1, tie a knotted rope and connect the two ends together. When can you untie all the knots and turn the rope into a circle without cutting it? 1949 In a differential geometry class at Princeton University, Professor Tucker raised this question to arouse children's interest. At the same time, he introduced the conjecture of Polish mathematician Bolsuke two years ago: In order to investigate the bending degree of knotted rope, the curvature of each point on the whole rope is integrated. If it is less than or equal to 4π, then all knots can be opened. What Tucker didn't expect was that a few days later, a 18-year-old student in the class handed him a complete certificate like handing in his homework. The total curvature of the knots on JSTOR 2, the child with overflowing IQ was not satisfied with mathematics during his college years, and he was also interested in many intellectual projects, such as Go, a game called Nash. After meeting the inventor of the game, they became good friends and published a series of articles on game theory. Recently, it was found that Nash's famous works were short and the references were less than one page. From 65438 to 0954, 23-year-old Milnor ended his doctoral career with nearly ten articles spanning many directions, became an assistant professor at Princeton University, and began a journey of subverting all topologists' values again and again. 4, masterpiece: the strange ball that has lost all three views. 1956 The structure of the milestone in the history of mathematics was given by 25-year-old Milnor, and six and a half pages of articles were enough to qualify him for all the mathematics awards in the world. Actually, it's true. At present, he is one of only four grand slam players who swept Fields, Wolff and Abel Abel. At the age of1962,31year, Milnor got Fields: "It is proved that a 7-dimensional sphere can have a serious differential structure; This led to the field of differential topology. " 5, published by Wolf Prize Committee, contains four articles by Milnor. The first is, of course, a strange ball, and the second is a new proof that divisible algebra in 1958 has only four kinds: real number, complex number, quaternary number and octal number. It quotes five articles by Mr. Wu Wenjun and writes: ". 6. The second subversion: the counterexample of Hauptvermutung. Hauptvermutung is German, which literally translates as a guess, pointing to the root of combinatorial topology, that is, is the triangulation of the same topological space unique? More precisely, do any two triangulation have the same refinement? At 196 1, 30-year-old Milnor directly constructed a counterexample. Like the strange ball, this subversion has ended countless people's research plans but opened a new direction for more people. Triangulation and hauptvermutung 7, once again subverted: can you tell the shape of the drum from its sound? This problem is very influential, and Kac won two awards for it: the sound we hear in our ears is determined by the frequency, which is the eigenvalue of Laplace operator on the drum surface. Then the question becomes: If the spectra of Laplacian in two spaces are the same, are these two spaces the same? But the problem has suddenly become a direction that many people are studying at present: if the spectra are the same, under what conditions the space is the same. This is because "almost immediately", Milnor threw out two examples of 16-dimensional torus, which have the same spectrum but different spaces. 8. Subversion again: Smooth manifolds with homeomorphism can actually have tangents with different structures. . . If it is subverted again, it will be aesthetically tired. Some people in the comments pointed out that this article is the same as 4. That's true. )9, ? Poincare conjecture is the greatest pursuit of his life since he came into contact with mathematics. I don't know what it's like to see perelman solve the conjecture in his lifetime. 10 Generally speaking, it is difficult for a scholar with high IQ to understand the struggle of the scum in research, so many talented professors are in a mess when they give lectures and write books. But Milnor is a counterexample. He has a gift to explain a chaotic and complicated cutting-edge research very simply. " Usually when Milnor explains it, it's very simple. "Milnor's writing level is the unique skill in mathematics since ancient times, differential topology, indicator class, H-edge, Morse theory. . . These unfathomable frontiers at that time became postgraduate courses after Milnor finished writing this book. He is the only one who won the Leroy P Steele Award for Outstanding Mathematical Works awarded by the American Mathematical Society, which is an achievement that Serge Lang can't achieve. 1 1, I didn't know that Macduff was actually Milnor's wife until I read his book. She gave up her tenured position in the school and became a non-tenured assistant professor just to get closer to him.