Chen is one of the greatest geometricians of our time. Born on October 28, 1911 in Jiaxing, China, he died on December 3, 2004, tomorrow marks the 15th anniversary of his death. The following is a 1998 interview with Chen Shengshen, whose interviewer, Allyn Jackson, is a senior writer and associate editor published in the Notices of the American Mathematical Society.
When Chen Shengsheng was a child, it was when He xingxixue in China founded the Western University College, he entered Nankai University before he was 15 years old, and was deeply attracted to physics, but when he found that he was not very comfortable with experimental work, he eventually switched to majoring in mathematics. In 1930, Mr. Chen entered the Graduate School of Tsinghua University, where many mathematicians had received doctorates in Western society, including Professor Dan Sun, one of the pioneers in the study of differential geometry in China, who was a student of Professor E. P. Lane at the University of Chicago. About 20 years later, Mr. Chen became Professor Lane's successor. When Wilhelm Blaschke, a mathematician at the University of Hamburg in Germany, visited Peking University in 1932, his lectures had a huge impact on Chen.
Chen Shengsheng | Photo by Peg Skorpinski
JACKSON: After studying in China, you decided to go to the West to get your Ph.D.?
Chen Shengshen: I was sent to the West by Tsinghua University in 1934 for further study. I was a teaching assistant at Tsinghua university for one year in 1930 and then studied at graduate school for three years. I think it's more appropriate for me to go to Europe than to go to the U.S., and the usual situation is to come to the U.S., but I'm not interested in Princeton and Harvard.
JACKSON: Why?
Chen Shengshen: It feels like it's not quite right for my situation. I wanted to be a geometrist, and the U.S. didn't have the kind of geometry I wanted to do, so I wanted to go to Europe. At that time, although I was a fledgling student, I had my own strengths, and I had my own ideas about the state of mathematics I wanted to study internationally, who was the best mathematician, and where was the most prominent research center. My estimate may not be right, but I have my own thoughts. I decided to go to Hamburg, in fact, and later turned out to be a very good option. At the end of the nineteenth century, the center of science was in Germany, including mathematics, and the center of mathematics in Germany was in Göttingen, as well as in Berlin and Munich. Of course, Paris has always been a center of mathematics.
I graduated from Tsinghua University in 1934. When Hitler usurped power in Germany in 1933 and the Great Purge of German universities forced Jewish professors into exile, Göttingen was devastated and Hamburg became the best place. The University of Hamburg was a newly established university after the First World War, not to be famous, but its mathematics department was outstanding. So, I went there.
It was at the University of Hamburg that Chen was first exposed to Cartar's research work, which gave him a decisive influence on his research methods in mathematics. At that time, Keller of the University of Hamburg was one of the main expounders of Carter's thought: Keller had published a book in which the fundamental theorem expounded was the now-known "Cardain-Keller theorem". Keller organized a seminar at the University of Hamburg, and on the first day of the seminar, all the professors such as Braschke, Atting, Heck, etc. attended.
Chen: The seminar looked like a celebration, and the class was packed with people. The book was also published just in time, and Keller came in with a whole bunch of books and distributed one to each of them. But the content was too difficult, so after the seminar was held several times, no one visited it. I want to be the only one who will stick to the end. I think I was able to stick to the end because I was able to keep up with its theme. Not only that, but I was writing a paper on applying its method to another problem. So the seminar was very important to me, and I even went to See Mr. Keller. How many times have we eaten together. There is a restaurant near the college where we talk about all sorts of things while eating. My German was limited, and Mr. Keller didn't speak English at the time. Anyway, we got along very well. So, in the end I finished my thesis very quickly.
