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# Courses

No. | Subjects | Credits | Spring Semester | Fall Semester | |
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1. | Colloquium | 2 | 1 | 1 | |

2. | Real Analysis | 6 | 3 | 3 | |

3. | Ordinary Differential Equations | 6 | 3 | 3 | |

4. | Partial Differential Equations | 6 | 3 | 3 | |

5. | Topics in Discrete Mathematics | 2 | 1 | 1 | |

6. | Numerical Methods for PDEs | 6 | 3 | 3 | |

7. | Introduction to Financial Mathematics | 6 | 3 | 3 | |

8. | Mathematical Foundations of Cryptography | 3 | - | 3 | |

9. | Representations of Finite Groups | 3 | 3 | - | |

10. | Applied Stochastic Control | 3 | 3 | - | |

11. | Special Topics in Dynamical Systems | 3 | 3 | - | |

12. | Linear Programming | 3 | 3 | - | |

13. | Advanced Algebra | 3 | - | 3 | |

14. | Algebraic Combinatorics | 3 | 3 | - | |

15. | Algorithms | 3 | - | 3 | |

16. | Graph Theory | 3 | - | 3 | |

17. | Algebraic Graph Theory | 3 | 3 | - | |

18. | Design Theory | 3 | - | 3 | |

19. | Introduction to Combinatorics | 3 | 3 | - | |

20. | Scientific Computing | 6 | 3 | 3 | |

21. | Applied Mathematics Methods | 6 | 3 | 3 | |

22. | Machine Learning | 6 | 3 | 3 |

### Quick Links

### Graduate Program (*Click the title for more information*)

Colloquium |
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This course consists of weekly lectures scheduled by the coordinator.
The topics of lectures focus mainly on the topical programs in this department including combinatorics, differential equations, dynamic systems, scientific computations, probability and theoretical mathematics. Those interested in the selected topics are welcome to seat in the lecture, though the course is set for graduate students. No prerequisite is asked for this course....more |

Real Analysis |
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This is the course designed to acquaint the graduate students in the applied mathematics department with basic ideas and tools in modern analysis. This comprises the subjects of real analysis and functional analysis, the analysis developed in the 20th century and after. We will treat real analysis mainly in the first semester and functional analysis in the second. Hopefully, we can lay down a solid foundation for further usage in some other theoretical or applied area. After taking the course, the students are expected to have the general idea on the modern ways to attack the analysis problems....more |

Ordinary Differential Equations |
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ODE is a standard course in mathematical or applied mathematical graduate program. In general, nonlinear differential equations can not be solved into solutions of exact forms. However, it is still feasible to understand at least certain part of dynamics, through applying various mathematical ideas and concepts. Indeed, mathematical analysis leads to qualitative descriptions on the solutions; combining numerical computation with geometric delineation of phase portraits enhances the investigation of differential equations. The goal of this course is to learn the fundamental theories of ODE and classical applications of these theories. ...more |

Partial Differential Equations |
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This is the GRADUATE level partial differential equation. We will focus on the relation between mathematics and physics and show the students how to understand PDEs intuitively....more |

Topics in Discrete Mathematics |
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This seminar is held every semester and has to be taken at least twice by every student enrolled in the combinatorics graduate program. The main goal is to expose students from the combinatorics graduate program to different topics in combinatorics such as graph theory, analysis and design of algorithms, algebraic combinatorics, analytic combinatorics, applications of combinatorics in biology etc. The course is discussion-based with students preparing papers or book chapters and presenting them to the other participants. Occassionally, experts working in combinatorics are invited to give a talk about their current research work....more |

Numerical Methods for PDEs |
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This course addresses students who are interested in numerical methods for partial differential equations. After introducing basic numerical approximation of polynomial interpolation and Fourier approximation, we particularly focus on fundamentals of finite difference methods and important concepts such as stability, convergence, and error analysis. Other methods such as finite volume, finite element and spectral methods will also be introduced briefly. Partial differential equations will be solved in this course include Poisson equation, heat equation, wave equation, convection-diffusion problems, Maxwell equation and Navier-Stokes equations etc....more |

Introduction to Financial Mathematics |
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Let students understand and be familiar with mathmatical tools from finance....more |

Mathematical Foundations of Cryptography |
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In this course, students will learn the mathematical tools used in cryptography which are mainly coming from algebra and algorithmic number theory. Equipped with these tools, classical public key cryptosystems such as the Diffie-Hellman system, RSA, Elliptic Curve Method, etc. will be investigated. Morever, deterministic and probabilistic methods for generating primes and pseudoprimes will be discussed. The course is suitable for upper-level undergraduates and graduate students in mathematics or other fields with an interest in number theory and its applications....more |

Representations of Finite Groups |
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Representations and characters of finite groups are very important tools in the group theory. For instance, many proofs in the classification of finite simple groups involve delicate computations in characters. Representation and character theory also has numerous applications in many disciplines, such as number theory, combinatorics, geometry, crystallography in chemistry, and so on. In this course, we will discuss properties of representations and characters of finite groups. Some applications will also be presented....more |

