Guide
Born on October 28, 1911 in Xiushui, Jiaxing, Zhejiang Province, Chen graduated from Nankai University in 1930 and was one of the greatest geometrists of the 20th century.
He developed the Gauss-Bonnet formula, which was named the Gauss-Bonnet-Chen Provincial Body Formula, and proposed the "Chern Class", which became a classic masterpiece. He developed the theory of differential fiber bundles, whose influence pervades all areas of mathematics. He founded the theory of the distribution of values on complex manifolds, including the Bott-Chen theorem, which had an impact on algebraic theory. He laid the foundation for generalized integral geometry and obtained basic kinematic formulas. His introduction of the Chen indicative class and the Chen-Simons differential formula has penetrated into other fields other than mathematics and has become an important tool of theoretical physics. His outstanding contribution to integral differential geometry influenced the development of mathematics as a whole, and was hailed as another milestone figure after Euclid, Gauss, Riemann, and Cardon. He has presided over the establishment of the former Academia Sinica Institute of Mathematics, the National Institute of Mathematics of the United States, and the Nankai Institute of Mathematics, creating a group of world-renowned mathematicians.
This year marks the 110th anniversary of Mr. Chen's birth, and Mr. Sai hereby reprints Mr. Chen's self-report to commemorate it.
Written by | Chen Province
<h1 class="pgc-h-arrow-right" data-track="12" >01 Education in China in the early years</h1>
I entered Tianjin Rotary High School in January 1923. It was a four-year senior high school, and I was allowed to transfer to the first year for the second semester. The school's mathematics programs include:
(1) the first year, arithmetic, the use of Chinese textbooks;
(2) The second year, algebra, uses Hall and Knight's textbooks;
(3) The third year, Geometry, using Wentworth and Smith's textbooks;
(4) The fourth year, trigonometry and advanced algebra, using the textbooks of Wentworth-Smith and Hall-Knight, respectively.
My teachers were very capable and dedicated, and I did a lot of exercises. By the fourth year, I had been able to do many of the cambridge university honours degree examinations cited in Hall-Knight's book.
I graduated from Rotary in 1926; in the same year I entered Nankai University, actually jumping two levels, so I never took an analytical geometry class. To make matters worse, I had to take nankai university entrance exams, which weighed heavily on analytic geometry in its math tests.
Three weeks before the exam, I taught myself Young and Morgen's Mathematical Analysis, and if I remember well, my paper came in second. For a long time, however, the concept of "the focal point of the conical curve" was a big headache for me, and it wasn't until I learned projective geometry a few years later that I didn't open up.
After entering Nankai University, I soon found myself clumsy in doing experiments, so mathematics became my only option. I was fortunate to have Professor Jiang Lifu as a teacher, who received a doctorate in philosophy from Harvard University in 1918 under the supervision of J. Coolidge, the thesis title is about the linear ball contact transformation in non-Euclidean space.
So, in my fourth year of college, I spent a lot of effort studying geometry, reading Books such as Coolidge's Noneuclidean Geometry and Geometry of the Circle and Sphere, Solmon's conic sections, and Analytic Geometry Three Dimmensions), as well as Castelnuovo's Analytic and Projective Geometry.
What particularly fascinated me was Otto Staude's two-volume adaptation of Fadenkonstruktionen. The geometry of quadratic hypersurfaces is a beautiful chapter in mathematics. I was glad to see J. Moser continued his work in 1979 on the hamiltonian system and spectral theory. (See 3) Even today, the study of Salmon's stuff may still be valuable, at least in my opinion, interesting.
In 1930, I graduated from Nankai and went to Peking Tsinghua University to work as Professor Sun Rong. Mr. Sun was the only mathematician in China to publish mathematical research papers at that time. Sun's field of study is projective differential geometry, and he was a phD student at E.P. Lane at the University of Chicago.
Founded in 1901 by E.J. Wilczynsky, this theme is a natural product of projective geometry that has dominated geometry for nearly a century. I became familiar with the literature in this area and wrote several papers, including my master's thesis on projective line geometry.
