# Mathematics ( Ph.D. )

### 1. Training/Research Orientation

- Number Theory and Algebra
- Scientific Computing
- Differential Equations and Dynamic Systems
- Geometry and Topology
- Graph Theory and Combinatory
- Modern Analysis
- Probability and Statistics

### 2. Program Duration and Credit

3 years in general, the maximum duration should not exceed 7 years (including the extension time).

20 credits in total, at least 12 compulsory credits.

### 3. Core Courses and Introduction

**Optimization Theory and Methods**

Content: This course concentrates on recognizing and solving convex optimization problems that arise in engineering. Convex sets, functions, and optimization problems. Basics of convex analysis. Least-squares, linear and quadratic programs, semidefinite programming, minimax, extremal volume, and other problems. Optimality conditions, duality theory, theorems of alternative, and applications. Interior point methods. Applications to signal processing, control, digital and analog circuit design, computational geometry, statistics, and mechanical engineering.

Prerequisites: Good knowledge of linear algebra. Exposure to numerical computing, optimization, and application fields helpful but not required; the engineering applications will be kept basic and simple.

Time: Once a week in 160 minute lectures.

**Algebraic Number Theory**

Content: This course is a first course in algebraic number theory. Topics to be covered include number fields, class numbers, Dirichlet's unit theorem, quadratic fields, cyclotomic fields, local fields, valuations, decomposition, ramification and inertia groups, basic analytic methods, and basic class field theory.

Prerequisites:Module Theory: tensor product, exact sequences, projective modules, injective modules, tensor algebras, modules over PID; Field Theory: algebraic extensions,seperable extensions, splitting fields; Galois Theory: Fundamental theorem, finite fields, solvable and regular extensions, radical extensions, insolvability of the quintic.

Time: Once a week in 160 minute lectures.

**Algebraic Differential Equations**

Content: Algebraic differential equations are a widely accepted tool for the modeling and simulation of constrained dynamical systems in numerous applications, such as mechanical multibody systems, electrical circuit simulation, chemical engineering, control theory, fluid dynamics and many others. This course provides a systematic and detailed analysis of initial and boundary value problems for differential-algebraic equations. The analysis is developed from the theory of linear constant coefficient systems via linear variable coefficient systems to general nonlinear systems. Further contents on control problems, generalized inverses of algebraic-differential operators, generalized solutions, and differential equations on manifolds complement the theoretical treatment of initial value problems. Two major classes of numerical methods for differential-algebraic equations (Runge-Kutta and BDF methods) will be discussed and analyzed with respect to convergence and order. Index reduction methods that allow the numerical treatment of general algebraic differential equations will also be discussed. A survey of current software packages for algebraic differential equations completes the course.

Prerequisites: A prerequisite of this course is a standard course on the numerical solution of ordinary differential equations.

Time: Once a week in 160 minute lectures.

**Nonsmooth Analysis and Nonsmooth Optimization**

Content: This course provides a comprehensive development of nonsmooth analsis, and its rich applications in nonsmooth optimization, including duality, minimax/saddle point theory, Lagrange multipliers, and Lagrangian relaxation/nondifferentiable optimization. Aside from a thorough account of nonsmooth analysis and nonsmooth optimization, the course aims to restructure the theory of the subject, by introducing several novel unifying lines of analysis.

Prerequisites: Convex Analysis and Optimization.

Time: Once a week in 160 minute lectures.

**Domain Decomposition Methods**

Content: Domain decomposition methods are widely used for numerical simulations on parallel machines from tens to hundreds of thousands of cores. Contrary to direct methods, domain decomposition methods are naturally parallel. This course provides a detailed overview of the most popular domain decomposition methods for partial differential equations (PDEs), focusing on parallel linear solvers. The course covers all popular algorithms, both at the PDE level and the discrete level in terms of matrices, along with systematic scripts for sequential implementation in a free open-source finite element package as well as some parallel scripts. Also included is a new coarse space construction (two-level method) that adapts to highly heterogeneous problems.

