Xu Yichao mainly carried out the research work of complex homogeneous bounded domain, and obtained very rich research results, which have the international advanced level and created a new situation in the research of complex homogeneous bounded domain. The famous Riemann theorem in the theory of simple complex functions asserts that a simply connected domain with at least two points on the boundary is holomorphic and equivalent to the unit circle. This result cannot be generalized to the case of multiple complex variables. E Cartan introduces Hermite symmetric space, gives a complete classification from the perspective of homogeneous space, and proves that it is the topological product of four typical domains (which can be clearly defined in complex Euclidean space) and two exceptional indecomposable Hermite symmetric spaces (one is complex 16 dimension and the other is complex 27 dimension). Later Harish-Chandra proved that Hermite symmetric space can be completely embedded in Euclidean space and is a bounded domain (called symmetric bounded domain), but we still don't know what these two exceptions are. Because Hermite symmetric space is a homogeneous complex manifold, Cartan conjectures that any homogeneous bounded domain is holomorphic equivalent of symmetric bounded domain. Hua gave a weak conjecture: the holomorphic section curvature of any homogeneous bounded domain is always non-positive. From 1959 to 1963, Piatetski-Shapiro of the former Soviet Union denied Cartesian conjecture with two counterexamples, introduced Siegel domain, and proved that Siegel domain (unbounded domain) is holomorphic, equivalent to bounded domain, and the same as Vinberg and Titkin. In 196 1, Lu Qikeng and Xu Yichao denied the Hua conjecture with some counterexamples.

From the point of view of classification, the next problem is the classification of homogeneous Siegel fields. Xu Yichao turned this problem into an elementary matrix theory problem. Firstly, he defined a set of real matrices and complex matrices (called normal matrix sets), and introduced normal Siegel fields (homogeneous Siegel fields in complex Euclidean space) by using these matrices: Cj(z) and Qj(u) are both square matrices, and the definitions are clear. Then, he proved that any homogeneous Siegel field is linearly equivalent to the normal Siegel field, and there is holomorphic equivalence between the normal Siegel fields if and only if the normal matrix groups defining them are equivalent to each other under a special relationship. In this way, the classification problem of homogeneous bounded domains is reduced to the equivalent classification problem of normal matrix groups. Along this line of thought, he gave a complete classification under the assumption that all matrices in a normal matrix group are square matrices. These results unexpectedly include Cartan's results on Hermite symmetric space, that is, the specific expressions of the definition domains of those two exceptions are found. Xu Yichao's above achievement was made around 1965, but it was not published until 1976 because of the "four clean-ups" movement and the "Cultural Revolution" movement.

The so-called homogeneous space is a connected lie group G module and a special closed subgroup H, where G is an automorphism group on G/H, so the holomorphic automorphism group of homogeneous bounded domain is very important. Therefore, many mathematicians hope to find holomorphic automorphism groups and have done a lot of work for this purpose. This problem was independently solved by German mathematicians Dorfmaster and Xu Yichao in 1976. The former is difficult to further study the specific properties of holomorphic automorphism groups because of some characterizations of general homogeneous Siegel fields.

Using the concrete expressions of normal Siegel fields, Xu Yichao calculated their Bergman kernel functions, Bergman metrics, Cauchy-Segeg kernels and formal Poisson kernels, and proved that the necessary and sufficient condition for formal Poisson kernels to be Poisson kernels is the symmetry of homogeneous Siegel fields. In addition, he also discussed the second-order invariant differential operator of homogeneous Siegel domain, proved that the Bergman mapping of homogeneous Siegel domain is holomorphic isomorphism, and found out why there is no way to discuss the function theory on homogeneous bounded domain with the realization of Vinberg aligning sub-Siegel domain.

The realization of Xu Yi's super-homogeneous Siegel domain greatly promotes the study of functional and geometric properties of homogeneous bounded domain, and makes the study of these problems computable. He proved that the formal Poisson kernel of asymmetric homogeneous Siegel domain is not Poisson kernel, and then proposed how to establish the theory of harmonic function on asymmetric homogeneous Siegel domain, that is, to study the properties of the solution space of Laplace-Beltrami equation. On the other hand, he gave a set of standard bases and multiplication tables of lie algebras of holomorphic automorphism groups, which provided good conditions for studying this kind of lie algebras. Xu Yichao's work is internationally recognized as the most important work in Siegel field since 1975. J.L. Koszul, a famous French mathematician, commented: "In my opinion, Xu Yichao's work on convex cones and Siegel fields is the most important and fundamental contribution to this theory since 1975, which should promote new development in many directions. Although it is necessary to better understand the algebraic structure of normal cone after introduction, as Xu Yichao's outstanding work shows, once this method is mastered, it is a very effective tool. " This work of Xu Yichao won the second prize of China Academy of Sciences 1987.

