The clothes are seamless, and the ingenuity is spliced. It's all in one, it's vast and wonderful.
Creation loves geometry, and the four forces fiber can. Eternal thoughts, Ao Gao Li Jia Chen.
Presentations | Zhang Weiping
Organize | Xu Xuehao
2021 marks the 110th anniversary of the birth of Chern Shiing-Shen, and the Chern Institute of Mathematics has held a series of activities. Director Bai Chengming invited me to give a lecture as the "finale" of this commemorative event for Mr. Chen, and assigned me a proposition essay to talk about Mr. Chen's academic achievements.
Mr. Chern has made outstanding contributions to the development of world mathematics and Chinese mathematics throughout his life, and his position in the field of mathematics is naturally also very important. Here, I would first like to quote a poem by Mr. Yang Zhenning "Praise Chen's Grade":
The clothes are seamless, and the ingenuity is spliced. It's all in one, it's vast and wonderful. Creation loves geometry, and the four forces fiber can. Eternal thoughts, Ao Gao Li Jia Chen.
"Eternal Thoughts" comes from Du Fu's poem "Articles through the ages, gains and losses are known", and "Eugori Jiachen" is a tribute to Mr. Chen's historical status to catch up with the four great geometricians in front of him, Euclid, Gauss, Riemann, and Cardan. This also gives us a point of reference, and I would like to follow this line to talk about the contributions made by Mr. Chen and their historical status.
"Euclid" refers to Euclid, and when we talk about Euclid now, we are talking about his mathematical work "Geometry Primitives". In fact, Euclid's original text is called "Elements", which is the original, which encompasses the entire mathematics of the time. There are two most important theorems in it, one is that the sum of the internal angles of a plane triangle is equal to 180 degrees; There is also a number of prime numbers, which are infinite and are number theorized. Coincidentally, in the 20th century, two great mathematicians in China, one was Mr. N Shiing-shen and the other was Mr. Hua Luogeng, who made great contributions to geometry and number theory respectively.
In fact, there is a person named Descartes between "Europe" and "Gao", and his introduction of the study of geometry with coordinates is also a revolutionary progress, but today we will mainly talk about geometry, so I will not repeat him too much. "Gao" is Gauss, who did geometry when he was about 50 years old when he was the director of the observatory and when he was doing geodesy. He generalized Euclid's theorem about the inner angles of a plane triangle to a triangle on a curved (non-planar) surface. Later, Bonnet extended this formula to the case where polygons and edges can be arbitrary curves, and now calls this generalization the Gauss-Bonnet theorem.
The further generalization of the Gauss-Bonnet theorem after Gauss is to talk about Riemann, who is a master of number theory, and his Riemann conjecture is currently the first big problem in the millennial problem, and he is also a master of function theory. His contribution to differential geometry is mainly reflected in a teacher qualification examination he took that year, and after passing, he can bring his own students, which is equivalent to a doctoral supervisor in China. In this short article, he introduced the concept of high-dimensional Riemannian space and defined the generalization of Gaussian curvature in high dimensions, which we can now call Riemannian curvature. Riemann's motivation came from the theory of complex variable functions and electromagnetism.
Riemann proposed the concept of high-dimensional space, so the development of high-dimensional Riemannian geometry requires a strict description of the objects in high-dimensional space, which is what we now call manifolds, and the first to strictly define it was Hermann Weyl. He wrote a book called "The Concept of Riemann Surface", which is to apply the idea of Riemann parts to Riemann surfaces and make them whole, just like we originally made a local surface into a closed surface. I consulted the relevant literature, and the book was published in 1913, when Weyl was only 28 years old, which was very young. Weyl is certainly a great figure, and he is also a pioneer of gauge field theory.
Mr. Chen once wrote a popular report called "From Triangles to Manifolds", which divided geometry into several eras, one is called "primitive man", which refers to Euclidean geometry; Later, when Descartes came, there were algebraic tools, and those who had tools to do were called "clothed people"; Later, geometry on manifolds appeared, and it became "modern man". So in the 20th century, with the concept of manifolds, it is natural to ask, how to be a modern man, that is, how to develop geometry on manifolds?
The representative figure to be mentioned here is Elie Cartan (Jiatang), which is the "Jia" of "Eugoli Jiachen". One of his major contributions is to extend the theory of local calculus to manifolds, called the external differential calculus. After finishing his Ph.D. in Germany, Mr. Chen chose to go to Paris to follow Jiadang as a postdoctoral fellow, and stayed in Paris for a year, reading Jiadang's articles hard and getting his essence. Gardan's article was notoriously difficult to read, and even Hermann Weyl found Gardan's article difficult to read. Mr. Kadang's collection of essays consists of three volumes, each of which is very thick, and Mr. Chan says he can read at least seventy or eighty percent. So two days ago, I told my students that if you want to be a good student of the teacher, you must first read the teacher's important works well.
