A Sketch of What is to Come
An Unbridled Optimism knows no bounds...
Those who know me would likely describe me as a man of many interests. I expect this Substack to reflect that - therefore I expect it will be 50% mathematics, 50% philosophy, and 50% other things. Much of my writing will be mathematically oriented, but there will not be a dearth of writing more focused on the humanities end of the spectrum, either.
As I write this, I find myself at the university of Michigan working on a summer mathematics research project. The work I am doing is utterly fascinating. The issue is that it is so locked behind mountains of technical jargon and subtle ideas that I am loath to attempt to explain what I am doing to anyone not in this particular subfield. Such is the life of the mathematician. So, I have decided to start writing articles that give the necessary background for understanding what I am doing. Roughly, the outline I have is this:
Basics of Group Theory
Some notes on Ring and Field Theory
An important aside on Linear Algebra
Introduction to Representation Theory
Maschke's Theorem, Schur's Lemma, and Characters
Induction and Restriction of Representations
Representations of GL2
A Sketch of the Research
If you know all these words, great, and if you know none of these words, that's great too! I'm hoping to write all of these at a level accessible to a strong high school student. This is why I'm starting with groups - I don't want to assume that my audience already knows group theory.
Because of the nature of research, I will likely not be able to share many details about what our research is on in particular, but I hope through this project to at least give a sense of the types of things we are trying to prove. I have no idea how many articles will be in each section - it could be 1, it could be 7, who is to say?
Beyond this mathematical project, I have no solidified plans for what I will be writing. I have a few essays in the works on scattered topics; they do not follow any kind of plan. Indeed, one could say that they will be random essays by deterministic means. I would like to be releasing at least one post every week, but we will see how long my motivation lasts, and how much time they actually require of me.