Vinberg and Titkin suspect that homogeneous Keller manifold is holomorphic fiber bundle, the bottom space is homogeneous bounded domain, and the bundle space is compact homogeneous Keller manifold. Master Dolph proved this conjecture. On the basis of Nobuyuki Murakami's work, Xu Yichao gave a complete classification of Keller manifolds under the transitivity of reduced Lie groups.

He also considered the classification in bounded domain with Thullen condition in two-dimensional complex Euclidean space. Runtu and H. Cartan gave a complete classification of Reinhardt domain, circular domain and semi-circular domain. Xu Yichao and his students gave a complete classification of semicircle domain and positive (m, p) circle domain, and provided many meaningful standard domains. This method of constructing standard domain is very useful for studying standard domain under other Thuram conditions and extending it to multiple complex variables.

From 1958 to 1976, Xu Yichao undertook many different mathematical application tasks. 1958, the Institute of Mathematics dissolved the algebra, number theory and topology groups and established the operation group. Participated in the group to promote linear planning, and participated in the programming of transportation and national grain distribution. On this basis, Xu Yichao, Wang Yuan and others compiled the Theory and Application of Linear Programming, which was published by Higher Education Press with the number 1959. This is the first book on linear programming in China. 1969, he completed the calculation task of primitive polynomials over the field of feature 2; 1976 completed the calculation task of small-scale population forecast. These works have been well received by users.

Since 1986, Xu Yichao has actively participated in the middle school students' mathematics competition. He participated in the training of the first China Mathematical Olympic Training Team, selected six players, and achieved good results in the international mathematical competition. Since 1992, I have participated in the proposition group of China Mathematical Olympiad, participated in the selection of training team members and overseas representatives, made my due contribution to China's winning the first total score and a large number of gold medals in the International Mathematical Olympiad for many years, and won honor for the motherland. 65438-0998 was hired by chinese mathematical society Olympic Committee as the national coach of Mathematical Olympiad.

Although scientific research institutions have no teaching tasks, Xu Yichao is very concerned about college mathematics education; He has taught advanced algebra 196 1 and 1963, Nankai University 1986, Tsinghua University 1989 and Henan University in 2000. Among them, the teaching time of 6 1 and 63 grades in the Department of Mathematics of the Chinese University of Science and Technology is as long as 4 years, and the teaching contents include analytic geometry of plane and space, advanced algebra, linear algebra, abstract algebra and so on. Later, he compiled the lecture notes into Introduction to Algebra, which was published in Shanghai Science and Technology Press on 1966 on the recommendation of Professor Hua. This book, for the first time in domestic textbooks, makes full use of matrix tools to turn linear space problems into algebraic problems. The book contains a large number of difficult problems, which has become a necessary reference book for graduate students after the Cultural Revolution and influenced many advanced algebra textbooks published after the Cultural Revolution. 1992, in order to meet the new needs, he rearranged some chapters in Introduction to Algebra and rewritten it into a book, Linear Algebra and Matrix Theory, which was published by Higher Education Press. The book won the first prize of 1996 National Excellent Textbook. It can be said that Introduction to Algebra, as the basic teaching material and teaching reference book of linear algebra, has fully influenced several generations.

Xu Yichao is one of the few mathematicians in China who are really familiar with Lie Qun. 1983 collaborated with Professor Yan Zhida to publish the book Lie Qun and Lie Algebra in Higher Education Press, and won the second prize of 1990 National Excellent Textbook. In 2000, he published Lie Qun and Hermite symmetric spaces in Science Press. He has taught Li Qun as a graduate student in Peking University, University of Science and Technology China, Graduate School of China Academy of Sciences, Hangzhou University, Zhengzhou University, Zhejiang University, Nankai University and Henan University, which has made indelible contributions to the popularization of Li Qun in China. Xu Yichao's lecture is clear in thinking, thorough in reasoning, inspiring and outstanding in teaching effect, which is well received by students and teachers everywhere. In the lecture, he paid special attention to explain clearly what the idea of proof was and why he thought so. He is good at analyzing the course content, paying attention to basic training, the essence of the course and the application of mathematical skills, which can lay a solid foundation for students to engage in research work in the future.