On the evening of June 20th, several hundred physicists, including a Nobel laureate, assembled in an auditorium at the Friendship Hotel in Beijing for a lecture by the Chinese mathematician Shing-Tung Yau. In the late nineteen-seventies, when Yau was in his twenties, he had made a series of breakthroughs that helped launch the string-theory revolution in physics and earned him, in addition to a Fields Medal—the most coveted award in mathematics—a reputation in both disciplines as a thinker of unrivalled technical power. Yau had since become a professor of mathematics at Harvard and the director of mathematics institutes in Beijing and Hong Kong, dividing his time between the United States and China. He also mentioned Grigory Perelman, a Russian mathematician who, he acknowledged, had made an important contribution. But Perelman resides in St.
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April Russian mathematician Dr. Petersburg gave a series of public lectures at the Massachusetts Institute of Technology last week. 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 actually begins his second preprint with the note, "This is a technical paper, which is the continuation of [Perelman ].
Stripped of their technical detail, Perelman's results appear to prove a very deep theorem in mathematics known as Thurston's geometrization conjecture. Here, the three-sphere in a topologist's sense is simply a generalization of the familiar two-dimensional sphere i. 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.
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.
Dunwoody in April MathWorld news story, April 18, , 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.
Clay Mathematics Institute. Johnson, G. Perelman, G. Paris: Gauthier-Villars, pp. Robinson, S. MathWorld Book. Weisstein April Russian mathematician Dr. References Clay Mathematics Institute.
Every simply connected , closed 3- manifold is homeomorphic to the 3-sphere. An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence : if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it. The analogous conjectures for all higher dimensions had already been proved. After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in and on arXiv. The proof built upon the program of Richard S. Hamilton to use the Ricci flow to attempt to solve the problem.
After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. 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, 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. But at the moment of his putative triumph, Dr. Perelman is nowhere in sight. But there is no indication whether he will show up. And they acknowledge that it may be another years before its full implications for math and physics are understood.
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