Nearly half a year has passed since my last post, but at least I do remember that I have a website of my own. Many things happened, but they can be summarized as: I got a new job; I am still waiting for the results from graduate schools; I am learning mathematics systematically again.

Mathematical Physics

In the senior year of my undergraduate education, I tried to audit the course MAT265 Mathematical Quantum Mechanics by Professor Roger Casals. I didn’t finish the course because it was too dense for me at the time, while I had other course workloads to do. This time, I will follow the course schedule and more important the text A Brief Introduction to Physics for Mathematicians by I. Dolgachev and proceed to finish the course.

One main reason that I choose to study this is that the content of this subject motivates or forces me to learn other subjects as well. As I mentioned before, the text A Brief Introduction to Physics for Mathematicians is dense because it assumes graduate student level of mathematics knowledge. For instance, some concepts like Lie Algebras and Symplectic Forms are directly used in the book. For more information, please keep reading.

This week I read the first lecture of the book with three lecture notes of MAT265. The pace was on point because this course also had three lectures per week. I am still debating whether to also complete the exercises, but I may not have enough spare time to do so. Moreover, I am even not sure whether I can keep up this pace later on, but we will see.

The style of MAT265 is to talk about a general idea of the materials, while the textbook contains all the details. Therefore, I plan to make notes from MAT265 and write side notes while reading the book. During the process, I have to say that the apple pencil comes in a clutch, and it is too good to use to take notes. Anyways, the topic of this week is an introduction to Quantum Mechanics. Starting from Newton’s law, I took a look at Euler-Lagrangian equations, how it connects to Hamilton’s equations, and these two formalisms with any smooth manifold M.

Manifolds

Manifolds appear in different subjects that I learned before, but I didn’t actually learn this subject systematically. Although the concepts of the tangent bundle, fibers, and different forms are easy to follow, I felt like this subject is a must need for me.

The textbook I choose to use is An Introduction to Manifolds by Loring W. Tu. I just finished the first three sections of the first Chapter. The first two sections are the basics: smooth functions on a Euclidean space and tangent vectors as derivatives. The third section establishes the connections between a space with its tangent space through (multi-)linear functions.

I have been pretty impressed with my pacing so far, but it all comes with the fact that I am somewhat familiar with the ideas. However, the road is going to be tough once I enter the area of manifolds theory.

Riemann-Hilbert Problems

This is a very interesting direction that I am taking because it seems so out of the blue. However, I do want to finish the lectures from Professor Percy in RHP and take this as a review if I somehow get an interview chance from U-Mich. After finishing this, I would probably proceed to learn random matrix theory or PDE theory.

Algebra

After the reading course of the Riemann-Hilbert Problems, I learned during some period of studying mathematics, one needs to set up the goal without concerning the things left behind. What I meant by this is that instead of equipping every knowledge to begin learning a new subject, it is better to simply start learning the subject and go back to review what is needed. Indeed, I admitted that I did not learn well in the undergraduate Algebra courses, but I decided to begin my road of learning graduate Algebra.

The book I choose is Algebra by Serge Lang with the materials of A Companion to Lang’s Algebra (berkeley.edu). Let’s see what I can say about this next week.

I feel like reviewing my learning by writing a blog does help me organize, and more importantly, helps me summarize and review what I did. Again, wish me luck with the applications, and my future study in mathematics.