Basic mathematics:

Number theory: classical number theory, analytic number theory, algebraic number theory, transcendental number theory, model and modular function theory.

Algebra: linear algebraic group theory, group representation theory, Lie group, Lie algebra, algebraic group, typical group, homology algebra, algebraic K theory, Kac-Moody algebra, ring theory, algebra, body, lattice, ordered structure, domain theory and polynomial topological group matrix theory vector algebra tensor algebra.

Geometry: (global and local) differential geometry, algebraic geometry, manifold analysis, Riemannian manifold and Lorenz manifold, homogeneous space and symmetric space, harmonic mapping, submanifold theory, Yang-Mills field and fiber bundle theory, symplectic manifold, convex geometry and discrete geometry Euclidean geometry non-Euclidean geometry analytic geometry.

Topology: differential topology, algebraic topology, low-dimensional manifold, homoethics, singularity and catastrophe theory, point set topology, large-scale analysis of manifold and cavity complex, differential topological homology theory complex manifold.

Function theory: function approximation theory.

Functional analysis: (nonlinear) functional analysis, operator theory, operator algebra, difference and functional equation, generalized function, variational method, integral transformation integral equation.

Differential equation: functional differential equation, characteristic and spectrum theory and its inverse problem, qualitative theory, stability theory, bifurcation theory, chaos theory, singular perturbation theory, dynamic system, nonlinear elliptic (and parabolic) ordinary differential equation, partial differential equation, micro-local analysis and general partial differential operator theory, mixed and other singular equations, nonlinear evolution equation, infinite dimensional dynamic system.

Mathematical physics: gauge field theory, classical theory of gravitational field theory and quantum theory, soliton theory.

Probability theory: Markov process, stochastic process, stochastic analysis, random field, martingale theory, limit theory, stationary process, probability theory statistics;

Mathematical logic and mathematical basis: recursive theory, model theory, proof theory, axiomatic set proof, and mathematical logic category theory.

Combinatorial mathematics: combinatorial counting, graph theory.

Analysis: series, series, sum calculus, abstract measure theory of real variable function, approximation and expansion of special function (single or multiple), complex variable function theory, harmonic analysis, Fourier analysis.

Applied mathematics:

Marginal discipline: system theory; Cybernetics, operational research, potential theory

Computational mathematics and scientific engineering calculation: numerical calculation of partial differential equation, numerical solution of initial boundary value problem, nonlinear differential equation and its numerical solution, numerical solution of boundary value problem, numerical method of finite element and boundary element, numerical method of variational inequality, symplectic geometric difference method, numerical solution of inverse problem of mathematical equation, numerical solution of ordinary differential equation and its application, research of two-point boundary value problem, singularity problem, algebraic differential equation, uncertain mathematical theory, Fractal theory, large sparse matrix solution, algebraic eigenvalue problem and its inverse problem, nonlinear algebraic equations, general linear algebraic equations solution, fast algorithm.

Function approximation: multivariate spline, multivariate approximation, surface fitting, rational approximation, scattered data interpolation.

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