Wednesday, June 07, 2006

Poincare Conjecture Proven?

News from China reports that two Chinese mathematicians, Zhu Xiping and Cao Huaidong, have proven the Thurston Geometrization Conjecture and by implication the Poincare Conjecture. Proving the Poincare Conjecture has been one of the most difficult mathematical problems since Henri Poincare first proposed it in 1904.

From The Epoch Times:
In the June issue of the U.S.-based Asian Journal of Mathematics, the two scientists published a 334-page paper, "A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow."

The Poincaré Conjecture, first stated by French mathematician Henri Poincaré in 1904, is that, in topology, if in a closed three-dimensional space, any closed curves can shrink to a point, this space is topologically equivalent to the three-dimensional sphere. Like the Riemann Hypothesis, the Hodge Conjecture and the Yang-Mills Existence and Mass Gap, the Poincaré Conjecture has been rated as one of the seven "Millennium Prize problems"for proofs of which the Clay Mathematics Institute of Cambridge, Massachusetts was offering prizes of US$1,000,000 each, in May 2000.

By the end of the 1970s, U.S. mathematician William P. Thurston had produced partial proof of Poincaré Conjecture on geometric structure, and was awarded the Fields Prize for the achievement. Fellow American Richard Hamilton completed the majority of the program and the geometrization conjecture. In 2003, Russian mathematician Grigory Perelman made key new contributions.

Utilizing the Hamilton-Perelman theory of Ricci flow, Zhu and Cao have successfully provided the complete proof of the Poincaré Conjecture in the paper.

The paper must survive the scrutiny of the mathematics community for two years before the Clay Math prize is awarded.

The problem is near and dear to my heart since I had the opportunity to study it for a little while in college. In 2003, when Perelman’s papers were published on the Internet, I tried to learn a little bit about the state of the art. As you can readily imagine, I didn’t make much progress with that endeavor. However, it was fun trying.

As far as I can tell from the news, no imminent breakthroughs are on the horizon for the Riemann Hypothesis. But you never know whether a brilliant mind is working in secrecy and about to crack the nut.

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