Curriculum

Course

Introduce

Core courses

The Course of 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

General Topology

This course is to introduce the students the fundamental concepts, theories and methods of general topology, and make the students summarize and improve their known contents in their undergraduate study from a general viewpoint. Furthermore, it offers theoretic support to further studying of Topology, Geometry, functional analysis, the theory of complex functions, differential equation, etc.

Elements of Algebra

The main content of the course includes some basic concepts and theories in modern algebra. By studying this course students will know the developments of modern algebra, learn its basic methods and ideas, raise the abilities in logistic thinking and abstract thinking, and build the foundation for further study.

Complex Analysis

Modern algebra is a one-semester course intended for the first year graduates majoring in mathematics.

Complex analysisstudiesanalysis ofcomplex function analysisof complex independent variable， mathematics is the most important one of the branches，in the other branches of mathematics and aerodynamics， fluid mechanics， electrical， thermal，Theoretical physics has an important application.

A Foundation of Fuzzy Mathematics

Fuzzy mathematics is originated in 1965 when L.A.Zadeh published the first article about fuzzy mathematics. After that it experiences rapid development. Now, it has proven its usefulness in both theoretical and applied application. This course aims at introducing students the fundamental theory on fuzzy mathematics and the necessary math basis for further study and research.

Specialized courses

Modern Theory of Fuzzy Topology

This course is intended to present a systematic exposition of these different approaches. Students are required to be familiar with the fundamental notions of topology, fuzzy set theory and residuated lattices.

Introduction to Value Distribution Theory

This subject contains four chapters. Chapter 1 deals with Nevanlinna basic theory. Chapter 2 introduces some research works related to uniqueness theorems of meromorphic functions of finite order (or lower order). Chapter 3 is devoted to Nevanlinna’s five values theorem and the study for multiple values and uniqueness, and contains various improvements and generalizations of Nevanlinna’s five values theorem, L. Yang’s method dealing with multiple values problem, and H. X. Yi’s research work related to multiple values and uniqueness. Chapter 4 provides various improvements and generalizations of Nevanlinna’s four values theorem ; including Gundersen and Mues’swork and Reinders’ 4DM values theorem.

Modern Complex Analysis Conduct

This courseis intended to serve as an introduction to the geometry of the action of discrete groups of Mobius transformations. Becausethe course has been written with the conviction that geometrical explanations are essential for a full understanding of the material and that however simple a matrix proof might seen, a geometric proof is almost certainly more profitable. The subject matter is concerned with groups of isometrics of hyperbolic geometry, many publications rely on Euclidean estimates and geometry.

The Modern Theory of Variational Methods and its Applications

The course introduces the modern theory of variational methods and its applications in differential equations. The following theory will be included:

1. direct methods to solve extremal problems and its applications.

2. general minimax principle and its applications to elliptic equations.

3. category theory and index theory and their applications

4. Morse theory and its applications.

Introduction to L-Topology

The contents of this course are as follows:

1. Fundamental concepts

2. Induced L-topological spaces

3. Connected L-topological spaces

4. Countability of L-topological spaces

5. Separation axioms in L-topology

6. Different compactness in L-topology

The requisites are the general topology and lattice theory, and good mathematical understanding and ratiocination ability.

Introduction to Modern Analysis

Topics in fundamental analysis. Includes linear topological space, Banach space, Hilbert space, spectral theory.

Introduction to Operator Algebra

A course in operator algebra. Includes fundamental theory of C*-algebra, ideal and positive linear functional, von Neumann algebra, and representation of C*-algebra.

Optional courses

Introduction of category

The aim of this course is to introduce fundamental contents and methods of category to graduate students. By this course, they should understand the basic theories and methods of category, and should realize that the category theory provide a simple “sign language” to mathematical fileds and the relationships between them. Now, the category theory is becoming one important fundamental tool to study modern mathematics.

Modern theory of lattices

The purpose of this course is to completely present the latest theory on lattices, and offer necessary math base for further reference reading and research.

The contents are listed as follows:

1. Partial ordered sets and lattice

2. Galois connections

3. Heyting algebra

4. Locales and topological spaces

5. Specialization ordering and topology

6. Continuous lattices

The prerequisites for understanding this course are good command of general topology and algebra.

Basic Concepts of Algebraic Topology

The aim of this course is to introduce students the basic concepts and ideas in algebraic topology and make students grasp the essence, methods and applications of algebraic topology.

