Doctor of Philosophy in Mathematics

Course Description

Compulsory Courses

Literature Survey and Thesis Planning（3 credits）

The objective of this course is to provide students with a comprehensive understanding of the current research landscape in the field of mathematics. Through the means of literature review, it seeks to elucidate the unresolved issues within this domain and explore potential avenues for their resolution. The course aims to equip students with the requisite knowledge and skills to make choices regarding their research direction and to conduct a thorough selection of their doctoral research topic.

Topics in Frontier Mathematics（3 credits）

This course is delivered by instructors who specialize in various research domains within mathematics. They will present the content and methodologies pertaining to their respective areas of research. The primary aim is to provide students with not only a comprehensive understanding of their specific field of study, but also to acquaint them with advanced topics from other research domains. This approach serves to broaden their knowledge within the realm of mathematics and its related disciplines.

Advanced Specialized Reading Course（3 credits）

This course involves teachers guiding students to read and study books or papers in the research field, and explaining the research content and methods to the students. Through their studies, students will acquire the foundational knowledge, cutting-edge theories, and research methods necessary to conduct mathematics-related research.

Elective Courses

Harmonic Analysis in Euclidean Spaces（3 credits）

This course primarily focuses on introducing the fundamental concepts, theories, and methodologies of modern harmonic analysis, while also highlighting the latest research advancements and relevant cutting-edge issues. The specific course content encompasses classical theories of Fourier transforms, Fourier transform theories of generalized functions, general theories of harmonic functions in R^n spaces, as well as the boundedness of singular integral operators. Furthermore, we will explore the applications of Fourier transform theory and harmonic analysis in signal analysis, image processing, and present the latest research outcomes in these fields.

Hypercomplex Analysis（3 credits）

This course aims to explore the extension of complex analysis theory into higher dimensions, including quaternion analysis, Clifford analysis, and multivariable complex analysis. The course will cover the fundamental properties and related results of analytic functions theory in various settings, such as Hardy spaces and Bergman spaces in different configurations. Additionally, we will delve into the applications of hypercomplex analysis theory in signal analysis and image processing, as well as present the latest research findings in these domains.

Matrix Theory（3 credits）

This course provides a systematic introduction to the principal concepts and methods of matrix theory. The content encompasses topics such as linear spaces, inner product spaces, linear transformations, matrix decomposition, norms of vectors and matrices, as well as matrix functions and function matrices.

Theory of Dynamical System（3 credits）

Fractal theory is an emerging interdisciplinary field that encompasses scientific theories and methods for studying irregular and complex phenomena in the natural world. This course provides a comprehensive introduction to fractal geometry, covering fundamental concepts, computation of fractal dimensions, generation of fractal patterns, fractal growth models and simulations, fractal interpolation and simulation, stochastic fractals, and the fundamental principles of chaos theory closely related to fractals. Building upon this foundation, the course explores the application of fractal concepts, such as fractal patterns, fractal dimensions, fractal simulation techniques, and fractal image encoding and compression techniques, in various domains including natural sciences, engineering, technology, socioeconomics, and cultural arts. The course highlights the practical achievements and applications of fractal theory by examining real-world examples of fractal behavior in the natural world.

Analytic Number Theory（3 credits）

This course primarily focuses on the research methodologies and cutting-edge theories of analytic number theory. It provides a detailed exploration of various aspects within analytic number theory, including circle methods, sieves methods, Diophantine equations, and automorphic forms, etc. Zeros of Riemann zeta, the Goldbach conjecture, and the twin prime conjecture will also be introduced. Additionally, the course will delve into the applications of analytic number theory in cryptography.

Graph Theory（3 credits）

Graph theory studies graphs composed of vertices and edges that represent relationships between concrete entities. Graphs provide an abstract representation of the structural relationships among numerous entities. With the rapid development of computer science, graph theory, closely associated with it, has demonstrated vast application prospects. Apart from its significant role in classical problems such as data structures, communication networks, and circuit design, graph theory has gained increasing importance in emerging fields like big data and artificial intelligence. Novel technologies like complex network analysis, graph neural networks, and knowledge graphs rely on graph theory as their theoretical foundation. Many classical methods and theories in graph theory have shown immense value in these application domains.

Therefore, learning graph theory plays a vital role in fostering innovative talents in computer science, information technology, and other related fields to thrive in the new wave of technology. This course will primarily focus on the following topics and their relevant applications:

1.Basic concepts and theorems of graph theory

2.Connectivity problems

3.Coloring problems

4.Matching problems

5.Eulerian and Hamiltonian problems

6.Fundamental algorithms in graph theory

Combinatorial Optimization（3 credits）

The exploration of discrete research objects is an important branch in mathematics. In fields such as computer science, control theory, information theory, artificial intelligence, large-scale circuit design, economics, molecular physics, biology, and engineering, there are numerous problems that involve finding optimal solutions under complex constraints in a discrete background. Combinatorial optimization, as a discipline, uses discrete optimization methods and algorithms to provide optimal or approximate optimal solutions to such problems.

This project aims to systematically cultivate students’ ability to solve complex problems from the perspective and technical approach of combinatorial optimization, focusing specifically on the prominent challenges present in artificial intelligence, machine learning, big data, and complex networks. The core content of this project includes:

1. Optimal trees and optimal paths

2. Maximum flow problems

3. Minimum-cost flow problems

4. Optimal matching problems

5. Traveling salesperson problem

6. Optimization problems in matroids.

By developing a deep understanding of these topics, students will enhance their ability to apply combinatorial optimization principles and techniques to address complex problem-solving scenarios in various domains.

Advanced Statistics（3 credits）

This course provides a comprehensive introduction to the fundamental principles of statistics and related mathematical tools. It covers essential topics such as mathematical statistics, linear models, non-parametric statistics, multivariate statistical analysis, time series analysis, advanced inference methods, machine learning, and the application of statistics in various domains.

Advanced Numerical Analysis（3 credits）

This course is centered on the comprehensive study of a range of numerical methods, including direct methods for solving linear systems of equations, iterative algorithms for solving linear systems of equations, root-finding algorithms for nonlinear equations, interpolation and approximation techniques, numerical integration and differentiation methods, and numerical approaches for solving ordinary differential equations. The theoretical instruction in numerical analysis primarily relies on classroom-based teaching, emphasizing the fundamental principles and ideologies of numerical analysis, with significant attention given to error analysis, convergence properties, and stability theories. The integration of numerical experiments constitutes a pivotal component within the course, as it provides a vital platform for bridging the gap between theory and application. Through practical implementations, students not only deepen their comprehension of the course contents, but also enhance their analytical and problem-solving abilities, and foster their innovative thinking acumen.