After Plücker and Klein, line geometry has always been a favorite subject for geometrists. In fact, Klein's dissertation was about quadratic linear bodies, i.e. lines loci determined by quadratic equations under Plücker coordinates. Quadratic line bodies have many backgrounds and also have content of many line geometries.
My thesis looks at line sinks, the two-dimensional submanifolds of lines and their close osculation through quadratic line bodies.
Toward the end of my graduate studies, around 1934, I began to recognize the importance of integral differential geometry (then known as large-scale differential geometry). My main inspiration came from W. Blaschke's writings on differential geometry.
It is clear that algebraic topology is the foundation of the entire field. The algebraic topology itself was still in its development stages. Veblen's 1922 publication of analyticsis situs introduced " homology characters " , the Betti numbers and torsion coefficients derived from the correlation matrix. Lefschetz's Topology was published in 1930, but it was not helpful for beginners to enter the field.
I listened to a lecture by Emanuel Sperner (1933-1934). Sperner was visiting Peking University at the time, and his class contained a rigorous and detailed discussion of Erhard Schmidt's proof of the Yodang curve theorem.
I also heard a lecture on "Position Analysis" taught by Jiang Zehan, a former student of Marston Morse who served as Lefschetz's assistant. And my feeling was that I had just stood at the door of the great hall of algebraic topology. It wasn't until Seifert-Threlfall's book came out in 1934 and Alexanderroff-Hopf's book in 1935 that things changed dramatically.
In the spring of 1932, Blaschke visited Peiping and gave a series of lectures on "Topological Problems in Differential Geometry". This is true local differential geometry. He replaced lie groups in classical differential geometry with pseudogroups composed of whole differential isomorphisms and studied local invariants.
I was able to keep up with Blaschke's lectures and read the many papers published in the Hamburger Abhandlungen and other journals under this overall heading. This topic is now called web geometry. Because of this contact and previous knowledge of Blaschke's differential geometry books, I decided to study abroad in Hamburg when I received a scholarship in 1934.
<h1 class="pgc-h-arrow-right" data-track="133" >02 Study abroad life in Europe</h1>
I was in Hamburg from 1934 to 1936, received my Ph.D. in 1936, and spent a year in Paris with Elie Cartan as a postdoctoral researcher. The University of Hamburg has a strong mathematics department, where Blaschke, Artin and Hecke are professors, and the younger members include E. Berger. Kähler, H. Petersson, and H. Zassenhaus。
At that time, Blaschke's mathematical interest was shifting from net geometry to integral geometry. When I first met him in September 1934, he gave me a large stack of swastikas of the geometry of the net. I became interested in the concept of the rank of the net and the net with the greatest rank. As we all know, a d-net with a residual dimension of 1 in Rn consists of d hypersurface leaf structures in a general position.
Let x1 ,..., xn be the coordinates of Rn, and the leaf structure is determined by the equation
Given. Shaped like
The equation is known as the Abel equation. The maximum number of linearly independent Abel equations is called the rank of the net. If the d-net is defined by the hyperplane of the class d algebraic curve in Rn space , it has such Abel equations , which are obtained by applying Abel 's theorem to the Abel differential. Thus the rank of this d-net is at least the genus of the curve.
In a short article I determined the maximum rank of all d-nets with an afterte dimension of 1 in Rn. According to one of Castelnuovo 's theorems , this integer is equal to the maximum deficit of a d-times algebraic curve in an n-dimensional projective space Pn that does not belong to the arbitrary hyperplane Pn-1.
It is worth noting the fact that not all nets with the greatest rank are given by algebraic curves with the largest deficits described in the above way; there are strange nets with the greatest rank, and the leaves of these nets are not all hyperplane. These Abel equations are essentially functional equations, because in the classical case, these equations become well-known additive laws for transcendent functions.
