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Undergraduate Program(Click the title for more information)
|This course offers the general concepts and principles in physics. The main objectives of this course are to motivate students to explore the beauty of physics in various disciplines, and to train students the physics problem-solving ability.
|This course provides a first introduction into the theory of differentiation and integration. The course mainly serves as a bridge between high school mathematics and university mathematics. Its main goal is to make students acquainted with rigorous mathematical thinking. This is done via learning basic concepts such as limits, continuity, differentiability, etc. on the one hand and fundamental theorems such as the intermediate value theorem, the extreme value theorem, the mean value theorem, etc. on the other hand.Moreover, the course is intended to train students problem solving skills as well as writing and oral skills. Finally, the course equips students with the basic tools needed in the more applied sciences and is the entrance door to more advanced courses on mathematics.
|Introduction to Computer Science
|The goal of this course is twofold. On the one hand, the course examines the basic ideas of the science of computing. Lectures cover a wide variety of topics such as computer organization, different number systems and its conversions, operating systems.
On the other hand, the course introduces the theory and practice of computer programming. The emphasis is on techniques of program development using the C++ language. It aims to help students feel confident of their ability to write programs to solve problems in mathematics or engineering.
|This course provides a first introduction into the origin, theory and application of Linear Algebra.
|The main goal of the course Vector Calculus is to explore the mathematical operations which one encounters in carrying out concrete calculations in multivariable Calculus, as well as the fundamental concepts of vector analysis.
Indeed, the scope of the course is to let students become familiar with the ideas of taking integrals of scalar - valued (or vector – valued) functions over curves or surfaces, as well as the applications of the theorems of Green, Stokes and Gauss.
In addition, the course will also introduce students to the basic geometric concepts of curves and surfaces in either Euclidean spaces of dimension 2 or 3.
The prerequisites of this course are Calculus and linear algebra.
|Differential Equation is an immediate-connected course of Calculus and Linear Algebra, as its content consists of various applications of these basic mathematics. Differential Equation is originated from Newtonian mechanics and celestial mechanics, and still plays an important role in modern physics including quantum mechanics. On the other hand, mathematical modeling has prevailed in neural science, infectious diseases, gene networks, circuit theory, financial engineering, and traffic flows, etc. Although nonlinear differential equations can not be solved into solutions of exact forms in general, it is still feasible to apply various mathematical ideas and concepts such as analysis and geometry in understanding at least a certain part of the dynamics. Moreover, combining numerical computation with geometric delineation of or phase portraits enhances the investigation of differential equations.
|Introduction to Partial Differential Equation
|The main driving force which completely changed the outlook of the scientific-technological aspect of human civilization was the scientific revolution initiated by Newton. The scientific revolution of Newton was the climax of human scientific thought which began immediately after a series of previous great artistic and scientific achievements by the giants since the time of Renaissance. In general, Newton’s scientific revolution, through his invention of Newtonian mechanics, is generally regarded as one of the triumphs of human’s rational thinking in recent human history.
Basically speaking, the main principle of Newtonian mechanics can be realized mathematically as a second-order ordinary differential equation. To a certain extent, the past developments of mathematical tools, which are capable of addressing mathematical problems arising from this second-order O.D.E. in Newtonian mechanics, constituted the whole historical development of Calculus.
To put it in simple words, we can say that Calculus was invented for the sole purpose of solving differential equations. Indeed, being witnesses to the tremendous success of Newtonian mechanics, mathematicians and physicists since the time of Newton began to apply the main ideas of Newtonian Mechanics to all the other branches of Natural Science. In particular, in the studies of the physical phenomena of a vibrating string with fixed end-points, the first partial differential equation, which is known as the wave equation (or the d’Alembert’s equation), was derived for the very first time in the early days of the historical development of P.D.E.’s theory. A partial differential equation is, by definition, an equation satisfied by some of the partial derivatives of the unknown multi-variable function.
Partial differential equations arise naturally in the study of geometry, and in almost all physical laws of Natural Sciences or practical engineering problems. We can even say that, without the developments of the theory of partial differential equations, there could be no development of Science at all in recent human history. Indeed, some of the most famous partial differential equations which appeared in the history of Science are:Laplace equation, Wave equation, Heat equation, Maxwell equation, Schrodinger equation, Navier-Stokes equation and Euler equation, Boltzmann equation, KdV equation.
Basically, one can say that the studies of these equations encompass almost all the mathematical problems arising from Physics and different areas of engineering.
In order to solve these partial differential equations, mathematicians, physicists, and engineers had developed various kinds of mathematical methods, among these the most famous one is the Fourier analysis. As a matter of fact, the subject of Fourier analysis constitutes almost all of the modern mathematical analysis. However, as the development of the theory of partial differential equations became more mature and sophisticated, researchers began to realize the importance of acquiring a better understanding about various kinds of nonlinear phenomena. This naturally motivated the developments of the theory of dynamical systems in recent and modern times, including the recent studies of the theory of Chaos.
