abs: Richard Hamilton (University of Columbia, New York, USA) finished his plenary lecture yesterday, the first of the ICM2006, by saying that he felt incredibly happy and enormously grateful to Grisha Perelman for finishing his work:“In this way we actually get a proof of the Poincaré Conjecture”

In May 2003 Perelman startled the mathematical world by posting on a preprint archive three papers that appeared to contain a proof of the Poincaré Conjecture as well as the Thurston Geometrization Conjecture, a more far-reaching statement that actually contains the Poincaré
Conjecture as a special case. In the three and a half years since Perelman made his papers public, mathematicians all over the world have worked hard to understand and verify his work, and no serious errors have been found.
Hamilton’s statement at the end of his lecture could be understood as a confirmation that Poincaré’s Conjecture has finally been proven. Hamilton is one of the experts who has not only analyzed Perelman’s work but has also developed the tool that was essential for the Russian
mathematician’s proof – a technique called Rucci Flow.
In his statements to the press, Hamilton said that Perelman had “overcome the last obstacle to the Poincaré Conjecture”, but that he preferred to see the proof of the conjecture as “the work of many mathematicians over a long period of time”. “The proof of the Poincaré Conjecture is a problem people have been working on for a long time, and many have made a conntribution”. Hamilton began working on this problem thirty years ago, but he became “stuck”. Now, however, he admits in a jovial tone, “I look back and wonder why I didn’t
manage to solve it”. Hamilton said that he had a “profound admiration” for Perelman’s work, and that he would be “delighted to work with him in the future”. He said that he had met
Perelman personally, but he was not prepared to comment on Perelman’s refusal to accept the Fields Medal conferred on him on Tuesday at the ICM2006. However, Hamilton did say that “it is not fair to criticize his position”.
Hamilton was also asked abut the Chinese mathematicians Xi-Ping Zhu, from the University of
Zhongshan (Canton, China), and Huai-Dong Cao, from the Lehigh University in Pennsylvania (USA), who last June published a paper in the Asian Journal of Mathematics. In the abstract of this paper the authors state that they present “a complete proof of the Poincaré and Geometrization Conjectures”. Hamilton is sure that “there is no controversy” because both mathematicians are “great researchers”. According to Hamilton, the controversy surrounding the proof of Poincaré’s Conjecture was caused by the press. He went on to say that Perelman’s work “is difficult to understand” and at some points even Perelman himself employs the term “sketch”. “A sketch is an invitation to complete a finished work, to find a way of doing it better. But no criticism is implied in this, only the wish to help to solve a problem.
There is no controversy involved. Grisha is a model of decorum and there is no dispute about who did what”.

At the ICM2006 there is also John Morgan (Columbia University, New York, USA), who together with Gang Tian (Princeton University, USA), has written a book presenting a complete account of the proof of the Poincaré Conjecture based on Perelman’s ideas. The book is available on the web and has been submitted for publication. Other key players contributing to the understanding and verification of Perelman’s work include Bruce Kleiner
(Yale University, New Haven, USA) and John Lott (University of Michigan, USA) who, shortly after Perelman’s papers appeared, started a web page in which they presented their notes on the papers as they worked carefully through them. In May 2006, Kleiner and Lott
posted on a preprint archive a paper containing their complete notes, and have submitted this paper for publication in a journal.
In a highly technical lecture given by professor Richard Hamilton from the University of Columbia, he began with the well-known idea of using Ricci flow (Hamilton program) for evolving metrics over a 3-manifold variety to arrive at a stationary solution belonging to Thurston’s list (thus proving the Geometrization Conjecture). Hamilton provided a quick overview of his most important results in the study of the evolution of a variety by Ricci flow: the existence of a solution, bounds of the curvature and its derivates, Harnack inequalities,
compactness theorem for flows, etc.. In his talk, Hamilton mentioned the difficulties he had
encountered in completing the proof. They consisted of the possible formation of singularities in finite time in the solutions to the equation (collapse phenomena). Hamilton achieved results on the classification of singularities and provided a good definition of Ricci flow with surgery in
dimension 4, thus carrying out a vital part of the work required to complete the proof. During his lecture, Hamilton referred to the fact that much of Perelman’s work (good definition of flow with surgery in 3-manifolds, collapse, kappa solution density theorem, theorem of the canonical neighbourhood, etc.) had enabled the last difficulties in the Hamilton program to be overcome to arrive at a solution to the Geometrization Conjecture. Hamilton also took pains to simplify some of the points in Perelman’s work, avoiding the use of Aleksandrov spaces, an enormously difficult technique but one well-known to Perelman. At the conclusion of his
talk, Hamilton expressed his thanks to Perelman for all his work.


Mario García Fernández
Becario Predoctoral del CSIC
Voluntario ICM 2006

All from http://www.icm2006.org/dailynews/dailynews24.pdf