- A Concise Course in Arakelov Geometry. + PDF
This is the note for my Arakelov geometry course, our main goal is to introduce the arithmetic Riemann-Roch theorem for arithmetic varieties. Due to class hours, the last two sections were not completed. This note was written in English.
- Representation Theory I: The Case of Finite Groups. + PDF
This note aims to introduce the representation theory of finite groups, including unitary representation, character theory, Fourier transformation, Burnside theorem, induced representations, etc. It can be read on the basis of linear algebra and abstract algebra, but it may be a little boring to read. (Update: 2020-08, Language: Chinese)
- Algebraic Number Theory I: Galois Theory. + PDF
This is a note to introduce Galois theory, including polynomial theory, the fundamental theorem and its applications, infinite Galois extensions and transcendental extensions. You can read it with the foundation of algebra and topology. (Update: 2020-08, Language: Chinese)
- Algebraic Number Theory II: Elliptic Curves and Modular Forms. + PDF
This note is the special introduction of "Riemann Surfaces", it does not involve proof, but mainly introduces the concept of elliptic curves, modular forms, Mordell theorem, B-SD conjecture and other basic contents. (Update: 2022-05, Language: Chinese)
- Algebraic Number Theory III: Tate's Thesis. + PDF
This is the lecture note of the course "Arithmetic Geometry" at CNU, 2021. The main contents include locally compact groups and their representation theory, abstract harmonic analysis, Fourier transformation and Gelfand transformation, Pontryagin duality and Poisson summation formula, profinite groups, adeles and ideles, Tate's thesis, etc. (Update: 2024-03, Language: Chinese)
- Algebraic Number Theory IV: Class Field Theory. + PDF
Class field theory uses the arithmetic properties of a global field to determine all its abelian extensions. There are many ways to prove it, we only introduce Tate's idea, and will not give serious proofs. This is the lecture note of the course "Class Field Theory and Reciprocal Law" at CNU, 2021. (Update: 2022-09, Language: Chinese)
- Algebraic Number Theory VI: Arithmetic Dynamics. + PDF
Dynamical system is a subject that studies the iteration by means of statistics, it also connects to the number theory. This note consists of four topics: ergodic theory, Oppenheim conjecture, entropies and geometric topology of moduli spaces. This is the lecture note of the course "Dynamical Systems" at CNU, 2021. (Update: 2022-10, Language: Chinese)