Courses

This is a collection of lecture notes, assignments, links, and some other related materials of the courses I have taken.

1. Algebra

Thanks to Albert Peng for his permission to edit his tex file of Math5031 and also J.S. Milne’s tex file.

Math5031-32 Lecture Note, with homework incorporated as solved exercises

snake lemma. I add some colors to the code in https://commons.wikimedia.org/wiki/File:Snake_lemma_complete.svg

2. Algebraic Geometry

We use Algebraic Geometry by Robin Hartshorne as textbook but not follow it closely for some parts of the course. There will be homework on most weeks, and some of them will be chosen from Hartshorne. Some resources:

The red book of Varieties and Schemes, David Mumford. Springer Lecture Notes, 1999.

Lecture note by Ravi Vakil.

Lecture note by Andreas Gathmann.

3. Lie Algbera and Representation Theory

We used the following textbooks:

Introduction to Lie Algebras, by K. Erdmann and M. J. Wildon
Introduction to Lie Algebras and Representation Theory, by J. Humphreys
Representation Theory: A First Course, by W. Fulton and J. Harris
Symmetry, Representations, and Invariants, by R. Goodman and N. R. Wallach

Math547 Lecture Note

4. Measure Theory and Functional Analysis

Math5051-52 Lecture Note

5. Complex Analysis

I transcribed Prof. Martinkainen’s written manuscripts into latex file, with several tikz illustrations.

I also add some useful remarks made by Prof. Steven Krantz in undergraduate complex analysis course Math416.

Math5021-22 Lecture Note

Above is the image of a cat by Cayley map $\frac{z-i}{z+i}$ visualized through online conformal mapping viewer. Cute cat becomes a bear.

6. Differential Equations

ODE: ODE lecture note, Lebl’s online tex file, Teschl’s Ordinary Differential Equations and Dynamical Systems.

Dynamical system and Chaos: Strogatz’s Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry, and Engineering.

PDE: Evans’s Partial Differential Equations.

pianist Emaneul Ax's signature on Lee's Intro to SM

7. Differential Topology

We used John M. Lee’s Introduction to Smooth Manifold (e2) and Tu’s Introduction to Manifold (e2). Some other texts are also very interesting, including a classical little book Milnor’s Topology from the Differentiable Viewpoint and Smooth Manifolds and Observables by a group named themselves Jet Nestruev.