At this point in mathematics, the next step is to generalize Gauss-Bonnet, which is the most important in two-dimensional geometry, to higher dimensions. There is a problem in this, to generalize the concept of Gaussian curvature to the higher dimensions, and then to find a way to prove the desired equation. The first to succeed were Allendoerfor and André Weil. André Weil, founder of Bourbaki and one of the greatest mathematicians of the 20th century, Allendoerfor was his colleague. In a sense, the research they have done can be said to have completed the generalization of the Gauss-Bonnet theorem to high dimensions, but it is not satisfactory, and it can be described as "knowing what it is, not knowing why it is".
At this time, Mr. Chern completely solved this problem, which he considered to be the most important work he had done in his life. Mr. Chen uses the so-called unit spherical bundle, and each point takes the tangent vector of the unit length, which is related to the manifold itself, and does not need to be embedded in another Euclidean space. Any Riemannian manifold itself has a unit-length vector at each point, which is equivalent to having hair on your head, and you cut a unit-long inch of hair at each point, and take this out to partially become a unit-long spherical cluster. Because the underlying manifold is too difficult to do, Mr. Chen put this problem on the spherical plexus of this unit. For the first time in history, Mr. Chen brought this problem to the spherical cluster of units, and then solved the problem. This idea is called transgression, and it was the first in Mr. Chen's history to introduce it. Later, his student Wu Guanglei named this idea "Chaodu", that is, if you can't do it at the bottom, you go to the top to do it, and then pull the problem back after you finish it, which is very vivid. The application of the idea of transcendence to specific techniques is the external differential calculus that Mr. Chen learned from Jiadang. From a philosophical point of view, Mr. Chen's work (first embodied in the Gauss-Bonnet theorem) is the development of the intrinsic correlation between the part and the whole in the high-dimensional situation.
Later, in the process of this work being promoted, Mr. Chen defined the demonstrative class Chern class named after him. According to Mr. Chen's own words, this Chern class was a sudden inspiration when he went to the library one weekend, which may be the humble words of the master, but the influence of the Chern class is obvious to all. For example, it has played a fundamental role in the Atiyah-Singer metric theorem, which is considered one of the most important mathematical theorems of the 20th century, the Fields Medal-winning work of Chengtong Yau (i.e., solving the Calabi conjecture), and the recent pioneering work of Fu-Chengtong Yau on non-Keller manifolds. The fundamental importance of Mr. Chen's mathematical contributions is self-evident.
In addition to the Chern class, another of Mr. Chen's most influential and groundbreaking work is the definition of the Chern-Simons schematic formula. Chern-Simons has had a profound impact on both physics and algebraic geometry, most notably Edward Witten. He proposed the so-called Chern-Simons quantum field theory, the Jones polynomial used to study kink theory, which won the highest prize in mathematics and is currently the only physicist to win a Fields Medal in mathematics. Witten's article was first published in the book edited by Mr. Yang Zhenning and Academician Ge Molin, and later published in CMP, so our Institute of Mathematics can be regarded as an indirect contribution.
To sum up, Mr. Yang Zhenning's famous sentence "Eternal heart, Ao Gao Li Jia Chen" is very in place, and Mr. Chern Shiingshen's position in the history of world mathematics is beyond doubt.
Mr. Chern's contribution to Chinese mathematics
Next, I would like to talk about Mr. Chen's contribution to the development of mathematics in China. During the three years from 1946 to 1948, Mr. Chen trained a group of young talents at the Institute of Mathematics of the Academia Sinica in Shanghai, the most prominent of which was Mr. Wu Wenjun.
According to Mr. Chen's own recollection, the Institute of Mathematics of the Academia Sinica was established with Mr. Jiang Lifu as the director, and after Mr. Jiang went to the United States, Mr. Chen served as the acting director. He believed that the first priority was to train new students, so he sent letters to the mathematics departments of various famous universities, asking them to recommend the best students in three years, and many people responded, which shows that Mr. Chen's prestige at that time was already very high.
