Mathematics

This course provides a panorama on the relationship and interplays between discrete mathematics, often called combinatorics, and other areas such as representation theory and algebraic geometry. A particular focus is on learning algebraic, geometric, and probabilistic methods in combinatorics. Specific topics are selected based on current research. Topics discussed include probabilistic methods and extremal combinatorics, algebraic methods and formal power series, and geometric combinatorics and discrete geometry.

Linear Algebra is one of the most widely used topics in the mathematical sciences. At lower levels students are taught standard techniques for basic linear algebra tasks including the solution of linear systems, finding eigenvalues/eigenvectors, and orthogonalization of bases. However, these techniques are usually computationally too intensive to be used for the large matrices encountered in practical applications. This course introduces students to these practical issues, and presents, analyzes, and applies algorithms for these tasks which are reliable and computationally efficient. The course includes significant lab work using an advanced programming language. The course studies three main topics: the solution of linear systems of equations, the solution of least squares problems and finding the eigenvectors and/or eigenvalues of a matrix.

This course addresses exploratory data analysis and graphs such as histograms, stem plots, measures of center and spread of a distribution, normal distribution, scatter plots, least squares regression (correlation), producing data (design of experiments, sampling design), probability (probability rules, random variables, probability distributions), and statistical inference (confidence intervals, tests of significance, nonparametric methods, categorical or count data). 

The six-week summer lab research program at National Taiwan University places students in various science, engineering and social science research labs and/or projects under the supervision of faculty. Students spend approximately 30 hours per week in lab activities.

This course covers ordinary differential equations and partial differential equations. Topics include: first-order differential equations; second-order linear differential equations; linear systems of differential equations; nonlinear systems and stability; method of separation of variables; Sturm-Liouville Problems; Inhomogeneous Problems.

Pre-requisites: Calculus I, Calculus II, and Linear Algebra.