After working in China for more than a year, I have met many middle school students, college students and graduate students. In order to attract outstanding students to mathematics, I have had many dialogues and exchanges with them, which has triggered my thinking on mathematics education from all aspects. So far, many articles criticize our education system, thinking that it stifles students' imagination. However, I think our education has put too much emphasis on skills since middle school, and the fundamental drawback is that students' knowledge has not been developed at all. Seeing more can make you know more, but without extensive knowledge, imagination is passive water. The tactics of asking questions on the sea based on Olympic Mathematics in middle schools make students forget that the purpose of doing questions is to understand knowledge. In universities, some teachers' knowledge is too narrow, which leads students to a dead end, and it is even more impossible to broaden their knowledge. I think that for students majoring in mathematics, we should first broaden our horizons, not only among various disciplines of mathematics, but also among related disciplines such as physics. All kinds of feelings have contributed to this article, and I hope my experience and experience can play a role in attracting jade.

I will combine my academic experience to discuss the importance of knowledge and the relationship between knowledge, skills and imagination. Since I was a graduate student, my work has revolved around geometry and topology in physics. Physicists need mathematics as a tool, and in turn put forward mathematical conjectures with the help of physical theory. Although physicists' deduction is often not rigorous, these conjectures are often proved to be correct in the end. This is very surprising!

In order to solve the mathematical conjecture put forward by physicists, we developed a brand-new mathematical theory and found unexpected connections between different branches of mathematics. These mathematical revolutions provide a strict theoretical basis for the continued development of physics. The interweaving of mathematics and physics has led to many revolutions in the history of science. The famous ones are calculus and Newton's laws of mechanics, general relativity and Riemann geometry. There are countless examples in recent years, such as the elliptic genus stiffness theorem obtained by combining quantum field theory with exponential theory, Verlinde formula of module space given by * * formal field theory, Yang-Mills field, 4-dimensional topology and so on. Chen-Simmons theory and three-dimensional topology, knot theory, mirror formula of mirror symmetry in string theory and Calabi-Hill space, marino-Wafa conjecture of Chen-Simmons theory, Calabi-Hill space and gromov-Witten invariants, the relationship between string theory and Ricci flow, three-dimensional topology, mirror symmetry and number theory. In the past 20 years, half of the prize winners' work in the field of mathematics has been related to quantum field theory and chord theory. This gives us reason to guess: God created the world according to mathematical formulas? But there is no doubt that mathematics is the key to nature.

It should be pointed out that physicists' contribution to mathematics is not limited to predicting mathematical conclusions. Many times, they also use rigorous mathematical language to point out important research objects in mathematics for us. Witten and Wafa are two outstanding representatives, and their mathematics is even better than most mathematicians. Some people describe them as if they came back from the future time and space, only remembering the fragmented scenes of future mathematics, and telling them from memory has become a guess that challenges contemporary mathematicians. The way physicists study mathematics may be worth learning. Witten probably never does math exercises, but they learn the math they need as quickly as possible. Taubes, a professor of mathematics at Harvard University, once said, "Physicists learn exponential theory first, and then Riemann geometry". I think we mathematicians should not only pay attention to the development of physics at all times, but also pay attention to physicists' skills of mastering knowledge, that is, learning in research and learning in study.

Physicists especially favor infinity, even at the expense of strictness, such as SL(2, z) symmetry, Chen -Simons theory of large N limit and path integral. Although Feynman's path integral still lacks a strict mathematical foundation, the theory has a far-reaching influence in modern quantum physics because of its intuitive and convenient physical form. As the saying goes, "beauty is endless, and beauty is useful." This lack of strictness also gives them endless imagination.

So how should we study math?

I went to study in the United States and only brought two books. One is the "differential geometry" of Qiu Chengtong and Schon, and the other is the "second order elliptic partial differential equation" of Gilberg and Te Rudinger. I want to show my skills in analysis and geometry. At the end of September, I walked into Mr. Qiu Chengtong's office and began my study life at Harvard. He asked me whether I wanted to start doing research or continue to study more mathematics. I replied that I wanted to start doing research. But Miss Qiu said to me, "You should study math as much as possible, because it is not easy to learn new things after graduation." He asked me to study algebraic geometry, algebraic number theory, geometric analysis ... There are many things that I still can't fully understand until today. But it has profoundly influenced my academic career and life track. After becoming a professor, the heavy pressure of teaching and scientific research made me realize how earnest Mr. Qiu's words are.