In Mr. Wu Wenjun's article reminiscing about Mr. Chen, he wrote that topology was recognized as difficult to learn at first, but after he and Mr. Chen completed the first year of study, he made an article on topology and published it in Annals. Many people thought it was incredible, but he said it wasn't surprising and it just showed that Mr. Chan was good at mentoring (and of course, Mr. Wu was talented). Then Mr. Wu pointed out that everything must be done from the fundamentals, although Mr. Chen's main goal is to specialize in a wide range of differential geometry, but during his three years at the Institute of Mathematics of the Academia Sinica, he did not talk about it to young people, because he wanted to do something more basic and devote himself to the cultivation of algebraic topology. Mr. Wu also pointed out that during the three years of the Institute of Mathematics of the Academia Sinica, Mr. Chen trained a group of topology backbones for the mainland, which was a contribution to Chinese mathematics before the liberation.
After returning to China in the 70s of the last century, Mr. Chen put forward a series of suggestions to help the progress of mathematics in China. In 1978, Mr. Chen personally invited Wang Qiming and Peng Jiagui of the Chinese Academy of Sciences to visit Berkeley, which opened a precedent for young mathematicians to visit the United States.
Later, the country decided to select 50 people to go abroad, and Mr. Chen recommended Zhang Gongqing to New York and Jiang Boju to Princeton, and finally 52 people made the trip, and Mr. Chen shoehorned two in. Mr. Zhang Gongqing and Mr. Jiang Boju later became leading figures in the field of mathematics in China.
Mr. Peng Jiagui once told me about the experience of going abroad, when two young people took a plane to go abroad for the first time, and they were very nervous, but as soon as they got off the plane, they saw Mr. Chen coming to pick them up, and the big stone in their hearts fell to the ground all of a sudden. Perhaps it was the first time that a Red China scholar had visited, and the local media really publicized it. Later, Mr. Chen also wrote a note on this matter: "In September 1978, Jia Gui came to California to do research together, hoping to make a meaningful historical fact for Chinese mathematics. In addition to his own outstanding research work, his student Tang Zizhou won the 2020 Mathematics Award of the Academy of Sciences for the Developing World (formerly known as the Third World Academy of Sciences). We can see it as a beautiful echo of Mr. Chen's original initiative.
The second step is that after the reform and opening up, for 7 consecutive years, from 1980 to 1986, the so-called double micro-years (differential equations + differential geometry years), invited a large number of relevant foreign experts to communicate with their domestic counterparts, which greatly promoted the development of domestic differential geometry and differential equations.
At that time, Mr. Chen invited Fields Medal winners Atiyah and Bombieri, Wolf Prize winners Bott and Griffiths, Abel Prize winners Lax and Nirenberg, as well as famous mathematicians Singer and Kohn, etc., and Mr. Hua Luogeng Hua Lao, Mr. Wu Wenjun and Mr. Gu Chaohao made speeches at the conference. At that time, these people were not invited to make a report and it was not just a matter of making a presentation, Bombieri wrote more than 100 pages of articles during this period, Bott wrote more than 50 pages of articles, and Mr. Chen's own articles were more than 100 pages long.
I would like to mention Nirenberg, who had a good relationship with Mr. Chen, and after Mr. Chen's death, the International Mathematical Union established the Chern Medal at the International Congress of Mathematicians, and Nirenberg was the first person to receive the award. His memories of Mr. Chen are very interesting, he thinks that everything he does with Mr. Chen is good, even eating with Mr. Chen in a Chinese restaurant, it is a wonderful experience. A famous story is that during the first Shuangwei Conference, Mr. Chen invited a group of mathematicians, including Nirenberg, to the front door for dinner, Nirenberg said that he had never eaten such a delicious meal in his life, and he never forgot to take someone to eat again next week, and the result was that I ate the same meal and ordered the same dish. How the taste of the dish is completely different.
Mr. Chen himself is quite proud of eating, and does not deny that he is a "foodie". When I was a postdoc at Berkeley's Institute of Mathematics, Mr. Chen once joked with me, "Weiping Zhang, mathematics, I don't know how much you can learn from me; Eat, you have to study hard. ”
Mr. Chern's initiative at Nankai University
In August 1984, Mr. Chern was officially appointed as the director of Nankai Institute of Mathematics. After the appointment ceremony, Comrade Deng Xiaoping hosted a banquet for Mr. Chen in the Great Hall of the People, along with Mr. Ding Shisun, Minister of Education He Dongchang, Mrs. Chen, and Mr. Hu Guoding.