In the plane ( n = 2 ) , the maximum rank of the 5-net of the curve is 6 , and there is a strange net ( Bol net ) with a binary logarithm in the Abel equation of the net. In 1978 Griffiths and I studied the d-net problem with the largest rank in Rn and a residual dimension of 1, but we did not get the final result. I think identifying such a weird web is a very interesting and important question.
From 1934 to 1935 my main energy was devoted to Kähler's seminars. The workshop was based on Kähler's recently published famous pamphlet, Einführung in die Theorie Systeme von Differentialgleichangen. The main result was what came to be known as the Cartan-Kähler theorem.
All of them, including Blaschke, Artin and Hecke, attended the first seminar and were each given a copy of the above-mentioned booklet. But the number of participants decreased rapidly, and I was one of the very few people who persevered. I apply this theory to the 3-net of r-dimensional manifolds in R2r. Both Blaschke and Kähler thought that this result was sufficient to write a dissertation from my previous result on maximum rank. By the end of 1935 my dissertation was ready.
Blaschke and his school of thought were primarily concerned with integral geometry, and Blaschke taught courses on integral geometry. The prettiest results of this theme were discovered by L.A. Santalò. One result is to represent the equal weekly deficit of the plane convex curve in terms of the infinity sum of the positive terms, where each positive term has geometric significance. Santalò's work has made him a world-class leader in integral geometry. Originally from Spain, he later immigrated to Argentina.
Another of my alumni was the algebraic geometrist Weiliang Zhou, who came to Göttingen from Chicago to do research with Hermann Weyl. But changes in the political situation in Göttingen and throughout Germany made this wish come to naught, and he moved to Leipzig to work with Van der Waerden. For some reason, he lives in Hamburg and sometimes comes to seminars.
Zhou Weiliang was developing his "zugeordnete Formen", later known as the "Zhou coordinates". Zhou is an inventive mathematician. He made important contributions to algebraic geometry, including his compact sub-cluster theorem and intersection theory. Zhou was born into a family of high-ranking officials and eunuchs in China, and it recognized the need for Westernization very early on, so this family produced many outstanding figures. Weekly habit of working at night. When he visited I had to sacrifice some sleep, but I learned some math.
Anyway, I'll go and listen to Artin's lectures whenever possible. His lectures over the course of two years included complex variable function theory, algebraic topology, relativity, and diophanto approximation. I also heard Hecke teach algebraic number theory, mainly in his book. My academic career in Hamburg was ideal, but the political situation did not allow it to continue.
From 1936 to 1937, I was able to do postdoctoral research for one year. When I consulted Blaschke, he suggested that I either stay in Hamburg and study number theory with Artin, or go to Paris to follow Elie Cartan. Both options were attractive, and I chose the latter in the end.
This choice is ideal. That year Cartan gave a course on differential systems; lecture notes were later published in book form. The "young" French mathematicians who would later become Bourbaki became active. They organized a "Julia Workshop", which met every two weeks to study a topic selected each year. The topic from 1936 to 1937 was "The Work of E. Cartan".
Cartan is an excellent mentor. Some of the "small" questions he asked became the subject of my papers. Probably because of my answers to his questions, he allowed me to visit his house about once every two weeks. The day after we met, I usually received a letter from him, which often said, "I thought about his problems after you left. ...... It seems like an interesting question..." The year was fun and memorable.
I also listened to Montel's lectures on multiplicity and attended Hadamard's seminars at the Académie française. Hadamard always summarizes at the end of each seminar, which is usually clearer and richer than the presentation itself.
After learning the news of the outbreak of the Sino-Japanese War, it was with a heavy heart that I bid farewell to Paris and returned to China on July 10, 1937.
<h1 class="pgc-h-arrow-right" data-track="138" >03 mathematically isolated</h1>
When I left Europe and returned to China in the summer of 1937, I had planned to go to Beiping to take up the post of professor at Tsinghua University, but due to the Sino-Japanese War, I did not achieve this goal until ten years later. Tsinghua University moved first to Changsha and then to Kunming in 1938, where it remained until the end of the war in the summer of 1945.