Which is more important, knowledge or skills? In my opinion, knowledge is more important to young people! Knowledge makes us stand higher and see the right direction. Because the direction is wrong, all efforts will have no result. However, we must also admit that the key breakthrough in research often comes from the innovation of skills. For example, a martial arts expert has learned a lot of martial arts, but if his internal skills are not good, he is easily possessed. As we all know, Mr. Qiu has made pioneering work in many fields of mathematics with his strong analytical ability and extensive knowledge. At present, the fierce math competition for middle school students in China places too much emphasis on skills. In fact, our students should be influenced by all kinds of knowledge from middle school, so that children can read more biographies of celebrities and cultivate their curiosity about science. The biography of Newton I read recently is very wonderful. It was out of curiosity that Newton asked himself dozens of questions about nature in his sophomore year. In order to solve them, he developed calculus as the foundation, and then developed the four laws of physics.

Next, based on my own experience, I will discuss the importance of extensive knowledge, the necessity of crossing mathematics with physics and engineering disciplines, and the benefits of academic exchanges with friends.

When I was studying in the Graduate School of China Academy of Sciences, my classmates included Zhang Weiping and Zhou Xiangyu, who have now become the most outstanding young mathematicians in China. At that time, there were few opportunities to listen to cutting-edge courses. We organized our own discussion class, reported to the old class meeting, exponential theory, Modal conjecture ... At first, we couldn't fully understand it, but we broadened our horizons and at least knew what "good" mathematics was worth learning. This is very important for everyone, and we need to cultivate our ability to appreciate mathematics. If you are not sure what good mathematics is, it is always right to read the works and articles of great mathematicians and follow the master. The idea of localization later became an important tool for my research, and it was also mastered during my study and master's thesis in China. Later, I used the idea of localization to understand all the mathematical knowledge I had learned, just like stringing many beads together with a thread.

Since I came to Harvard University, what impressed me most was the diligence of professors and students there. What China lacks most now is such an atmosphere. A first-class university is actually such a first-class atmosphere. What drives them to be so involved is their curiosity and love for mathematics and their thirst for knowledge. Harvard holds various discussion classes and the students are very active. There are not enough seats, even sitting on the floor. I feel as if I have plunged into the ocean of knowledge and feel different sunshine every morning. This is an exciting day.

Witten's article "Supersymmetry and Morse Theory" has the greatest influence on my work, and the research style of Professor Bot of Harvard University has also benefited me a lot. Bert said, "Go downstream, not upstream". That is to say, when doing mathematics, we should always go downstream, don't be too laborious and reluctant, and pursue the fluency of "canoe has crossed Chung Shan Man", but don't go with the flow, and both sides should coordinate well, otherwise there will be no innovation.

Every change in mathematics is inseparable from new ideas and methods and the integration of different branches. This requires us to think more creatively on the basis of mastering rich knowledge in order to occupy a place in the forefront of mathematics development. The interaction between mathematics and physics will undoubtedly be the mainstream branch of mathematics research for a long time to come. To give several interdisciplinary examples, differential geometry is created by combining calculus with linear algebra; Faltings proved Mordell's conjecture with Arakeloff's theory which combined algebraic number theory and algebraic geometry. Starting from symmetric functions, or more generally, from the theory of compact group representation, we can get Chen class, K- theory, Riemann-Roch formula and exponential theory. Integrated modular form, representation theory and elliptic genus of topology; Many applications of duality in string theory revealed by physicists in mathematics and other aspects.

Mathematicians have made great contributions to the whole society and people's daily life. From computers and the Internet to life sciences and finance, mathematics can be found everywhere. Although it is not easy to find a job in the United States now, Wall Street still recruits a large number of math graduates, who can be competent after three months of training. There are also several mathematicians among the winners of the Nobel Prize in Economics, including Nash, the hero of A Beautiful Mind, and Debru, who gave an hour's report at the International Congress of Mathematicians. It can be said that mathematics is the most selfless and potential major. Advance can strive to be a great scientist, while retreat can lead a quality life. It is easy to transfer mathematics to other majors, but on the other hand, it is not easy for other majors to transfer to mathematics. Mathematics can give you a good logical thinking training, even if you don't do mathematics in the future, you can do well in other fields. My classmate 150 in the Department of Mathematics of Peking University, although I am the only one doing pure mathematics now, is doing well now.

Einstein said, "Imagination is more important than knowledge". But without profound knowledge, imagination can only be a castle in the air. The so-called "genius" means that you always have seven or eight questions in your mind. While reading the literature, you constantly analyze these problems with newly learned skills and methods to see if you can find a breakthrough. As long as you persist with your heart, you can always solve two or three of them, then others will think you are a genius.