The main topics are as follows:

Simplicial homology: simplicial complex and polyhedron, simplicial group and its topological invariance, simplicial approximation, exact sequence, Lefschetz fixed-point theorem.

Introduction of singular homology: axioms of singular homology, CW-complex. Homotopy theory: fundamental group, covering space and classification of covering spaces, higher homotopy groups, etc.

Some Special Topics in Complex Analysis

This subject contains four chapters. Chapter 1 mainly introduces some complex oscillation theory of linear differential equations. Chapter 2 mainly introduces normal families of meromorphic functions, including normal families ofholomorphic functions, normal families of meromorphic functions and the singular directions of meromorphic functions. Chapter 3 mainly

the theory of algebroid functions. Chapter 4 mainly introduces fractorization theory of meromorphic functions

Nonlinear Functional Analysis

The course introduce the pre-estimate methods for the second order elliptic equations with the Dirichlet BVP，including theory, Schauder theory, theory, De Giorge-Nash theory etc; the existence and regularity of weak solutions for linear and nonlinear elliptic systems.

Nonlinear Problems in Abstract Cones

The course introduce cone theory and the semi-order theory: the fixed point theory of increasing operators and decreasing operators; the fixed point theory of concave or convex operators; the fixed point theory of cone mapping; the multiplicity of solutions for the nonlinear equations; the existence of positive solutions.

Modern Theory of Partial Differential Equations

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

1. Introduce the space Hs(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.

Riemann Surfaces

The course introduces compact Riemann Surface and No- compact Riemann Surface. The following theory will be included:

1. Covering Spaces.

2. Compact Riemann Surfaces.

3. No- compact Riemann Surface

Uniqueness Theory of Meromorphic Functions

This course consists of five chapters. Chapter 1 deals with Nevanlinna basic theory. Chapter 2 introduces some research works related to uniqueness theorems of meromorphic functions of finite order (or lower order). Chapter 3 is devoted to Nevanlinna’s five values theorem and the study for multiple values and uniqueness, and contains various improvements and generalizations of Nevanlinna’s five values theorem, L.Yang’s method dealing with multiple values problem, and H.X.Yi’s research work related to multiple values and uniqueness. Chapter 4 provides various improvements and generalizations of Nevanlinna’s four values theorem. Chapter 5 introduces various kinds of sharing three values theorems.

Nevanlinna Theory and Complex Differential Equations

This course consists of five chapters. Chapter 1 mainly introduce some results from function theory. Chapter 2 mainly introduce Nevanlinna theory of meromorphic functions. Chapter 3 mainly introduce Wiman-Valiron theory； Chapter 4 mainly introduce the basic results of linear differential equations； Chapter 5 mainly introduce the zero distribution in the second order linear differential equations；Chapter 6 mainly introduce some results of complex differential equations and the Schwarzian derivative；Chapter 7 mainly introduce some results of higher order linear differential equations.

Introduction to K-Theory for C*- Algebras

A course in K-theory for operator algebras. Includes K_0 and K_1 groups，K_0 and K_1 functors，index map，Bott periodicity, etc.

Selected Topics in Modern Analysis and Algebra

Advanced topics in Modern Analysis and Algebra. Includes extension theory of C*-algebra, KK-theory, quantum group and Hopf algebra. The content of course will vary from time to time, reflecting current trends and recent developments of Modern Analysis and Algebra.

Artificial Neural Network

Artificial Neural Networks represent a technology that is rooted in many disciplines: neuro-sciences, mathematics, statistics, physics, computer science, and engineering. As such, neural networks find applications in such diverse fields as system modeling, time series analysis, pattern recognition, signal processing, and control. The primary purpose of this class is to provide a comprehensive introduction to ANN. Introduction of ANN, learning processes, perceptrons, radial-basis function network, support vector machines, principal component analysis, and self-organizing maps are all included in the class.

Introduction to Artificial Intelligence

Artificial Intelligence is a subject on simulating human’s intelligence by the computer. It develops intelligent machines or intelligent systems and lets them simulate, extend and expand human’s intelligence by artificial methods and technologies in order to realize some intelligent actions. An introduction to artificial intelligence introduces basic concepts, principles, methods and simple applications of artificial intelligence. It lays a foundation for graduate students to further research on the related fields of artificial intelligence.