Everyone knows that Cartan was the greatest differential geometrist, but his articles are very difficult to understand. One reason for this is that he uses so-called outer differentiation. And in our topic of differential geometry, where you talked about manifolds, one of the difficulties is that geometry is described in terms of coordinates, but coordinates have no (intrinsic) meaning because they allow transformations. And in order to deal with this situation, an important tool is the so-called tensor analysis or Ritchie calculus, which is new to mathematicians. In mathematics you have a function, you have to write down this function, you calculate, or add, or multiply, or you can differentiate, you have something very specific. In geometry, geometric positions are described in numbers (coordinates), but you can arbitrarily choose your numbers (coordinates), so in order to handle these, you need the Ritchie calculus.
Mr. Chen had a three-year scholarship, but it took him only two years to complete his academic qualifications. In the third year, Braschke arranged for him to go to Paris to work with Cardain. The gentleman did not know much French and Kadang only spoke French. At the first meeting, Carlton gave his husband two problems for him to solve. After some time, they occasionally met on the stairs of the Poincaré Institute, and Chen told Jiadang that he had not yet been able to solve the two problems. Kadang asked Mr. To come to his office to discuss them together. When Chen Arrived on time for his office hours in Kadang, he saw that the famous mathematician's office was crowded with many seekers. A few months later, Carlton invited his husband to his house to discuss issues with him.
Chen: Usually after meeting with Kadang, I always get a letter from him. He always said, "Since you left, I've thought more about your questions... He'll come up with some results, and there will be more questions, and so on. He was familiar with all the papers on single-plum groups, Lee algebra, and was familiar with them. When you meet him on the street or when a report is published, he always pulls out some old envelopes, writes something, and gives you an answer. To get the same answer, sometimes it even takes me hours or days. I saw him about every two weeks. Obviously, I had to study hard. I did this for a year, and it wasn't until 1937 that I returned to China.
When Chen returned to China, he became a professor in the Department of Mathematics at Tsinghua University. The brutal Japanese-Kosovar invasion of China limited his contacts with foreign mathematicians. He wrote to Cardain explaining his situation, and Cardain sent him a box of his reprints, including some previous papers. Mr. Chen spent a great deal of time reading and thinking about them, and despite his isolation from the outside world, Chen continued to publish his papers, which attracted international attention. In 1943 he was invited by the geometrist Professor Wiblen to go to Princeton College to continue his in-depth research. As a result of the war, it took him a week to arrive in the United States on a U.S. military plane. During his two years at the Academy, he completed his implicit proof of the Gauss-Bonnet theorem, which expressed the Euler indicative numbers of arbitrary dimensionally closed Riemannian manifolds as the integral of curvature over the entire manifold. This theoretical combination of local geometric properties and integral topographic invariants demonstrates a deep theme in Mr. Thompson's work.
JACKSON: What do you think is most important in your mathematical research?
Chen Shengshen: I think it is the differential geometry of fiber space. You know, mathematics is going in two different directions. One is general theory, for example, everyone must learn point set topology, learn some algebra, thus laying a general foundation, that covers almost the entire basic theory of mathematics. However, there are also some topics that are special, and they play an important role in applied mathematics. You have to know a lot about these things, such as linear groups in general, or even unitary groups. They permeate everywhere, whether you're studying physics or mathematical theory. Therefore, there is both a general basic theory and something wonderful in mathematics. Fiber space is one of them. You have a space whose fibers are fairly simple, which is the classic space, and you have to put them together in some way. This leads to a very basic concept. In fiber space, the concept of connection is extremely important, which is the entry point of my research work. In general, the best mathematical research work is to combine some theories with some very specific problems, and to promote the development of general theories in special problems. I am the first proof of the Gaussian-Bonnet formula by applying the idea of contact.
The Gauss-Bonet formula is one of the most important and fundamental formulas, not only in differential geometry, but also in the entire field of mathematics. I had thought about this before I came to Princeton in 1943, so in a sense, my development at Princeton was quite natural. After arriving at Princeton, I met Wey, who and Alondorfer had published their papers. Wey and I quickly became friends, so naturally, we discussed the Gauss-Bornet formula, and I got my proof afterwards. I think it's one of my best jobs because it solves an important and fundamental classic problem and the ideas are very novel. In order for your ideas to be put into practice, you need to have technical genius. It's not easy, and it's not something you can do as long as you have an idea. It's subtle. So I think it's a very good job.