Mr. Hu, I would like to emphasize here that Mr. Hu visited Berkeley three times and invited Mr. Chen to Nankai to establish the Institute of Mathematics, which was indispensable for the establishment of the Nankai Institute of Mathematics. Mr. Chen once said to Mr. Hu: "Nankai Mathematics Research Institute is promoted by you, the future is immeasurable, when it has a significant contribution to the country, if there is something you can do, you should try your best to help." ”
Later, Mr. Chen personally wrote the name of the office, Wu Daren proposed it, and Mr. Chen recognized it, and determined that the policy of the office was "based on Nankai, facing the whole country, and looking at the world". This means that the Nankai Institute of Mathematics is not only for Nankai, but also for the whole country.
At the inauguration ceremony of the Institute of Mathematics, the 75-year-old man asked everyone to imagine what it would be like to hold a one-year-old "baby" in his arms. He said, "For Chinese mathematics and Nankai mathematics, I will do my best and die." Mr. Chan kept his promise. From 1985 to 1995, under the impetus of Mr. Chen, the Nankai Institute of Mathematics held 12 far-reaching academic year activities. In 1987, Mr. Chen invited Mr. Yang Zhenning to Nankai to establish a theoretical physics laboratory, realizing a high degree of integration of mathematics and physics.
In August 1988, at the 21st Century China Mathematics Outlook Conference, Chern put forward the conjecture that "China will become a mathematical power in the 21st century", and won a special fund, called the Mathematics Tianyuan Fund. At the meeting, Mr. Cheng Minde of Peking University made a keynote speech on behalf of the Chinese mathematics community on "Let Chinese Mathematics Take the Lead in Catching Up with the International Advanced Level", in which he mentioned that Professor Neuing-Shen, a contemporary mathematician, has repeatedly proposed that "as long as everyone works hard and the road is right, then mathematical science can take the lead in catching up with the world's advanced level". Mr. Cheng Minde said that we are willing to see Mr. Chen's wish become a reality.
When it comes to promoting the progress and development of mathematics in China, we have to mention the most important conference in the field of mathematics, the International Congress of Mathematicians (ICM). In 1993, Mr. Chen and Mr. Yau jointly proposed to hold the International Congress of Mathematicians (ICM) in China. In 1998, China won the right to host the International Mathematical Union Conference held in Germany. The 2002 International Congress of Mathematicians went down in history as the first time a country's top leader attended the opening ceremony.
I still remember that on January 12, 2000, Mrs. Chen passed away in her Nankai apartment, and the next day, I went to see Mr. Chen. He said that I should do two things now, the first is to do a good job in the International Congress of Mathematicians, and the second thing is to do a good job in the Nankai Institute of Mathematics.
Mr. Chen proposed to hold a commemorative meeting of Chen Guocai and Zhou Weiliang in Nankai in October 2000. Chen Guocai and Zhou Weiliang are both the most famous mathematicians in China and have a high status in the world.
In addition to Griffiths, Secretary General of the International Mathematical Union, as a delegate to the conference, Mr. Chen also invited Palis, a dynamical systems scientist and President of the International Mathematical Union. In the invitation letter, Mr. Chen wrote at the end that President Jiang and I are good friends, and said nothing else. After reading the letter, Palis came to the conference.
When Comrade Jiang Zemin received foreign guests, Palis, on behalf of the International Mathematical Union, invited him to attend the opening ceremony. Comrade Jiang Zemin said that if he was in Beijing during the opening ceremony, he would definitely attend the meeting. This meeting paved the way for the successful convening of the International Congress of Mathematicians in Beijing, and all subsequent departments gave the green light to the Chinese mathematical community. This incident also shows Mr. Chen's foresight.
In his speech at the 2002 ICM Conference, Mr. Chen, honorary president of the International Congress of Mathematicians, concluded: "Confucius's Confucianism has had an impact on China for more than 2,000 years. Its main doctrine is 'benevolence', which literally means 'two', which means to value interpersonal relationships. Modern science is highly competitive. I think that if we inject a human element, it will make our subject healthier and more interesting. ”
If we think about it from a human perspective, the "Chern class" is the "Chen class", which covers the students that Mr. Chen has cultivated throughout his life, as well as the students' students, and so on. Liu Kefeng has an article commemorating Mr. Chen, entitled "We All Belong to the 'Chen Category'".
Mr. Chen's two most famous students, as everyone knows, one is Wu Wenjun and the other is Yau Chengtong, these are the two most representative figures of the "Chern class". In Nankai there is actually a "Chern class" as well. Hou Zixin and the old principal posted a commemorative article in Tianjin Daily, which talked about this Nankai mathematics pilot class. In fact, in order to help the development of Nankai University, Mr. Chen first established the Institute of Mathematics and proposed the establishment of a pilot class of mathematics. Mr. Chen went hand in hand with two fists, not only to establish a mathematics institute, but also to make great contributions to mathematics education and undergraduate education in Nankai.