My doctoral thesis mainly studies elliptic genus, which is a combination of exponential theory and modular form and can be regarded as exponential theory in cyclic space. After Witten put forward the rigid conjecture of elliptic genus inspired by quantum field theory, Bott and Taubes spent a lot of energy to study this problem, but their proof was too skillful and complicated. I attended the seminars of Harvard and MIT on elliptic genus. I noticed the symmetry of elliptic operators in cyclic space under the action of module group SL(2, z), and then gave a concise proof of rigid conjecture within a few months, in which Jacobi-theta function and module form in number theory were used. SL(2, z) symmetry is also a basic principle in string theory. The idea of proof first germinated on the way to Princeton to participate in table tennis competition, and the last step of proof was when watching movies. I remember that there are always loopholes in the first few proofs, which is very distressing. But I firmly believe that such a wonderful idea must be right, otherwise mathematics is not interesting at all, maybe I have given up doing mathematics long ago. This is why the mathematical feeling I have learned and cultivated in many aspects plays a key role. Later, I continued to generalize the rigidity theorem and combined it with infinite dimensional Lie algebra. Through this new method, I not only found new extinction and rigidity theorems, but also used the knowledge of number theory and algebraic geometry to understand the rigidity phenomenon through the geometry of die surface. These methods are still very useful now, which completely exceeds my expectations. This brand-new method also contributed to my cooperation with friends such as Ma Xiaonan, Zhang Weiping and Dong Chongying, and combined the theory of vertex operator with the rigidity of ellipse operator.

The first step of my research career is to benefit from extensive knowledge accumulation. In the process of research, I also deepened my understanding of what I have learned. After studying at Harvard for several years, I think the most important gain is my understanding and ability to grasp "good mathematics".

In the late 1980s, when studying the two-dimensional * * * shape field theory, physicist Verlinde put forward a famous conjecture to calculate the holomorphic cross-section dimension of regular line bundle in the module space of stable bundle on Riemannian surface, which was a very popular research topic in the early 1990s. The module space of stable bundles on Riemannian surfaces has been studied in many branches of mathematics, especially algebraic geometry and topology. Mathematicians have tried many methods to calculate the holomorphic cross-section dimension of its canonical line bundle, but all failed. However, string theorists have unexpectedly given a very concise closed formula. Soon Witten put forward a conjecture about the closed formula of intersection number on the module space of principal bundle on Riemannian surface when studying the two-dimensional gauge theory. In principle, Welinder formula can be obtained by combining Witten formula with Riemann-Roach formula or exponential formula. At that time, I taught at MIT, participated in many discussion classes on this issue, and tried many different ways to understand Witten formula, which is the infinite sum of all irreducible representations of compact Lie groups. During that time, I also had a deeper understanding of symplectic geometry.

Until one day in the library of MIT, I was browsing the interested literature as usual, and I accidentally came across the expression of thermonuclear on Lie group, which was given by an infinite sum formula of the same type as Witten formula. I was immediately convinced that I had found a tool to prove Witten's formula, that is, the thermonuclear of Lie group. Having an idea is only the first step, and there are still many technical difficulties to overcome. It took me months to write down all the details of the proof.

Inspired by my work, Bismut can prove the general Verlinde formula by my method.

The English word "research" of "research" is repeated search, which well embodies the essence of research. Qiu Chengtong and Mr. Yang Zhenning both have a good habit of reading magazines in the library. They don't want to know, just to be well informed. Like other disciplines, every progress in mathematics is based on the work of predecessors. It can be said that "opening books is beneficial"!

I received an acceptance letter from Stanford University. Just an hour before I was about to drive away from Boston, Mr. Qiu Chengtong called me to discuss mirror symmetry. 1990, British physicist Candeira and others put forward five conjectures about the counting formula of rational curves in Calabi-Hill space on the basis of mirror symmetry. In the past hundred years, algebraic geometricians have been trying to count the number of these rational curves, but they can only get the number of rational curves no more than three times. Candeira's formula gives the number of rational curves of any degree by calculating a very simple third-order ordinary differential equation, namely Picard-Fuchs equation, which has caused a great sensation. Many mathematicians tried to prove this formula, including witten, kontsevich, giventhal and other famous mathematicians. I didn't pay attention to the research field of mirror symmetry before, so I began to read more literature. Sometimes I can't figure it out after thinking hard for a few days, and even get an experience of "this road is blocked" after complicated calculations. Later, inadvertently, when I noticed the importance of recursive structure in the stable mapping module space, the problem seemed to suddenly become clear. This wonderful feeling is hard for others to understand. Soon, Qiu Chengtong, Lian Wen Hao and I gave the first complete proof of Candeira's mirror image conjecture. The key to the proof is functor localization technology, which is also a very important tool in my future research work. Since then, we have extended the mirror image theorem to a very wide range of situations. This is a very pleasant cooperation, and our advantages are combined with each other, so that difficult problems can be solved quickly.