Kunming is a beautiful city. Although the country in the midst of war is short of supplies and the situation is turbulent, it is pleasant in other aspects of life. Tsinghua University joined forces with Peking University and Nankai University to form Southwest United University, and Kunming immediately became the center of intellectual circles in wartime China. My math colleagues include Hua Luogeng and Xu Baolu. I gave courses and seminars on algebraic topology, Lie groups, sphere geometry, and outer differential systems, and attracted a group of students.
The main thing is that the connection with the outside world has been cut off: for a time even the "Myanmar Channel" has been closed, and the only way to contact the outside world is by air. I have a private library of small books. At first, I did what I wanted to do before but didn't have time: I read some books, I thought about problems, and I thought it was funny.
But setbacks soon came and had to be overcome. I send this love letter to E. Cartan, he sent me many of his swathes, including some past papers. I spent a lot of time studying these papers, considering their implications and applications. It really benefited me a lot.
In the 1930s, the importance of Cartan's work, such as Weyl, Blaschke and Kähler, was recognized, but almost no one read Cartan's old papers (with the exception of those on Lie algebra). I was fortunate to be able to read all of these papers because of the circumstances.
Dr. Hu Shi, the Chinese ambassador to Washington, D.C., air mailed a book on "The Theory of Dimensional Numbers" by Hurewicz-Wallman. People who are accustomed to electrostatic photocopying today may have a hard time imagining that I copied the entire book except for the last chapter. In the last chapter, the author deals with the problem of orthogonal sequences without the concept of orthogonal sequences, which I find difficult to understand. In fact, at that time, it was very common to read papers and take notes. Photocopying a large amount of information does not show how much progress you have made.
I began to have some students, including Wang Xianzhong and Yan Zhida. Wang later made many contributions to topology, although his most famous achievement was the king sequence. Yan was the first to give the correct value for the Betti number of all exceptions to the Li group.
Looking back, I don't think I had a perfect view of mathematics as a whole. I am aware of some of my shortcomings and desire to be enriched. My mathematical prowess lies in my ability to do the math. To this day I don't care about complicated calculations, and until a few years ago I did such calculations with few mistakes. Training in this area is now less popular and not encouraged, but it still has great benefits in dealing with many issues.
I was fascinated by the Gauss-Bonnet formula, and I know that its most conceptual proof is the external differentiation of the contact form through structural equations. When I went to Princeton in 1943, it had opened the door for one of the most proud papers I had ever written in my mathematical work.
<h1 class="pgc-h-arrow-right" data-track="139" >04 Princeton is sunny</h1>
I arrived in Princeton in August 1943. The change in atmosphere is unforgettable. The Institute for Advanced Study was quiet in those days, and most of them had left to serve the war.
Hermann Weyl is interested in my work. Before my visit, he had reviewed one of my papers on fan-oriented surfaces for the Annals of Mathematics and written a long, positive report. He personally leaked it to me. The report made suggestions for improvement, which showed that he had carefully read the entire text. We talk a lot. One of Weyl's deep insights is that predicting algebraic geometry has a very good future.
Andre Weil was at lehigh University nearby, and we met very quickly and had a lot to talk about. Weil had just published a paper on the Gauuss-Bonnet formula in collaboration with Allendoerfer, and it immediately became the subject of our discussion.
Based on my burying of two-dimensional situations, I know that the correct proof should be based on what we now call transgression. There are two difficulties:
(1) At the time I was not very clear about the Poincare-Hopf theorem about the singularity of vector fields;
(2) The excess must be achieved in the unit tangent bundle rather than in the main bundle, which involves an extraordinary technical difficulty.
I overcame both of these difficulties in a short period of time, and things had a satisfactory result. I still think it's the best job I've ever done.
It is natural to extend this result to the Stiefel-Whitney class. At that time, even in Princeton, when talking about fiber clusters, we had to start with definition. There were no vector bundles, only spherical bushes. I noticed that the repeatability class is simpler, allowing for local curvature representations. The work was not difficult, but it was not a fashionable subject in topology of the time.
