Curriculum

Course

Introduce

Core courses

A Course in Functional Analysis

1. nonlinear operators in infinity dimensional normed linear spaces

2. the basic theory of topology linear spaces

3. Frechet-derivative and Gateaux-derivative of nonlinear operators.

4. topology degree theory and its applications

5. Variational methods and its applications

Differential Manifold

Differential manifold is the nature generalization of the smooth curve (1-dimensional differential manifold) in 3-Euclidean space and the smooth surface (2-dimensional differential manifold) in 3-Euclidean space. On high-dimensional manifold, the famous Green formula, Gauss formula, Stokes formula etc. are unified to be one form by exterior differential. The main content of this course is as follows: differential manifold, differential forms, exterior differential, integral on manifold, Stokes theory, Riemann manifold etc

Stochastic Mathematics

1. Fundamentals of measure and stochastic processes;

2. Poisson processes and generalized Poisson processes;

3. Markov processes;

4. Martingales and Brownian motion;

5. Stochastic analysis;

6. Bayesian statistics.

Geometric Theory of Ordinary Differential Equations

This course intent to let students acknowledge following contents:Liapunov stability,Lagrange principle, Poincare index of vector fields, Index of planary vector fields,Bendixion formula for index,Poincare-Birkhoff twist theorem, Poincare,s last geometric theorem, improvement of Poincare-Birkhoff twist theorem,limit set, P-type recurrent motion, quisi-minimal set,B-type recurrent motion, the interval of recurrent, minimal set and its existence, almost periodic motion, Liapunov stability of almost periodic motion, Poincare-Bendixson theorem, Seifert’s conjiecture, some remained open problems, quisi-periodic motion, elements of Chaos, necessary condition for chaos, geometric structure of chaos set, chaos and fractals, etc.

Specialized courses

Theory of Functional Differential Equations

In natural and social phenomena, the development trends or future state of many systems have something to do not only with present condition, but also with the past development trends more or less. This phenomenon is called delay or inheritance. The mathematical models brought forward from engineering, physics, mechanics, cybernetics, chemical reaction, biomedicine, heredity, epidemiology, circular systems of animal and plant and kinds of economic phenomena in social science have obvious delay arguments.

Nonlinear Evolution Equations

Learning through this course to make students master some methods and techniques for the handling of the existence of solutions, uniqueness, asymptotic behavior (blow-up, decay rate) of typical type equations in nonlinear evolution equations, and build the good foundation for their further study and research.

Stochastic Differential Equations

The main content of this course is as follows: measure theory, Markov process and diffusion process, Winner process and white noise, stochastic integral and stochastic differentiation, the well-posedness and asymptotic behavior of solutions of stochastic differential equations, the application of stochastic differential equations etc.

Dynamical Systems

The universal result of invariant sets and attractors, the introduction of Sobolev space theory, attractors in dissipative evolutionary equations, attractors in dissipative wave equations, Liapunov index, Hausdorff dimension, fractal dimension of attractors, the estimation of dimension for attractors in some physical systems, instable flow; Liapunov functional and the estimation of dimension lower bound, cone and contraction property,inertial manifold, inertial manifold and slow manifold under non-self-adjoint cases, the approximation for attractors and inertial manifold.

Modern Theory of Partial Differential Equations

The course introduces the Sobolev space H^s(W) and the partial differential equations in H^s(W), and the main course is the following:

1. introduce the space H^s(W) and its main characteristic: completeness, reflexivity, imbedding theory etc.

2. the existence of solutions for partial differential equations under suitable conditions, and the regular estimate, where we will introduce variational methods, semi-group methods, Fourier transformation, Galerkin methods etc.

Optional courses

Seminar on Partial Differential Equation

By this course to enable students to master the basic knowledge of reading and writing papers, and solve the problem of lack of applicable ability. It is helpful for students to choose areas of research and develop the independent research capacity.

Include the following:

1. To introduce the reading skills of the papers

2. Teacher’s forefront lectures

3. Students’ lectures and the discussion between teacher and students

Theory of Stochastic process

The course is focusing on basic concepts and basic theory of stochastic processes and provides basic access knowledge to further study and research stochastic processes.There are five chapters.

1. Introduction2. Markov processes3. Second moment stochastic process and random analysis4. Stationary process5. Time series analysis

Topics on Nonlinear Partial Differential Equations

Nonlinear partial differential equations is very important contents of modern mathematics.Nonlinear partial differential equations are often arise from modern sciences and technology as mathematical model. This course focus on some resent topics such as: Viscosity solutions of Hamilton-Jacobi equations, Homogenization theory, p-Laplacian equations etc. This course will be helpful for graduate students to decide the topics for their theses.

Special Topics on Stochastic Functional Equations

The infiltration between stochastic differential equations and functional differential equations produces stochastic functional differential equations naturally. Its study started from 1960s, and did not develop until 1970s. The stochastic functional differential equations have developing foreground either in theory or in application. Especially since 1980s, many scholars have made use of semi-group theory, linear or nonlinear functional analysis theory, and stochastic analysis to study stochastic functional differential equations, and have made great progress in well posedness, boundedness, stability, attraction region and attractors became the hotspot in international researches. it needs many mathematical tools，and has great difficulty, but it is a good opportunity to overtake the international advanced level, and therefore，opening the course It is necessary for students to open the course.

Special Topics on Hamiltonian Dynamics

Hamiltonian dynamics is active joint field of modern mathematics and physics. This course help students to hold basic knowledge of Hamiltonian dynamical systems. And then we focus on some special topics such as: perturbation theory, KAM thery, Aubry-Mather, weak KAM theory, and some connection with nonlinear partial differential equations etc. So this course will benefit for students to decide the topics for their thesis.

Recurrent Neural Networks

The content of the course is as follows：

1. Introduction: the basic concept of RNN.

2. RNN theoretical basis with the introduction of some Recurrent Neural Networks models.

3. The qualitative analysis of Hopfield neural networks.

4. The qualitative influence of parameter perturbation.

5. The qualitative influence of time delay.

6. Some synthesis methods of associative memory

7. The influence of interconnect confinement.

8. Great scale NN theories and methods.

9. Differentiable manifolds methods in RNN.