Besides standard tools borrowed from mathematical analysis, numerical analysis is another important methodology which is currently used in the studies of solutions to partial differential equations. Through carrying out numerical simulations with the aid of high-speed computers, the use of “Numerical Analysis” in the studies of P.D.E.’s can be regarded as carrying out mathematical experiments. These mathematical experiments through numerical simulations not only help us to gain a deeper understanding about the equation being studied, but even may provide us with clues which can lead to rigorous proofs of mathematical assertions related to the equation being studied.
Finally, but not the least, we should never forget that the sole purpose of a differential equation is to describe natural phenomena. Even priori to our efforts devoted in solving a specific differential equation, the equation itself has already revealed some of the secrets of the Nature to us. In other words, Differential equations themselves naturally speak the language of the Nature. So, it is absolutely true that knowing the basics of Physics will be extremely helpful for the studies of partial differential equations.
|Computational mathematics is the study of algorithms use to solve mathematical problems. In this course, we introduce numerical methods for solving systems of linear equations and nonlinear equations. Approximation theory including polynomial interpolation, numerical differentiation and numerical integration will also be covered. Numerical errors arisen from various approximations in some of the above mentioned topics will be analyzed. Moreover, round-off errors from computations under floating point arithmetic will be discussed. Finally, students will learn methods for solving ordinary differential equations numerically and understand the stability/consistency of the numerical methods they use.
|This course is intedned as an elementary introduction to the theory of probability for students in mathematics. It attempts to present not only the mathematics of probability theory but also, through numerous examples, the many diverse possible applications of this subject. The prerequisite for this course is very fundamental, including the basic counting in high school and a one-year course in calculus.
|The origin of statistics can be traced back to the urge of governments to gather information about economic development and population. However, modern statistics is not only restricted to these two areas anymore, but also studies problems from psychology, medicin, anthropology, epigenetics, agriculture etc. Below, a more detailed description of the course is given.
|Discrete Mathematics, or called Combinatorics, is an important branch of Mathematics, and its influence continues to expand. Part of the reason for tremendous growth of Combinatorics is the development of computer, which needs combinatorial thinking to perform and analyze its correct and efficient functioning. Another reason for the recent growth of Combinatorics is its ideas and techniques applicable to many areas, for example, the physical sciences, the social sciences, the biological sciences, information theory and so on. This course prepares students the ability concerned with the existence, enumeration, analysis and optimization of discrete structure.
|Introduction to Analysis
|The purpose of this course is to lay down the rigorous foundation for the study of real number system, real sequences, real series and real-valued functions. The course provides a bridge from “freshman” calculus to graduate courses that use analytic ideas, e.g. , real and complex analysis, partial and ordinary differential equations , numerical analysis, fluid mechanics , and differential geometry.
|Abstract algebra is the subject area in mathematics that studies algebraic structures of mathematical objects, such as integers, complex numbers, matrices, roots of polynomials, symmetries of polyhedrons, and et al. Contemporary mathematices make extensive use of abstract algebra.
For example, algebraic number theory and algebraic geometry apply algebraic methods to problems in number theory and geometry, respectively. In recent years, abstract algebra also figures prominently in coding theory and cryptography, which are essential in telecommunication nowadays. In this one-year course, we will study three kinds of algebraic structures that appear most often in mathematics, namely, groups, rings, and fields, culminating in the proof of the theorem that a general quintic equation has no solutions in radicals.
|Complex Analysis is an important area with many application in Physics, Number Theory, Combinatorics, etc. The main goal of complex analysis is to study analytic functions. In this undergraduate course, student will learn the basics of the theory. Topics covered include analytic functions, line integrals, singularities, the residue theorem, conformal mappings, analytic continuation, etc.
Graduate Program(Click the title for more information)
|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.
|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.
|Ordinary Differential Equations
|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.
|Partial Differential Equations
|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.
|Topics in Discrete Mathematics
|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.
|Numerical Methods for PDEs
|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.
|Introduction to Financial Mathematics
|Let students understand and be familiar with mathmatical tools from finance.
|Mathematical Foundations of Cryptography
|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.
|Representations of Finite Groups
|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.
|Applied Stochastic Control
|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.
|Special Topics in Dynamical Systems
|This course introduces Patterns Generation and related problems.
| 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.
|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.
|This will be a self-contained course focusing on some subjects and methods used in the field of combinatorics.
|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.
|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.
|Algebraic Graph Theory
|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.
|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.
|Introduction to Combinatorics
|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.
|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.
|Applied Mathematics Methods
|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.
|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.