During these years at UCLA, I have gained a lot in research, and we have also proved the Hori-Vafa mirror conjecture of Glassman manifold. In addition to the functor positioning formula, complex combinatorial techniques and algebraic geometry are also used. Only after discussion with Liu Jianhao can these difficulties be overcome. I was deeply impressed by Hao Jian's hard work and courage not afraid of all difficulties. Therefore, it is very important to communicate with good friends, especially those who know each other's work and ability.

In the early 1990s, kontsevich proved Witten's famous conjecture that the generating series of some classical integrals (called Hodge integrals) in the modular space of algebraic curves satisfy infinite KdV differential equations. I have been paying close attention to kontsevich's work and related development for a long time, and the popularization of mirror symmetry in high genus also needs to calculate more extensive Hodge integral. In 200 1, Marino and Vafa speculated that the generating series of a wider class of Hodge integrals in curvilinear modular space can be expressed as a combined closed formula about symmetric group representation, that is, Chern-Simons knot invariants, based on the dual relationship between Chern-Simons theory and Calabi-Yau space.

I was quickly attracted by a beautiful guess and realized that I needed to find a breakthrough in the combination method first. In the summer of 2002, the International Congress of Mathematicians was held in China. Zhou Jian and I discussed many problems such as mirror symmetry in Beijing-Hangzhou and Shanghai-Hangzhou planes, and of course mentioned the marino-Wafa conjecture. Since then, many fruitful discussions have continued by email. Soon, Zhou Jian clarified the combination part of marino-Wafa formula, that is, the combination formula represented by symmetric groups. He noticed that this combination formula satisfies a so-called "cut-join" equation. Because this "cut and union" equation is equivalent to a set of ordinary differential equations, from the uniqueness theorem of the solution, the remaining problem is only to prove that the geometric part of Marino-Vafa formula, that is, the generating series of Hodge integral also satisfies this "cut and union" equation and has the same initial value as the combined part.

The proof of the geometric part of Marino-Vafa formula is quite tortuous and difficult, and we have made many attempts with functor positioning technology. In April 2003, Liu Qiuju came to UCLA and attended a seminar hosted by me. I told Liu Qiuju about the research progress with Zhou Jian, and with the joint efforts of the three of us, the proof of the geometry part was soon completed. I remember that all three of us were puzzled by extremely complicated expressions, and we were puzzled. We once thought about giving up and only wrote down some results. Finally, Liu Qiuju came back from Arrow Lake to discuss with me, and made one last helpless attempt, using a method similar to our proof of the mirror formula, which succeeded! The feeling at that moment is unforgettable. When this conjecture is proved, there is really a feeling of unity between heaven and earth, which is a wonderful feeling of soul stirring. The pre-printed version of the certificate was published in June 2003, which caused great international repercussions.

Compared with Witten-kontsevich formula, marino-Wafa formula not only has more extensive Hodge integral, but also is a non-recursive closed formula. More importantly, our proof is a wonderful combination of geometric methods and combinatorial skills, which has a good methodological reference for the proof of similar formulas in the future. We continue to use our method to establish the mathematical topology of vertex theory. From this, we can draw many more interesting conclusions, including our connection with the exponential theory, and my student Pan Peng used this new theory to prove the famous Gopakumar-Vafa conjecture on the circular Calabi-Hill manifold.

Near Zhu Zhechi, with a group of smart people, you will become smarter.

The geometry of the module space and Teichmuller space of Riemannian surfaces is an old problem. In the early 1980s, in cooperation with Zheng and Zheng, the existence of Kahler-Einstein metric on Teichmuller space was proved. Then he guessed that Kahler-Einstein metric on Teichmuller space of Riemannian surface is equivalent to classical Teichmuller metric and Bergman metric. Recently, Qiu Chengtong, Sun Xiaofeng and I proved Qiu Chengtong's conjecture by studying two new complete metrics-Ricci metric and perturbed Ricci metric in detail. In addition, it is proved that all classical complete metrics are equivalent to our newly introduced metrics, which clarifies many old problems in this field. More importantly, we further obtain the algebraic geometric result that the log cotangent bundle of module space is stable. This result is still unknown to algebraic geometricians.

When I was a student, I was interested in the geometric problems of module space and Teichmuller space. Participated in various discussion classes, wrote two papers, and proved several important results in algebraic geometry by using the curvature property of Weil-Petersson metric on module space. I think this is the most effective way to learn a new course, which is more beneficial than doing exercises and has a deeper understanding of problems and concepts.

I met Sun Xiaofeng when I was teaching at Stanford. At that time, he was a doctoral student at Schoen and had some reading classes with me. He is clever and persistent, and he is a rare mathematician. With Mr. Qiu, we had a lot of fruitful discussions on Riemannian surface module space, which enabled this work to be successfully completed. Our work is an important contribution to the space geometry of Riemannian surface modules. We are still studying many interesting problems and will write many results soon.