# Mathematics ( M.S. )

### 1. Training/Research Orientation

- Pure Mathematics
- Scientific Computing
- Applied Mathematics
- Operations Research and Control Theory
- Probability and Statistics
- Mathematics Education

### 2. Program Duration and Credit

3 years in general, the maximum duration should not exceed 5 years (including the extension time).

30 credits in total, at least 20 compulsory credits.

### 3. Courses Requirements and Introduction

**Modern Analysis**

Content: This course provides coverage of real and abstract analysis, offering a sensible introduction to functional analysis as well as a thorough discussion of measure theory, Lebesgue integration, and related topics. This significant study clearly and distinctively presents the teaching and research literature of graduate analysis by: providing a fundamental, modern approach to measure theory; investigating advanced material on the Bochner integral, geometric theory, and major theorems in Fourier Analysis, including the theory of singular integrals and Milhin's theorem; offering exceptionally concise and cardinal versions of all the main theorems about characteristic functions; and, containing an original examination of sufficient statistics, based on the general theory of Radon measures. The contents span basic measure theory in an abstract and concrete form, material on classic linear functional analysis, probability, and some major results used in the theory of partial differential equations.

Prerequisites: Real Analysis, Functional Analysis.

Time: Once a week in 120 minute lectures.

**Abstract Algebra**

Content: Algebra is the language of modern mathematics. This course introduces students to that language through a study of groups, group actions, vector spaces, linear algebra, and the theory of fields. Topics include binary operations, groups, subgroups, homomorphisms, cosets, Lagrange's theorem, permutation groups, the general linear group; rings, polynomial rings, Euclidean rings.

Prerequisites: Linear Algebra.

Time: Once a week in 120 minute lectures.

**Basis of Measurement and Probability**

Content: Measurement plays a fundamental role both in physical and behavioral sciences, as well as in engineering and technology: it is the link between abstract models and empirical reality and is a privileged method of gathering information from the real world. This course focuses on a single theory of measurement for the various domains of science and technology in which measurement is involved by addressing the following main issues: What is the meaning of measurement? How do we measure? What can be measured? Measurement, which played a key role in the birth of modern science, can act as an essential interdisciplinary tool and language for this new scenario. A sound theoretical basis for addressing key problems in measurement is provided. These include perceptual measurement, the evaluation of uncertainty, the evaluation of inter-comparisons, the analysis of risks in decision-making and the characterization of dynamical measurement. The course presents a unified probabilistic approach to many fields which may allow more rational and effective solutions to be reached.

Prerequisites: Functional Analysis, Probability and Statistics.

Time: Once a week in 120 minute lectures.

**Differential Manifolds**

Content: Differentiable manifolds are among the most fundamental notions of modern mathematics. Roughly, they are geometrical objects that can be endowed with coordinates; using coordinates one can apply differential and integral calculus, but the results are coordinate-independent. Examples of manifolds start with open domains in Euclidean space , and include multi-dimensional surfaces such as the -sphere and -torus , the projective spaces, and their generalizations, matrix groups such as the rotation group, etc. Differentiable manifolds naturally appear in various applications, e.g., as configuration spaces in mechanics. They are arguably the most general objects on which calculus can be developed. On the other hand, differentiable manifolds provide for calculus a powerful invariant geometric language, which is used in almost all areas of mathematics and its applications. In this course we give an introduction to the theory of manifolds, including their definition and examples; vector fields and differential forms; integration on manifolds and de Rham cohomology.

Prerequisites: Standard calculus and linear algebra; familiarity with the statement of the implicit function theorem (students may consult any good textbook in multivariate calculus); some familiarity with algebraic concepts such as groups and rings is desirable, but no knowledge of group theory or ring theory is assumed; some familiarity with basic topological notions (topological spaces, continuous maps, Hausdorff property, connectedness and compactness) may be beneficial, but is not assumed.

Time: Once a week in 120 minute lectures.

### 4. Supervisors

Haitao Cao, Ercai Chen, Jinru Chen, Yonggao Chen, Xin Chen, Lixia Dai, Yujun Dong, Qikui Du, Xiuli Du, Shitai Fu, Hongjun Gao, Qibing Gao, Fei Guo, Deren Han, Wei He, Chungang Ji, Yutian Lei, Zhilin Li, Zhibin Liang, Guoxiang Liu, Yue Liu, Lianhua Ning, Yakui Song, Yongzhong Song, Wenyu Sun, Yuehong Sun, Lixin Tian, Hui Wan, Li Wang, Xiaoqian Wang, Yushun Wang, Jiaqun Wei, Ningjie Wu, Jianguo Xia, Xiaoming Xu, Yan Xu, Baogang Xu, Conghua Yan, Mingsheng Yang, Zuodong Yang, Yi Yao, Huicheng Yin, Ping Yu, Jihui Zhang, Xiaoyan Zhang, Xuebin Zhang, Zhiyue Zhang, Haiyan Zhou, Xiuqing Zhou, Huaiping Zhu, Jiandong Zhu, Quanxin Zhu.