1, geometric analysis

"Research content" takes the theory of partial differential equations as the main tool to study the geometry, topology and analytical structure of differential manifolds.

"Prepare knowledge" of partial differential equations and differential geometry.

2. Algebra and its application

Research contents: the structure of finite groups, the study of solvable groups, the normal conditions of groups and the quantitative characterization of groups.

Preparatory knowledge, finite group theory, modern algebra

3. Functional differential equation theory

"Research content" uses group theory and nonlinear analysis theory as tools to study the structure and qualitative properties of functional differential equations.

Basic theory of differential equation, functional analysis.

4. Partial differential equation

"Research content" Theory and application of partial differential equations and related topics. At present, we mainly study the free boundary problem of tumor growth, and study the global existence of the solution of nonlinear evolution equation, the asymptotic behavior of the solution of reaction-diffusion equation, oscillation integral and Fourier integral operator theory in Fourier analysis. The general theory of linear partial differential equations, Fourier analysis and invariant partial differential equations on nilpotent Lie groups, the existence of solutions of singular elliptic partial differential equations, the comparison principle and uniqueness theorem of nonlinear elliptic and parabolic partial differential equations, and the global existence of solutions of nonlinear parabolic partial differential equations are studied. In the next few years, we will mainly study the oscillation integral and Fourier integral operator theory in Fourier analysis and the well-posedness and global existence theory of solutions of various nonlinear evolution equations related to it.

"Preparatory knowledge" includes partial differential equations, ordinary differential equations, functional analysis, harmonic analysis, etc.

5. Number theory and its application

"Research content" Diophantine approximation and Diophantine equation: mainly study the effective algebraic approximation of algebraic numbers and the solutions of some Diophantine equations, and use Diophantine equations to study the class number of quadratic fields. At the same time, the irrationality and transcendence of the sequence are also studied. Difference set theory: the nonexistence of some difference sets is mainly studied by algebraic number theory. Theoretical basis of cryptography: Some problems in cryptography are mainly studied by using the theory of finite field and cyclotomic field.

"Pre-knowledge" number theory, algebra and complex analysis. You need a good foundation in number theory and algebra, or in number theory and complex analysis.

6. Symplectic topology and mathematical physics

The main problems studied in the "research content" are the blow-up formula of gromov-Witten invariants of symplectic manifolds, the change of quantum homology groups under double rational operations, the relationship between gromov-Witten invariants and integrable systems, and mirror symmetry.

"Preparatory knowledge" functional analysis, partial differential equation foundation, abstract algebra, differential geometry, topology, algebraic geometry.

7. Set theory and mathematical basis

The "research content" uses various technical means developed in contemporary set theory, such as compulsory method and large radix method, to solve problems in infinite group theory and various topological spaces.

"Preparatory knowledge" measurement theory, group theory, basic abstract algebraic knowledge, set theory.

8. Differential geometry

"Research content" focuses on Ricci flow theory and its application in differential geometry, and studies the problems in real complex differential geometry such as gap theorem, univalence theorem and function theory on manifold of curvature Pinching phenomenon.

9. Nonlinear partial differential equations

The "research content" mainly involves the theories and methods of nonlinear waves, nonlinear evolution equations and infinite dimensional dynamic systems. Well-posedness, blow-up and global existence of strong solutions, global existence and uniqueness of weak solutions, stability of special solutions (such as equilibrium points, periodic solutions, soliton solutions, etc.). ), regularity of solutions, global existence of classical solutions and long-term behavior of solutions.

"Preparatory knowledge" functional analysis, partial differential equations, differential geometry.

10, the method of partial differential equation function theory

"Research content" studies the boundary value problems of singular integral operators and equations, analytic functions and their practical applications.

The mathematical basis of "preparatory knowledge" mainly includes calculus, linear algebra, ordinary differential equations, partial differential equations, complex variable functions, real analysis and measure theory, functional analysis and so on.

1 1, asymptotic analysis

"Research content" studies the Stokes phenomenon of integral, the uniform asymptotic expansion of integral and orthogonal polynomial system, Riemann-Hilbert analysis, Painleve function and the application of asymptotic analysis method in mathematical physics.

The mathematical basis of "preparatory knowledge" mainly includes calculus, linear algebra, ordinary differential equations, partial differential equations, complex variable functions, real analysis and measure theory, functional analysis and so on.

12, harmonic analysis

"Research content" mainly focuses on the theory of singular integral operator with non-smooth kernel and its application, function space related to differential operator, functional calculus of operator, etc.

"Preparatory knowledge" harmonic analysis, functional analysis, and the basis of partial differential equations.

13, functional differential equation theory and its application

The theory and application of ordinary differential equations, functional differential equations and time-scale dynamic equations are the "research contents".

"Preparatory knowledge" is mainly the basic theory of ordinary (functional) differential equations. It is better to have difference equation foundation and good functional analysis foundation.