Controversy of Poincare Conjecture Proof Sukhbir Singh 11/9/2014

The Poincare Conjecture was known as a millennium problem, there were 7 problems total, including the Riemann hypothesis and the P vs. NP problem. Fermats last theorem was supposed to be on the list, but it was solved in 1995 by Andrew Wiles. The prize for a millennium problem such as the Poincare Conjecture is one million dollars and the recognition and fame that would come from solving the problem. It was conjectured in 1904 in a work by Henri Poincare (29 April 1854 17 July 1912.

The conjecture was proved by Grigori Perelman (1966-present) in 2002 and made public in the following year. He turned down the millennium prize of one million dollars, he turned down a Fields medal, and he left the pro- fessional mathematical community. Grigori Perelman was a Russian math- ematician who mainly worked for the Steklov Institute of Mathematics in St. Petersburg. He is a very reclusive man with few friends and currently lives in an apartment with his mother in Siberia. Perelman had little self interest as he did refuse a one million dollar prize. The min reason for this was that he only cares for the advancement of mathematics. In fact, he sent the first installment of his work on the conjecture to more than sixty mathematicians in November of 2002. This was a major risk, since there was a large chance of someone stealing the work and using it as their own. There was also the chance of there being an error in the work which would be embarrassing for Perelman and risky as well since someone could correct the mistake and receive even partial credit for the proof.

There are two main ways that people receive credit for mathematical proofs. The first is to come up with an original proof, which Perelman had done. The second is to find an error or gap in proof by someone else and fix the mistake or add in the missing information so that the proof makes sense. If someone were to revise the wording of a proof so that it is easier to understand, they would not receive credit for the proof as they have contributed nothing new to the work. The difficulty to distinguish whether or not new work was added can be high and cause debates between members of the mathematical community.This was the case in this controversy.

Shing-Tung Yau (1949) is a famous Chinese mathematician who received a Fields medal in 1982. Unlike Perelman, Yau was proud and had ideas

1 beyond the advancement of mathematics. He wanted to be the best of his field in mathematics ad sought to reform education in China in universities since it had been heavily degraded after the cultural revolution. He had many students who also excelled in mathematics.

”At Harvard, he ran a notoriously tough seminar on differential geometry, which met for three hours at a time three times a week. Each student was assigned a recently published proof and asked to reconstruct it, fixing any errors and filling in gaps. Yau believed that a mathematician has an obligation to be explicit, and impressed on his students the importance of step-by-step rigor” according to The NewYorker. Yau’s educational skills were indeed phenomenal.

Two students of Yau, Xi-Ping Zhu and Huai-Dong Cao went over the proof by Perelman. They found it to be confusing and so rewrote the work in a much more detailed way so that most people would be able to understand it. When they published the work known as ”The Hamilton-Perelman Theory of Ricci Flow: The Poincar and Geometrization Conjectures” and claimed credit for the work with the support of Yau, many mathematicians who had read the original proof were enraged and argued that there was nothing in the original proof that contained error and so the students should not receive credit. Perelman had no aggressive stand in the position although he stated in one interview that he saw nothing new added in the students’ work.

It was found that others had written nearly identical works before Cao and Zhu which lead to complaints about copyright. Cao and Zhu took back their paper and revised it to make it so to avoid conflict. The new paper stated that it was a detailed exposition of a complete proof, meaning it was simply more clear although nothing new was added. No undeserved credit would be taken by Zhu and Cao. This was resolution to the controversy.

References http://www.newyorker.com/magazine/2006/08/28/manifold-destiny http://en.wikipedia.org/wiki/Manifold Destiny

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