General Progress
Since my last Blog post, I have gotten the feeling of leaving the windy hills of notation-land and taking my first steps into the arid dunes of group theory proper. Mainly, the lecture course has finally started to get into some interesting things on the more abstract, algebraic side of the subject. For example, it started off with an explanation of subgroups, which then led into cosets and normal subgroups, which paved the way for the normalizer of a subgroup. They’ve also taken forays into direct products and quotient groups. Overall, I feel an impending sense of progress, much more so than the introductory period of the project.
I shan’t say I’m all satisfied however. Some of those topics above seem like very algebraic concepts inherently. For example, direct products are defined as the Cartesian product of the elements of the two groups, which forms a set. However, they still attempt to explain these concepts using visual tools such as Cayley graphs, perhaps obstinately. Don’t get me wrong, I love visualizing math, but when the tools in question (Cayley graphs) are as flimsy and non-fundamental as they are, it leaves you wanting a more detail-oriented explanation for how these structures are operating. On the other hand, I found their visual explanation of left and right cosets, and what it means for the two to be equal, to be very enlightening. The standard path through group theory with no visual aids also has much more friction than the lectures do (trust me). Overall, I’d just wish they didn’t shy away from diving into the algebra every now again for some of the more abstract things like direct products.
Apart from the lecture series, I have been reading Adventures in Group Theory by David Joyner. This book has been an interesting experience alongside the lectures, as it takes a more traditional approach to the subject. However, it is still apparently much more accessible than regular group theory courses. Anyhow, it takes me more time to progress through the book, as it isn’t afraid whatsoever to present lengthy strings of symbols and sprinkle lemmas throughout each chapter, which can be troublesome at times when you’re trying to pick apart every unknown sentence for bits of understanding. They have recently started to venture into the territory of permutations (I have learned here how to represent any finite group as a square matrix :)) and have been introducing me to many properties of permutations. I hope that the eventual connections to group theory are robust and fruitful.
Mentoring Logistics
As expressed in a previous blog, my mentor has had a busy week or two at his day job, so the boat has been rocky, though thankfully not capsized. We are aiming to meet every two or three weeks for about one or two hours at a time. We plan to meet at his house, as it provides a comfortable abode for thinking and discussion, contains his textbooks which have proved helpful, and has readily built “whiteboards” installed in the walls. It has so far been unreasonable for me to expect too much time commitment from him outside of meetings, but I’ve been preparing to pack each one to the brim with content since our first. In terms of a minimum, finishing at least one section of the lecture series for every meeting would be acceptable. However, this is bare minimum performance, and I don’t think I could help myself from doing more. I could speed through all the lectures very quickly, but I’ve been trying to give ample time between each one to give myself time to digest ideas.
Outside of the meetings, we have been keeping contact online through a specially made Discord server for the project (yes). We have, obviously, been using it to set meetup times and details, and also send miscellaneous information about the subject that the other might find interesting. That’s a little undefined, so for example, the last one he sent talked about character tables’ role in Representation Theory, and how the “number of irreducible representations is equal to the number of conjugacy classes”. It is safe to say I do not know what that means.
Mentoring Sessions
I think the biggest strength we had in our meeting was the dialogue we had. Though it might’ve been far-between at times, when it was present, it was high quality. For questions he knew the answer to, his explanations were direct and intuitive. When I was the only one who knew the answer to something, he was receptive to my ideas, and perhaps more notably, for questions neither of us new the answer to, we explored possible arguments together to arrive at a hypothesis. It at times felt non-trivially genuine. Unfortunately, one large learning challenge did rear its ugly head. Given the introductory nature of the course at that time, my question bank wasn’t very full, so we fumbled about on occasion. To hold ourselves accountable for our learning, my mentor has been continually refreshing himself on group theory to provide more answers to my questions, and I have been at work absorbing as much information as possible in order to generate many questions.
Plug
If you haven’t already, check out my Log series where I go into more detail about what I’ve been learning, as well as intimate snapshots of my result. If you want to read me genuinely trying to explain this subject to best of my mediocre capability, and my laughable attempts at mathematical invention, I’d suggest you give it a read. I hope the series gets the interested reader acclimated to the subject and perhaps learn something new and compelling. The first Log is linked below.
https://mygleneagle.sd43.bc.ca/pavelv2021/2022/02/28/in-depth-2022-group-theory-captains-log-1/
Logs are posted in anti-phase to the required Blogs.