggeration to say that the approximately 5,000 mathematicians from all over the world who are due to attend the ICM2006 are eagerly awaiting a verdict on the solution posed a few years ago by the Russian Grigori Perelman.

Although the ICM2006 is not about to issue an official statement, those who have been hard at work on Perelman’s solution are expected to present their conclusions. Since Perelman posted the results of his work on the Internet three years ago, no one has so far drawn attention to any errors.  

One of the experts attending the ICM2006 is Richard Hamilton, who developed a tool used by Perelman to prove the Conjecture. Another expert who will be present at the ICM2006 is John Morgan, a renowned specialist who has also worked on the Conjecture.

The attendance of these experts at the ICM2006 has recently been the object of even greater interest since the Chinese mathematicians Xi-Ping Zhu, from the University of Zhongshan (Canton, China), and Huai-Dong Cao, from Lehigh University in Pennsylvania (USA), announced in the latest issue of the Asian Journal of Mathematics, published in the U.S.A., that they had arrived at “a complete proof of the Poincaré and Geometrization Conjectures”.

Zhu and Cao base their work –according to their abstract- “on the accumulative works of many geometric analysts in the past thirty years”, and they specifically mention Hamilton and Perelman himself. However, their paper is still not available on the Internet, and is therefore still not accessible to the mathematical community at large.  

Since its formulation in 1904, Poincaré’s Conjecture has been one of the problems that has consumed most “brain power” in mathematics, to such an extent that the issue has been referred to as “Poincaritis”, a species of contagion affecting those who have spent decades in an attempt to resolve the problem. Furthermore, Poincaré’s Conjecture is one of the seven Millennium Problems, the solution of which carries an award of one million dollars each from the Clay Mathematical Institute (Cambridge, Massachusets).

This particular problem belongs to the field of topology. As Vicente Miquel, professor of Geometry and Topology at the University of Valencia, explains: “Topology considers as equal shapes or spaces obtained by deforming one into the other without cutting or tearing, since the only thing that changes with deformation is distance, something we have tended to overlook”. A topologist does not distinguish between a pyramid and a ball - one shape can be transformed into the other without being torn or broken – just as a cup with a handle and a donut are considered to be the same. Topology is not concerned exclusively with objects in the three-dimensional space we are all familiar with, but also with objects in 4, 5, 6 and n dimensions.  

Poincaré’s Conjecture is concerned with closed objects. In non-mathematical terms, the Conjecture amounts to saying that objects in four dimensions with the same topological properties as a sphere in our three-dimensional world are also spheres. However, Poincaré himself was unable to prove this assertion, for which reason we speak of a “conjecture”. In the mid-20th century the problem was generalized to all dimensions. If and when it is definitively verified it will be known as a theorem.

相关内容：
上文提到的最近一篇关于庞加莱猜想的论文是发表在Asian Journal of Mathematics第十卷第二期（第165－498页）上的"A Complete Proof of the Poincaré and Geometrization Conjectures--Application of the Hamilton-Perelman Theory of the Ricci Flow"（庞加莱猜想暨几何化猜想的完全证明：汉密尔顿－佩雷尔曼理论的应用）。这篇长达334页的论文是由中山大学数学与计算科学学院院长朱熹平教授和美国Lehigh University数学系的曹怀东讲座教授合作完成。该文摘要如下
“In this paper, we give a complete proof of the Poincaré and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow. ”
另外一篇关于庞加莱猜想的论文是2006年5月25日在网上公布的“Notes on Perelman’s Papers”(关于Perelman论文的注记)。这篇长达192页的论文是由美国Michigan University 数学系的Bruce Kleiner教授和John Lott教授合作完成的。

相关链接：

Grigori Perelman 发布在网上的论文: http://arxiv.org/find/math/1/au:%20Perelman_Grisha/0/1/0/all/0/1

Grigori Perelman 简介:
http://grigori-perelman.biography.ms/

Eric W. Weisstein关于Perelman工作的介绍:
http://mathworld.wolfram.com/news/2003-04-15/poincare/

Mark Brittenham 关于Perelman工作的介绍:
http://www.math.unl.edu/~mbrittenham2/ldt/poincare.html