Applied Stochastic Control |
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This is an one-year introductory course on applied stochastic control. In the first semester, we will start from the basic of stochastic calculus and its applications. In the second semester, applications to optimal stopping and stochastic control will be treated fully for the processes with continuous paths. Some topics in mathematical finance will also briefly be mentioned....more |

Special Topics in Dynamical Systems |
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This course introduces Patterns Generation and related problems....more |

Linear Programming |
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This course introduces a unified view that treats the simplex, ellipsoid, and interior point methods in an integrated manner. To expose graduate students interested in learning state-of-the-art techniques in the areas of linear programming and its natural extension.
We will organize this course into ten subjects. We first introduce the linear programming problems with modeling examples and provide a short review of the history of linear programming. The basic terminologies are defined to build the fundamental theory of linear programming and tor form a geometric interpretation of the underlying optimization process. The next topic will cover the classical simplex methods and corresponding theories including the revised simplex, duality theorem, dual simplex method, the primal-dual method. After discussing the sensitivity analysis, we will look into the concept of computational complexity and show that the simplex method, in the worst-case analysis, exhibits exponential complexity. Hence, the ellipsoid method is introduced as the first polynomial-time algorithm for linear programming. From this point onward, we focus on the nonsimplex approaches. The next subject is centered around the recent advances of Karmarkar’s algorithm and its polynomial-time solvability. We then study the affine scaling variants, containing the primal, dual, primal-dual algorithms, of Karmarkar’s methods. The concepts of central trajectory and path-following are also included. The eight topic reveals the insights of interior-point methods from both the algebraic and geometric viewpoints. It provides a platform for the comparison of different interior-point algorithms and the creation of new algorithms. We extend the results of interior-point-based linear programming techniques to quadratic and convex optimization problems with linear constrain. The important implementation issues for computer programming are addressed in the last subject....more |

Advanced Algebra |
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Abstract Algebra, sometimes also called modern algebra or algebra in short, is the branch of mathematics concerning the study of algebraic structures such as groups, rings, modules and fields. This course is an advanced study of the above structures....more |

Algebraic Combinatorics |
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This will be a self-contained course focusing on some subjects and methods used in the field of combinatorics....more |

Algorithms |
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This course is a fundamental course in Computer Science. Its purpose is to learn the techniques for designing an algorithm and the techniques for analyzing an algorithm....more |

Graph Theory |
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Graph is the model of many problems, e.g. computer programming, experimental designs, or even pure mathematical problems, and its theory is a delightful playground for the exploration of proof techniques in discrete mathematics. This course prepares students for algorithmic, constructive, probabilistic and algebraic abilities in dealing problems....more |

Algebraic Graph Theory |
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Algebraic Graph Theory is the branch of mathematics that studies graphs by using algebraic properties of associated matrices. More in particular, spectral graph theory studies the relation between graph properties and the spectrum of the adjacency matrix or Laplace matrix. And the theory of association schemes and coherent configurations studies the algebra generated by associated matrices. This course, also called Linear Algebraic Graph Theory, emphasizes on the first part, and another course, called Algebraic Combinatorics, includes the second part....more |

Design Theory |
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This is a course in the study of combinatorial designs originated from studying the combinatorial structures of experimental designs. Many different designs will be introduced in this one semester course: Latin squares, Steiner systems, t-designs, Hadamard matrices, finite geometries, association schemes and pooling designs. Also, related applications will be mentioned, for example group testing and special networks....more |

Introduction to Combinatorics |
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This is a required course of the Combinatorics Graduate Program. The syllabus of the course will contain half of the topics from the Ph.-D. qualification exam of combinatorics (the other half will be taught in the course "Graph Theory"). Moreover, if time permits, selected topics from Analytic Combinatorics, Enumerative Combinatorics, Combinatorial Design Theory, Design and Analysis of Algorithms, Algebraic Combinatorics, etc. will be presented as well....more |

Scientific Computing |
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Many problems in science and engineering can’t be solved analytically or empirically. Problems such as weather forecasting, financial and economic forecasting and big data processing etc., can only be studied using computer simulations. Moreover, with advance computer technologies, industry also relies on a lot of simulations for product design and manufacturing. Scientific computing as one of the core element inside every computer simulation plays an important role in the development of science and technology nowadays. The emphasis of this course is on understanding and using numerical methods that are foundations of scientific computation. Students who finished this course are expected to be able to solve the following types of problems: solutions of linear equations, nonlinear root finding, optimization, curve fitting, numerical integration and the solution of differential equations by using computer simulations....more |

Applied Mathematics Methods |
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Understand the basic ideas and tools used in the formulations and solutions of problems.
The object of this course is the construction, analysis, and interpretation of the mathematical models to help us understand the world we live in. We shall introduce key ideas in mathematical methods and modeling, with an emphasis on the connections between mathematics and applied natural sciences. The course covers both standard and modern topics, including scaling and dimensional analysis, regular and irregular perturbation; calculus of variations and continuum mechanics, etc. ...more |

Machine Learning |
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We introduce the concept of machine learning and several useful learning methods including linear models, nonlinear models, margin-based approaches, structured models, dimension reduction, unsupervised learning (Clustering), ensemble classifiers. Also some special topics and applications will be discussed....more |