Although I am a member of the Institute for Advanced Study, I spend a lot of time in Princeton University's Fan's Building. Chevalley was writing his book about Li Qun at the time. Lefschetz was stubborn, unwilling to study differential geometry using the conventional methods that prevailed at the time. When I was asked to review a paper for the Mathematical Chronicle and suggest a rejection, he asked me to serve as the associate editor of the journal.
I am very comfortable with the environment and the beat of work in Princeton. My view of mathematics is much more mature. Staying at Princeton made me feel a great pleasure. Scientific competition in recent years has made the lives of scientists a mess, although it is much better in mathematics. I don't think there's a need to produce results so quickly, and I'm not moved by the discovery of email.
At the end of 1945 I bid farewell to Princeton and returned to China. Immediately after setting foot in his homeland, he was commissioned to form the Chinese Academy of Sciences, the Academia Sinica, at a time when World War II was over, but China was in a state of division due to the civil war. I extended an invitation to Hermann Weyl to visit China, and he gladly accepted. However, the situation in China at that time made this visit impossible.
At the end of 1948, the Nanjing government was in the midst of a collapse, thanks to the Institute for Advanced Study for taking the initiative to arrange for me to leave China. In the winter semester of 1949 I was at the Institute for Advanced Study and was a keynote speaker at Veblen's Differential Geometry Seminar. The speech was supplemented two years later and circulated widely.
These lectures are now included in the fourth volume of my selected papers, which has been published. The main result is the Weil homomorphism. This is a generalization of the Chen class from unitary groups to arbitrary Li groups. I knew this result in 1944 when I was writing a paper on the reproducible class; I could not prove it at the time because I was not proficient with Li Qun. Weil provides a key idea by considering the contact family. I call this result the Weil homomorphism. Friends think I should share this honor, and I naturally have no problem with this.
<h1 class="pgc-h-arrow-right" data-track="140" >05 Mathematically enter the year of confusion</h1>
After World War II, Marshall Stone was called upon to reorganize the Mathematics Department at the University of Chicago and became its chair. His first two offers to be sent to Hasler Whitney and Andre Weil were proof of his insight into mathematics and mathematics. Whitney declined, and Weil accepted after several consultations.
When I was in China, Stone wrote to me about offering me an interrogation position in Chicago. After I came to the United States in 1949, the Mathematics Department of the University of Chicago decided to hire me for a long time. I think the University of Chicago is the only university in the United States whose primary goal is "intellectual progress" rather than education. I had many friends in the mathematics department there; in the summer of 1949 I became a member of that department. This led to a pleasant and rewarding period of cooperation.
In the 1949-1950 school year, I taught a course called "Large-Scale Differential Geometry" and had a group of talented students. I myself am forging my own path, and my students correct my many mistakes and omissions in a timely manner, which is a lively and interesting combination. I remember Arnord Shapiro, who presided over many of these discussions.
In retrospect, my understanding of differential geometry was rudimentary. Some of the debates in this discipline are still unresolved, perhaps reflecting its power. For example, what is a surface? Is it embedded or immersed, or defined by equations that may have singularities? On the other hand, many of the topics covered in my classes have also gained new and multifaceted developments.
I have a close relationship with Weil. He is always ready and ready to cooperate. Among the many mathematicians I have discussed mathematics with, Weil is one of the very few mathematicians who can quickly grasp my thoughts and give useful comments. We used to walk along lake Michigan for long periods, which were safe at the time.
I'm also interested in algebraic topology, and occasionally I take a course in this area. I've collaborated with Ed Spanier on the study of the sphere. One of the results obtained was to write Gysin's work as a positive sequence. Rene Thom made it clearer, a result now commonly referred to as Thom isomorphism.
I find both Chicago and Hamburg very pleasant. I think both are the right size. Unfortunately, the development of mathematics has inflated everything.