JACKSON: One of your most important jobs is the development of the indicative class.
Chen Shengshen: The indicative classes are not so impressive: the indicative classes are very important because these are the basic invariants of the fiber space. Fiber space is very important, so the indicative class is produced. But that didn't hurt my brain much. They appear frequently, including first-order stereotypical class c: because in electromagnetism you need the idea of a complex bundle. The complex bundle leads to c1, which is found in Dirac's paper on quantum electrodynamics. Of course, Dirac did not call it c. When c is not zero, it is related to the so-called magnetic monopole. The importance of indicative classes is obvious because they arise naturally in specific and fundamental problems.
Chen Shengshengshi | Source: Lin Jiexuan
JACKSON: When you first developed the stale class theory in the 1940s, were you aware of Pontriakin's work and the fact that a solid fiber bundle of Pontria gold can be regained from its compounded stale class?
Chen: My main idea is that people should do topology and overall assimilation in the plural situation. The complex case has more structure and is in many ways simpler than the real case. So I've introduced the Chen class in the compound case, whereas the real situation is much more complicated. I've read Pontryagin's paper, but I don't see his detailed paper, but I think he published an abstract in the English-language edition of The Chronicle of Science. I learned about the relationship between the Chen class and the Pontria gold class from Hijzherbruch. Chen class can be expressed in local invariant curvature. I am mainly interested in the relationship between the nature of the part and the nature of the whole. When you study empty questions, all you can measure is local properties. Importantly, some local properties are related to the overall properties. The simple case of the Gauss-Bonnet formula is that the sum of the three inner angles of a triangle is 180 degrees. It appears in very simple facts.
Jackson: You are considered one of the leading experts in integral differential geometry, and you enjoy using tools such as Cartan's differential forms and connections. The German school, such as Klingenberg, studied the geometry of the whole in different ways: they did not like to use differential forms, they used geodesics and comparative theorems. How do you see this difference?
Chen Shengshen: There is no fundamental difference, this is a traditional development. For example, to study geometry on manifolds, the standard trick is the Ritchie calculus. The basic problem is the question of form, which was solved by Lippschitz and Christopheel, especially the latter. Christopheel's idea goes back to Ritchie, who wrote about Ritchie's operations on tensors. So all people, including Weyl, learned mathematics through the Ritchie calculus. Tensor analysis plays such an important role that everybody learns, and everybody starts with tensor analysis in differential geometry. In any case, in some respects, differential forms should be introduced. I usually like to say that the vector field is like a man, and the differential form is like a woman, and society must be made up of both sexes. If there are men alone or women alone, society is incomplete.
From 1943 to 1945, Chen visited the Princeton Institute for two years before returning to China. He also spent two years in China, where he helped establish the Institute of Mathematics at academia Sinica. In 1949 he became a professor of mathematics at the University of Chicago. In 1960, he taught at the University of California, Berkeley. He retired in 1979 and remained active, notably helping to found the National Institute of Mathematics in Berkeley. From 1981 to 1984, he was the first Director. Chen has trained 41 doctors. That number does not include the many students he has been in contact with during his frequent visits to China. As a result of the Cultural Revolution, China lost many talented mathematicians, and the tradition of mathematical research was almost lost, and Chen did many things to restore this tradition. In particular, in 1985, he played an important role in the establishment of the Nankai Institute of Mathematics in Tianjin, China.
JACKSON: How often do you return to China?
Chen Shengshen: In recent years, I have to go back to China every year. Usually stay for a month or more. I founded the Institute of Mathematics in Nankai, and most importantly, I have a group of outstanding young mathematicians rooted in China, and we have achieved success in this regard. Our new researchers include Long Yiming (Dynamic System), Chen Yongchuan (Discrete Mathematics), Zhang Weiping (Index Theory), Fang Fuquan (Differential Topology) and a number of outstanding young people. I think the biggest obstacle to mathematical research in China is mainly that the remuneration is too low. By the way, the International Mathematical Union has selected the next International Congress of Mathematicians to be held in Beijing.
