Fundamental mathematics

Fundamental mathematics is one of Beijing municipal key subjects, and obtained the authority to award doctor's degree in 1996. The main research directions include: expression of finite group, group and graph, Kac-Moody algebra, Hopf algebra, expression of the kernel function in functions of several complex variables, modular space of complex structure, domain of holomorphy, geometry and analysis on complex manifold, harmonic analysis, analysis of Lie group and its special functions, singular integral, representation and approximation of multivariate periodic function, ordinary differential equation and dynamical system, nonlinear functional analysis and application, conformal approximation of multivariate linear operator, full distributive lattice, Domain theory, set theory topology and infinite combination theory, theory of topological fixed point, symplectic geometry, geometry and topology of positive curvature manifold, etc.

Applied mathematics

Applied mathematics is one of Beijing municipal key construction subjects. The main research directions are as follows: mathematical problems in physics, quantum information and quantum computation, partial differential equations and their applications, and operational control. The first direction mainly studies: 1) topological property of quantum gauge theory, 2) infinite-dimension symmetry in quantum field theory, and 3) quantum group, quantum algebra, and Yang-Baxter equation. The second, quantum information and quantum computation is a new field brought forward by the concepts of quantum mechanics to information processing and computing methods, and belongs cross study of physics, mathematics, information theory and computing method. The third, partial differential equations and their applications, mainly researches several classes of nonlinear partial differential equations, especially the existence, stability and gradation of the traveling wave solutions and equilibrium solutions of some diffusion equation sets simulating linear reaction with staggered diffusion as well as parabola & hyperbola coupled equation sets, adaptability of Euler equation's solution, and multi-solution property of some ellipse equations. The last, operational control, mainly studies nonlinear optimization theories and their applications.

Computational mathematics

Computational mathematics is a key subject of Capital Normal University, and the main research directions include: finite element and boundary element method, high-precision numerical solution of partial differential equations, computational fluid dynamics, numerical approximation and numerical algebra, structure-keeping symplectic algorithm of dynamic systems, linear and nonlinear optimization. The first direction, finite element and boundary element method, mainly discusses the numerical computation method using finite elements and boundary elements for differential equations. The method is computing method that has the most vitality and receives much attention in the field of computational mathematics. The second, high-precision numerical solution of partial differential equations, mainly probes into different kinds of high-precision numerical methods for solving partial differential equations, to provide accurate computing methods for the calculation of actual scientific works. The third direction, computational fluid dynamics, mainly discusses the numerical computation and numerical simulation methods for various dynamic systems in fluid dynamics. The fourth, numerical approximation and numerical algebra, mainly studies various computing methods used in numerical computation. The fifth, structure-protection symplectic algorithm of dynamic systems, mainly explores the symplectic algorithm for the differential equations in dynamic systems. And the last, linear and nonlinear optimization principally researches the optimization algorithms used for linear and nonlinear optimization problems.

Probability theory and mathematical statistics

The probability theory and mathematical statistics subject includes such research directions as filtration theory, random operational research, etc. Filtration theory is an emerging mathematical subject. In 2001, S. Smirnov published the article on the conformal invariance of penetration probability of critical filtration on triangular grid points, which implies the coming of the age when mathematicians study critical phenomena. After that, G. Lawler and Yu Zhang et al, based on conformal invariance, solved a series of power law and critical exponent problems of triangular gird point filtration model in succession. These advancements have made filtration studies become the mainstream direction of particle system (also called infinite particle Markov process) and even random process studies. The contents concerned by random operational research are mainly concentrated in such fields as queue theory and its applications, performance analysis and optimization of random network, optimal design of optical fiber communication network and wireless communication network, system reliability analysis, and so forth. This direction attaches importance to actual problems, aims at the solution of actual problems, and focuses on the important topics concerning the national economy and the people's livelihood to carry out innovative studies.

Theory of course and teaching (mathematics)

The theory of course and teaching (mathematics) subject began to recruit students in 1982. The main research directions include: theory of mathematic courses and teaching, study of mathematics learning psychology, mathematic modeling and mathematic education, mathematic methodology, and history of mathematic education. This subject has once assumed the training of state-level high school backbone mathematic teachers, and as one of the main sponsors, participated in the constitution of national course standards for high school and junior high schools, and has participated in the compiling of not a few middle school mathematic textbooks. Just now, two national education "10th five-year plan" projects of the Ministry of Education are being researched here.