Analysis:

Henri Poincaré, a Frenchman, is called "the last all-rounder in the field of mathematics". In his great scientific legacy, there is a proposition of fundamental significance in algebraic topology, which is the "Poincare conjecture" that has puzzled mathematicians for a whole century.

Poincare put forward this conjecture in a set of papers published in 1904: "A simply connected three-dimensional closed flow is like an embryo on a three-dimensional sphere." Later, it was summarized as: "Any N-dimensional closed manifold that is homotopy with an N-dimensional sphere must be homeomorphic with an N-dimensional sphere." We might as well make a shallow analogy with two-dimensional examples: a rubber film without holes is topologically equivalent to a two-dimensional closed surface, while an inflatable balloon can be regarded as a two-dimensional spherical surface, and the points between them are one-to-one correspondence, while the adjacent points on the rubber film are still adjacent points on the inflatable balloon, and vice versa. Interestingly, the high-dimensional inference of this conjecture was solved in the 1960s and 1980s respectively, but the three-dimensional situation still lay there like a stumbling block, challenging the best topologists in the world.

Algebraic topology is one of the most active fields in mathematics today. The proof of Poincare conjecture and its consequences will deepen mathematicians' understanding of manifold properties, and even have an impact on people's description of the universe in mathematical language. The statement of this conjecture is so concise that the Clay Institute of Mathematics in Boston listed it as one of the "seven-thousand-year puzzles" in 2000, and offered a reward of $6,543.8+0,000 for this conjecture. Because of this, when the news that this conjecture may be proved spread in American media and internet, it caused a sensation in the whole mathematics field.

Perelman, a middle-aged mathematician from Russia, made an important contribution to this conjecture. He is a researcher at the Skolov Institute of Mathematics in St. Petersburg, and has been working on differential geometry and algebraic topology for the past 10 years. From June 2002 to October 2002 10, perelman published a research report through the Internet, claiming to prove the "geometric conjecture" about three-dimensional manifolds put forward by American mathematician thurston 25 years ago, and "Poincare conjecture" is a special case of the latter. Since a new "proof" will appear every few years and then be overturned, the mathematical community has been very cautious about such reports. Four months later, perelman published a second report on the Internet, introducing more details of proof. At the same time, he also communicated with several experts in this field by email.

In April 2003, at the invitation of China mathematician Tian Gang, perelman gave three speeches at MIT, which were very successful. He seems to be ready for all the questions and questions-either answer them fluently or point out that they are trivial. Professionals who have heard the speech think his works are extremely creative. "Even if it is proved to be wrong, he has developed some tools and ideas. Is this enough to lead to geometric conjecture? Exquisite treatment and extremely exciting things, "said Jim Carlson, director of the Clay Institute.

A few days later, on April 15, * * * disclosed the news to the public for the first time with the title of "Russian Report, Solving Famous Math Problems". On the same day, MathWorld, an influential mathematics website, published the headline article "Poincare conjecture proved, this time it is true". Perelman soon became a news figure, but he was not used to it. Two weeks later, when he was invited to give a speech at the Courand Institute of new york University, the lecture hall was packed with reporters and non-professional listeners. The enthusiasm of perelman's speech was greatly reduced. He refused to answer the reporter's question about "what's the application" and loudly stopped the attempt to take pictures of him. He also disdains to accept telecom interviews with famous magazines such as Nature and Science. Later, people couldn't find him at all, and even his colleagues in St. Petersburg didn't know where he was and what he was doing. At the end of 2003, he held two seminars on his work in California, which he did not attend.

Perelman is not only shy by nature, but also maverick. When he visited the United States about 10 years ago, his work attracted people's attention and he got the opportunity to work in an American university. However, contrary to many of his brilliant and talented compatriots, he soon returned to Russia and lived a life close to a hermit. "What he needs is mathematics, not rewards, funds and positions." This is the hint used in an article about him published in Nature 427 in June+10 this year.

Perelman's certificate is now under strict examination by several qualified experts. Tian Gang has read most of the contents of the second report and found no loopholes so far. He hopes to finish the rest this summer. Mathematicians' systematic review may last until 2005, when we can know whether the Poincare conjecture has been proved and whether perelman will win the Millennium Prize.