JACKSON: Do you think China will improve tremendously in math?
Chen: Well, yes. My concern is that there will be too many mathematicians in China.
JACKSON: China is a big country, and probably they need a lot of mathematicians.
Chen: I don't think they need too many mathematicians. China is a big country, and naturally it has many talents, especially in some small places. For example, in the international mathematical Olympiad for middle school students, Chinese students have performed very well. However, in this case, in order to win the competition, these students need training, and as a result, other topics are ignored. In China, parents always want their children to learn English, do business, and make more money. And competitions don't give money. I think that if they invest less in this kind of training in any year, China's performance may immediately decline.
When he chose to go to Germany for graduate school in 1943, geometry was still a fledgling field in the United States; by the time he retired in 1979, geometry had become one of the most brilliant directions. Much of this change is attributable to Chen Shengsheng. However, Chen was extremely humble about his achievements.
Chen: I don't think I'm far-sighted. I'm just doing some small problems. There are so many concepts and new ideas popping up in mathematics that you just ask a few questions and then try to get simple answers and expect to give simple proofs.
JACKSON: So you observe and then you generate ideas?
Chen: Right. In most cases, you don't have any ideas. And in many cases, your idea doesn't work either.
JACKSON: Do you describe yourself as a problem solver, not a theorist?
Chen Shengshen: I think this difference is very small. Every good mathematician should be a problem solver. Otherwise, if you only have vague ideas, how can you make an outstanding contribution? You solve certain problems, you use certain concepts, and the judgment of mathematical contributions may not be seen until the future.
It is difficult to evaluate a mathematician or an aspect of mathematics. Such as the concept of differentiation. Twenty or thirty years ago, many people didn't like negotiability. Many people say to me, "I'm not interested in anything in mathematics with the concept of differentiation." These people want to make math simple. If you refuse to accept ideas that involve differentiation, you exclude a lot of mathematics. This is not enough, Newton and Leibniz should play their role. But this is interesting, because there are many controversial ideas in mathematics.
JACKSON: Can you give a few controversial examples?
Chen Shengshen: One thing is that some of today's papers are too long. For example, with the classification of "finite monogroups", who intends to read more than 1000 pages of proof? Or there is also proof of the four-color problem. I think people have to make math interesting.
I don't think mathematics is going to die out anytime soon. It's been around for a while, because it has so many wonderful things to do. Studying mathematics is the behavior of an individual. I don't believe that it's possible to do math by a group of people, basically it's a personal act and thus easy to do. Mathematics doesn't need much equipment, it's not like other sciences, they need more material equipment than mathematics. So mathematics can last for a while. I don't know how long human civilization can last, but that's a much bigger problem than math. But as far as mathematics itself is concerned, we have to spend some time with it.
At the age of 86, he continues to work on mathematics. In recent years he has been particularly interested in Finsler geometry. He had discussed this in this issue two years earlier ("Finsler geometry is a Riemannian geometry without quadratic restrictions," September 1996, 9 59-9 63).
Chen: Finsler geometry is much more extensive than Riemann geometry and can be treated in a uniform way. It will be part of the university's basic course in differential geometry for the next decade.
I don't have any difficulties in math, so when I do math, I appreciate it. The reason I've been doing math is because I can't do anything else. I've been retired for years and have been asked if I'm still studying math. I think my response is: it's the only thing I can do, there's nothing else I can do. I've been like this all my life.
This article is revised and edited according to the translation of the "Interview with Mr. Chen Shengshen, the Greatest Contemporary Geometrist" in the Journal of Suzhou Institute of Education.
* This article is reproduced from the public account of "Return to Simplicity"