This fall I took a class, Math 790STC Characteristic Classes and K-Theory, and the final project involved writing an expository paper on a topic of our choosing that we then related back to the course. My choice of topic was spin structures, and I enjoyed writing the paper a lot. Once we wrote the first version of our papers, we went through the process of peer reviewing another student’s paper. It was a great exercise in both reading and commenting on someone else’s work, but also in taking someone’s comments into account and deciding how to improve our writing in a way that fully addresses any critiques.
Even though I enjoyed the process a lot, I think that my paper did not turned out as well as it should have. Part of this is probably due to my poor time management when going through the process of writing the paper – I spent too much time just reading, and not enough time taking notes to help with writing. I also think that I lost a lot of steam with the final rewrite towards the end, and as a result some parts are not as polished as they should be. Because of this, I would like to spend some time over the winter break working on improving the paper and turning it into something that is more in line with what I wanted the final product to look like. I think that using this blog to type up the process of fixing the paper will be useful in that it will serve as a way for me to get my ideas down somewhere.
Since I just finished the original version (which I will call version 0 or v0), I will probably take a few days off before coming back to work on it. It will be good to get some distance from the original work before going back to edit it so that I can look at it with a more critical lens. Here I will just jot down some overall feelings I had about the paper from finishing it so that I don’t forget them. I will also include a PDF of the original version at the bottom of this post.
- The overall structure is fine, although I need to decide if I want the discussion on the “zoo” of bundles should be in the appendix or the first section.
- The actual discussion of the “zoo” of bundles needs some work. This is the section I was writing when I “lost steam” and it feels very rough.
- The inclusion of the section on the classification of Clifford algebras needs to be reevaluated.
- Notation needs to be cleaned up throughout.
- The discussion on spinor representations needs to be completely redone.
- I’d like to add more discussion on what a spin structure is, how different ones can be the same as bundles but not as spin structures, etc.
- I’d like to talk more about when spin structures can be defined in the $(t, s)$ signature case and not just the $(0, n)$ case.
- Section 6 needs to be completely redone and fixed. I think I didn’t even finish the section fully.
- I think some explanation of the classical groups used might be useful.
Here is the PDF of the original version:
