"This is a one semester course at the third year undergraduate level. The main topic will be plane algebraic curves. These will be studied using basic techniques in elementary algebra and theory of one complex variable."

"Functions of Real Variables" is a basic course for all undergruaduate students in School of Mathematical Sciences, which concentrates the Lebesgue thoery of measure and integral, and provides the knowledge and training of modern analysis for students.

"This is an undergraduate course introducing Error Analysis, Polynomial Interpolaiton, Numerical Solutions of Nonlinear Equations, Numerical Integration, Numerical Differentiation, Numerical Solution of Ordinary Differential Equations as well as Fast Fourier Transform and Monte Carlo Method that are excluded in the standard Numerical Analysis course. The course is more focused on the mathematical analysis and principle of these algorithms and their implementations as well. Students will be expected to complete several computing projects in addition to other homework and the final examination"

Ordinary Differential Equations is a basic course for mathematical students. In this course, the students will learn the basic knowledge of ordinary differential equations, including how to solve some simple equations, the existence and uniqueness for Cauchy problem, boundary value problems as well as the theory of linear differential equations.

The course focuses on the basic theory of complex variable functions. The main contents include: complex numbers and the complex plane, complex functions and analytic functions, integrals, harmonic functions, the series of analytic functions, residues and its applications, analytic prolongation, the gamma function, conformal mapping, Laplace transformation.

Mathematical analysis is one of the most important courses for the students who wish to study the mathematics and related subjects. The course mainly includes the theory of Riemann integrals and the theory of series. The course is a basis for Mathematical analysis and for many courses such as differential equations; differential geometry, functions of one complex variable; real analysis, probability; basic physics, etc. The course provides the training for the mathematical thinking and skills.

Mathematical Statitics is a basic course with wide application, it mainly focuses on the analysis of randon sample and other data set, including how to effectively collect data, parameter estimation , hypothesis testing, linear model and statistical design. The purpose is to let the students to understand elementary ststistica concepts and ideas, to study the most commonly used statistical methods and to solve some practical problems, and to establish the way of statistical thinking.

Understand principle and theory of Data Structures and Algorithms, able to design and implement fundamental data structures and algorithms. Covers programming, data structures, algorithms. Topics include the organization and implementation of fundamental data structures such as list, binary tree, tree and forest, and graph; sorting and searching; data abstraction and problem solving.

The content of the course consists of polynomials, linear spaces and linear transformations. This course will train the students with mathematical thinkings and the preliminary ability for solving practical problems.

algebraic number fields, algebraic integers, ideal class group, elliptic curve, class fields theory, modular form, Weil conjecture, Iwasawa theory

Differentiable manifolds is a prerequisite course for many other courses in differential geometry, topology, Lie groups and others. Students will combine their knowledge in various fields and build a solid background for further study of modern mathematics.

Mathematical logic is an important mathematical foundation and is the only undergraduate course on logic of computer science and technology. The course introduces about:formal axiomatic system of propositional calculus (propositional language and formal infer-ence)､semantics and meta-theory (soundness and completeness of propositional calculus);formal axiomatic system of first order logic (first order language and formal inference)､semantics and me-ta-theory of predicate calculus(including soundness and complete-ness of first order predicate calculus); formal arithmetic and recur-sive function､G·del’s incompleteness theorems and decidable problem.

This course takes some examples from physics, ecological sciences, environmental science, medical science, economics and information science as the background, and introduces how to study the real problem by the mathematical modeling. It emphasizes the mathematical essential of the algorithms. At the same time, it encourages students to find the problem from the real life, and establish their own mathematical model to solve the real problem.

This is an undergraduate course introducing most of the basic and classical numerical algorithms as well as the newest research results for the system of linear equations and the matrix eigenvalue problem. The course is more focused on the ideas and implementation of these basic algorithms, besides the congergence analysis and error estimates. Students will be expected to complete several computing projects in addition to other homework and the final examination

"Combinatorics is a core course for all majors in the School of Mathematical Sciences, while it is also useful for students in (theoretical) computer science, logic, linguistics, philosophy, EE, chemistry, biology, physics, etc. Topics shall include (varying by instructors): Generating functions, enumeration methods, Polya`s theorem, combinatorial designs,Ramsey theory, fundamental graph theory, extremal graphs, special enumerating sequences, the probabilistic method, etc."

Elementary number theory is a basic course of studying properties of integers,the main contents include the divisible theory of the integer, the congruence theory of the integer, the fractional theory and some special indefinite equation.

Functional analysis plays increasing role in applied and pure mathematics. This course will familiarize the student with basic concepts, principles and methods of functional analysis and its application. This course is suitable for a one-semester course meeting three hours per week.

Geometry and its exercise class of School of Mathematical Sciences, PKU (As part of all undergraduate students of our department and Yuanpei experimental class of undergraduates) open the first door of the geometry curriculum. It is one of the most important basic courses in our department. Important task of the course is to charge with the cultivation of students` geometric thinking and to enhance the quality of students` geometry. This lesson mainly introduces the theory of analytic geometry of space and the basic idea of geometry properly, such as geometric in-variants, relations between groups and geometry. Curves in the space, the geometric properties of surfaces and in-variants are discussed in algebraic methods such that graphics and equations linked. Topics include vector algebra, plane and a straight line, the common surface, coordinate transformation, simplification of the quadratic equation and its properties, orthogonal transformation and affine transformation, projectile plane and projectile transformations. Therefore, the lesson is not only the extension and expansion of plane analytic geometry knowledge, but also lays a solid foundation for student diversity in undergraduate calculus, physics and other courses.

This course will introduce the basic knowledge of Sobolev spaces used in Partial Differential Equations. Based on Sobolev spaces, the well-posedness problems such as existence, uniqueness and regularity for weak solutions of linear partial differential equations of second order will be investigated. These linear partial differential equations include linear elliptic partial differential equations, linear parabolic partial differential equations and linear hyperbolic partial differential equations. The aim is to help students understand the elementary theory and modern methods of linearpartial differential equations.

Before midterm we will focus on basic concepts of point set topology and most commonly used topological properties such as compactness and connectivity. After midterm we will introduce some elementary algebraic topology, mainly focus on fundamental group and covering space. Students will not only be trained the ability of abstract reasoning, but also learn to do it together with geometric intuition.