Mr. Chen also took classes in person for the pilot class and put a lot of effort into it. The pilot class has also cultivated a group of talents, only two are listed here: one is Zhu Chaofeng, who is now in Nankai; One is Guan Qi'an, who is now at Peking University. Professor Zhu Chaofeng and Academician Long Yiming have a paper published in the top journal "Annals of Mathematics", if I remember correctly, this should be the first article in "Annals of Mathematics" after the reform Chinese mainland and opening up, and it has been widely talked about for a while.
Professor Kwan worked with Academician Xiangyu Zhou in an article in Annals of Mathematics to solve one of Demailly's so-called strong openness conjectures. The American "Mathematical Reviews" commented on their work as the greatest achievement of complex analysis and the "cross-sectional" aspects of algebraic geometry in recent years. General Secretary Xi Jinping delivered a speech at the 2016 Academician Conference of the Chinese Academy of Sciences and the Chinese Academy of Sciences, which mentioned that China's science, of course, mathematics ranked first, and that the theory of multiple complex functions, together with some other scientific breakthroughs, laid an important foundation for the mainland to become a big country with world influence. I think General Secretary Xi Jinping said this, not only to praise Zhou Xiangyu, but also to praise the inheritance of generations and generations of scientists in terms of multiple complex changes starting from Hua Luogenghua Lao. Of course, there are many, many more outstanding students of Mr. Chen, so I will not list them all here.
Finally, I would like to talk about the building of our Institute of Mathematics, which was hailed as the best mathematics building in the international mathematical community when it was completed, but unfortunately Mr. Chen did not see it when he was alive. At that time, Mr. Chen "forced" me to be the director of the Institute of Mathematics, and I said that I would only be the director of the Cherry N Institute of Mathematics, and we all hoped that the Nankai Institute of Mathematics could be renamed the Chern Institute of Mathematics, but Mr. Chen just disagreed.
It was not until after Mr. Chen's death that Comrade Jiang Zemin went to Tianjin on September 27, 2005, when he went to Tianjin, he had to go to the new building of the Nankai Institute of Mathematics (Provincial Building) to have a look. The 20-minute activity was originally scheduled to last for 40 minutes, and finally we prepared pen and ink and wanted Comrade Jiang Zemin to give an inscription, and he said that I was tired, so I should sign it. Seeing that the signature has been completed, I am in a hurry. Maybe it was also the courage that Mr. Chen gave me, so I had the courage to say, "Chairman, we would like to ask you to write about the Chern Institute of Mathematics." Comrade Jiang Zemin said that I would pay the "Chern Institute of Mathematics." Halfway through writing, I realized that I wanted to use his words, so I said: "Today I wrote it standing up, it's not neat, I'll go back and write another formal one." ”
Later, the Institute of Mathematics officially changed its name to Comrade Jiang Zemin's inscription. I think this can be regarded as a great achievement in my life, and I am finally worthy of Mr. Chen. Nowadays, the provincial building that towers on the edge of the Jin River looks particularly beautiful. I said to Director Bai Chengming that as long as this building is here, this name is there, no matter the vicissitudes and changes of the world, our Chern Institute of Mathematics will one day create more brilliance, and this especially depends on the efforts of the young people here.
We are pleased to see that Chinese mathematics has made tremendous progress since Mr. Chen's article entitled "Prospects for Chinese Mathematics" published in the journal Nature in 1981. As reported by China Newsweek in April 2021, the so-called "golden generation" of mathematics has emerged.
"Winter is coming, will spring be far away"? I believe that in the near future, the "chern conjecture" of "China will become a big and powerful country in mathematics" will definitely be solved, and I believe that Mr. Chen's soul in heaven will also be gratified.
"When the mountains are full of flowers, he laughs in the bushes", and we hope that our young people will work together to make their own contributions to the complete solution of the "Chern conjecture".
This article is reprinted from the WeChat public account "Mathematics Compound" and was originally published in Tianjin Daily (December 20, 2021). The "Mathematical Compound" was edited and organized when it was released. The speaker is an academician of the Chinese Academy of Sciences.
Special Reminder
1. Enter the "Boutique Column" at the bottom menu of the "Huipu" WeChat official account to view a series of popular science articles on different themes.
2. "Back to Park" provides the function of searching for articles by month. Follow the official account and reply to the four-digit year + month, such as "1903", to get the article index in March 2019, and so on.