Posted: March 07, 2020
Edited: March 07, 2020
I have gone to some length writing notes about Topology and spent less time writing on Physics models. This is not something I am extremely happy about. This post is to help me figure out what I have sit through and provide a general direction of where to go next. How easy it is to get disoriented when you study Math for Physics.
For Topology, I have had a working knowledge of Homology and Cohomology of topological manifolds. I have a working knowledge of point set topology although I did not write about it. I even studies a bit on Lie group and Lie Algebra.
For Abstract algebra, I am writing a note which takes a very heuristic path to introduce braid group from simple diagrams. This is what I am very proud of.
For Physics, I have done some calculations on Chern numbers but I got very technical in the end. For Topological Insulators, I calculated a model’s Chern numbers using projection which I can explain but not deduct. For Topological Quantum Computations, I have gone trough some details on a toy model.
As far as I can tell, the theories I have studies is going to meet very soon. In fact I should be able to apply many of the topics to actual models in this website with a few patches here and there, but I never did because 1, that would be a huge work themselves and 2, I do not have a nice model or I did not know where to find them.
Although writing the notes on this website is very rewarding, it’s kind of frustrating seeing my knowledge lying not being used and gradually fading away because of that. When I hear someone talk about the topics I studied, I got excited but soon discouraged because I really did not have much to offer. My level of math is far from rigorous, and my understanding of its application in physics is extremely limited.
And I intend to change that.
I think the most urgent job for me is to write a note on topological insulators while reviewing some of the techniques I used calculating Berry phase and Chern number, especially with randomness. That way I should be able to review winding numbers, berry phase and Chern number or even connections on fiber bundles.
After that I should go pursue the Topological K-theory. I do not have a very clear direction yet but I am hopeful that I will after the completion of notes on topological insulators. If I start straight with the theory it’s too easy to get lost, as is shown in the many paused or unfinished nots on this website. The hope is that I would have some concrete examples that I could always fall back to, and that is always the best thing to have studying math.
Finally my masterpiece would be the note currently entitled “The amazing world of diagrams”. I hope to provide an easy introduction to anyone who wants to be familiar with terminologies in abstract algebra or want to know about braid groups. I have always feared that this note will come as too shallow for math students or any colleagues, so it’s bound to go through many revisions.
During the seasons of Coronavirus, I hope I will have enough time for at least the first item on my list.