Prerequisites: Numerical Methods for Differential Equations; Parallel Computing; Iterative Methods.

Time: Once a week in 160 minute lectures.

**Modern Methods for Differential Equations**

Content: The modern landscape of technology and industry demands an equally modern approach to differential equations in the classroom. The course emphasizes a systems approach to the subject and integrates the use of modern computing technology in the context of contemporary applications from engineering and science. The focus on fundamental skills, careful application of technology, and practice in modeling complex systems prepares students for the realities of the new millennium, providing the building blocks to be successful problem-solvers in today’s workplace.

Prerequisites: Modern Analysis.

Time: Once a week in 160 minute lectures.

**Stochastic Partial Differential Equations**

Content: Stochastic Partial Differential Equation, which emerged in 1960s, is one of professional courses on Markov process and stochastic differential equations. This course mainly introduces the solution properties of stochastic parabolic, hyperbolic equations in bounded domain and the whole space, the existence and the asymptotic behavior of solutions of stochastic evolution equations in Hilbert space, stochastic nonlinear partial differential equations driven by Brownian motion and levy process, the abstract theorems on existence and the long time behavior of solutions of these SPDEs. Meanwhile, this course will be exclusively devoted to stochastic partial differential equations of Ito type. We will employ the familiar tools such as the methods of eigenfunction expansions, the Green’s functions and Fourier transforms, together with the conventional techniques in stochastic analysis. The abstract theorems on existence, uniqueness and regularity of solutions will be proved and applied later to study the asymptotic behavior of solutions. The purpose of this course is to ensure students to master basic methods and theories in studying the solution properties of stochastic partial differential equations.

The content of this course includes:

- preliminaries such as Brownian Motions, martingales, Ito foumula and B-D-G inequality
- the existence, uniqueness and regularity properties of solutions of stochastic parabolic equations in bounded domain and the whole space respectively
- the analyasis of stochastic hyperbolic equations by adopting a similar approach
- the existence and uniqueness theorems for linear and nonlinear stochastic evolution equations for two kinds of solutions: the mild solution and the strong solution
- the boundedness, stability and the existence of invariant measures for the solutions
- several examples arising from physical models to show that the theorems given above has a wider range of applications
- a brief exposition on the connection between the stochastic PDEs and diffusion equations in infinite dimensions

Prerequisites: Partial Differential Equations; Stochastic Process.

Time: Once a week in 120 minute lectures.

**Graph Theory**

Content: This course serves as an introduction to major topics of modern enumerative and algebraic combinatorics with emphasis on partition identities, young tableaux bijections, spanning trees in graphs, and random generation of combinatorial objects. There is some discussion of various applications and connections to other fields.

Prerequisites: Combinatorial Mathematics.

Time: Once a week in 120 minute lectures.

**Discrete Probability, Stochastic Processes, and Statistical Inference**

Content: The purpose of this course is to integrate the theory and applications of discrete probability, discrete stochastic processes, and discrete statistical inference in the study of computer science.

Prerequisites: Set Theory, Mathematical Induction, Number Theory, Functions, Equivalence Relations, Partial-Order Relations, Combinatorics, and Graph Theory.

Time: Once a week in 120 minute lectures.

### 4. Supervisors

Haitao Cao, Ercai Chen, Jinru Chen, Yonggao Chen, Yujun Dong, Qikui Du, Hongjun Gao, Deren Han, Wei He, Chungang Ji, Yutian Lei, Zhilin Li, Zhibin Liang, Yue Liu, Lianhua Ning, Yongzhong Song, Wenyu Sun, Lixin Tian, Li Wang, Yushun Wang, Jiaqun Wei, Yan Xu, Baogang Xu, Zuodong Yang, Huicheng Yin, Ping Yu, Jihui Zhang, Zhiyue Zhang, Huaiping Zhu, Jiandong Zhu, Quanxin Zhu, Zhilin Li, Huaiping Zhu, Junping Wang.