< h1 class="pgc-h-arrow-right" data-track="141" >06 settled on the West Coast</h1>
In 1960 I moved to Berkeley. This place is no stranger to me. My teacher in China, Professor Jiang Lifu, earned a science degree at Berkeley. In 1946 and 1949 I stopped at Berkeley twice and spent some time in berkeley's math department.
The Berkeley Department of Mathematics is first-rate and it is .C by G. Created by Evans. Evans has asked me on several occasions if I have any interest in going to Berkeley. Evans' brother was the owner of a famous Western bookstore in Tianjin. I used to buy textbooks there, and the prices were usually frighteningly expensive.
Evans was retiring, and I was getting serious about my work at Berkeley, and indeed, I sometimes thought that as I got older, Berkeley's warmer climate was attractive. Of course, the expansion of the Berkeley Mathematics Department, the development of air freight, and other factors that have made California less isolated than it once was, have also contributed to my move.
Berkeley has been improving its position in the mathematical community, attracting many outstanding students. There are 31 graduate students who have received PhDs under my supervision, and of course I have influenced other students. I started working "as a second author" with young people, such as With Bott, Griffiths, Moser, and Simons. In this case I feel less responsible. Life is getting more and more comfortable.
My close academic colleagues are Hans Lewy and Chuck Morrey, both creative and highly capable analysts. Lewy and the local equidistant embedding problem of the three-dimensional Riemannian metric in R6 were studied for a while. It directs us to three studies of asymptotic cones, and we figure out that it's hyperbolic, but that's about it.
The role of differentiation in mathematics is wonderful. Algebra and topology are often considered to be the two pillars of mathematics. But things weren't that simple; Newton and Leibniz were playing stunts. This period has seen differential geometry merge into the mainstream of mathematics.
<h1 class="pgc-h-arrow-right" data-track="142" >07 Pastime of the Old Age</h1>
My life is coming to an end, and my only consideration is how to spend it. The answer is simple, I'll continue fiddling with math. I've never been in sports, let alone now. Listening to music has always been a waste of time for me, and occasionally getting involved in it, purely for social reasons. Fortunately, there are still many fundamental problems with integral differential geometry, although in its development I will probably have been only a spectator.
I think that the object of study is limited to smooth manifolds only for technical reasons and is not satisfactory. Not only do non-smooth manifolds naturally exist, but even when starting with smooth manifolds, some geometric constructions such as envelopes will lead to non-smooth manifolds, and Whitney introduced the concept of stratifiad manifold, which allows for singularities and infinitesimal analysis to be applied.
Recent work by Robert McPherson has brought new hope. The Cheeger-Goresky-McPherson intersection and the McPherson chen class have revealed the nature of the concept. (see 2)
To me, it is unclear whether the Riemann structure is as largely as the latest developments suggest. After all, in that historic paper, Riemann allowed his metric to be a 4-fold form of 4-fold roots. The more general case is now called a Finsler measure. As I pointed out in a recent article, Finsler geometry can be expanded simply by taking the right point of view. Further development is inevitable.
As Griffiths has noticed, my love of algebraic techniques stems from my experience. Local differential geometry needs to be done in this way, but it is difficult to obtain a beautiful locality theorem. It is clear that the question of the net of the greatest rank discussed earlier is an important issue, and it will receive my attention.
concentrate:
This article was originally titled My Mathematical Education. Translation from a copy sent by the author to the editor of the Selected Writings of Chen Province on October 28, 1991. The original article was published in Chern-A Great Geometer of the Twentieth Century (1992), an essay edited by Yau Chengtong. This article is now included in Chern-A Gre at Geometer of the Twentieth Century, published by Jiaotong University Press.
bibliography:
[1] P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley, 1978.
[2] Robert McPherson, Global questions in the topology of singular spaces, Proc. ICM Warszawa, vol 1, 198 213-235.
[3] J. Moser, Geometry of quadrics and spectral theory, Chern symposium, Springer-Verlag, 1979, 147-148.
[4] S. Chern, On Finsler Geometry, Comptes Rendus, Academie des Sciences, Paris (1991).