Mathematical analysis is a fundamental course that aims to provide us advanced theorems and prepare us writing rigorous mathematical theorems. The main contents of this course in the first semester will cover most of the materials in the textbook. The contents in the second semester will cover some selected advanced analysis topics such as Lebesgue theory, Fourier analysis, Oscillatory integrals and their connections and applications to combinatorics and number theory. In particular, we will also introduce additive combinatorics and its connections to combinatorial number theory.