Posted: March 05, 2020
Edited: March 05, 2019
This note is an effort to tell the stories in the languages of mathematical concepts mostly used in condensed matter Physics.
I can remember when I found the terminologies used by the mathematicians frightening, as I was more than once deterred by the formidable appearances of Topology, of Abstract Algebra, of Lie Algebra. And I have found the easiest way for me to understand these concepts so that I could see what they have to offer for Physics is via examples and often visualizations. I had to watch hours of open courses and plow through the book just to find a few. That was not a very enjoyable experience, and it can sometimes be infuriating that some perfect examples (perfect that in my opinion would have helped me a lot) are left out. So this is my feat towards the mountain. I will try to cover the fields that I have found useful in my daily applications. These notes are almost bound to be limited, naïve and could be on the edge of misunderstanding by the pure mathematicians. I would love to hear any suggestions.
Much of the materials of this notes came from or inspired by a course taught by Dr. Emil Prodan in Yeshiva University.
This note is to be reorganized. I intended the note to start from sets and go over group and come to algebra. I will also try to add general topology and Lie group if possible. The order of the materials might be changed later.
I only assume the reader have basic knowledge of linear algebra. An introductory level of group theory is appreciated but definitely not required. This note do not require complex analysis.