Friday's class began with a definition of simplicial complexes. I knew a little bit about them from the presentation one of my classmates had given at the end of last semester. However, in his presentation, he'd jumped right into the realization instead of describing an abstract simplicial complex., so for the firsst 10 minutes, the little expeirence I had with them just confused me further.
The abstract definition is a lot like the first definition of a group. Great, I have a set satisfying these properties, but why is that important? But then we moved on to drawing a couple of simplicial complexes, which is pretty much all I needed to be convinced that they're worth studying.
Dr. Farley wanted to describe simplicial homology, but before he did that, we needed to backtrack so we could get an intuive understanding of homology. Perhaps, for some people, the understanding might intuitvely make sense. Without any of the surrounding knowledge of topological spaces, I really wouldn't know. But there wasn't the time for us to go into detail about homology, so we had to rely on a rough description that's supposed to make intuitve sense. I kept writing things down in the hope that later, I would gain that intuitve understanding and everything would suddenly make sense. (I'm not sure when the last time that's happened has been, but I seem to be expecting it to happen this summer.)
From homology groups we moved on to Discrete Morse theory. (Nothing in common with Morse code, unfortunately.) And, as I continued writing down definitions and properties, I realized that I finally understood how the people who had never taken an Abstract Algebra class must have felt last week.
Not understanding what's going on in topology is one thing. Yes, it's frustrating, but either I understand it exclusive or I don't. But in algebra, you can understand every single word of a definition and still have no idea how to use it. Most of math requires working examples and exercises, and I feel that's especially true for algebra. And that's also something we've been missing both in research seminar and the short course. It's really hard to understand how normal subgroups work when the only example of a non-Abelian group you have is S3 and S4.
Towards the very end of class, we started working through one example of the definitions and theorems that we'd put up on the board previously. And, amazingly, things started maing sense. The definitions were still pretty hazy, but they got slightly clearer. And the example itself made more sense to me than any other example for that class had. (Including the examples of Cayley Digraphs for the Symmetirc Group, which I'd known and drawn before.) I could understand what was going on. Not how to generalize it to other kinds of Thompson groups, or how to use that to find the homology group (our two homework problems for the weekend) but I did understand what was going on in class, so that was a significant step in a positive direction.
And then it was time for analysis. I was rather saddened to learn that most of my classmates didn't like it that much. Whether it was because of a lack of formal proof-based classes before now or a general disinterest in the subject area, analysis was not the part of their day that most of my clasmates looked forward to. In fact, most of them would have been happier if they didn't have analysis on Fridays.
Me: Oh...
Hannah: No one wants to be in analysis at 4:00 on a Friday afternoon. Except Sabrina.
Me: I have a friend who would also want to be in analysis.
In fact, since there's someone who does want to be in aanlysis at 4:00 on a Friday, and, given someone who does, they know someone else who does as well. Therefore, by a Justin-style inductive argument, you can prove that everyone wants to be in analysis. The people who say they don't are lying to the world and to themselves.
I mean, how could you not want to be in analysis? We learned more about measure, including the tools we needed to build an unmeasurable set. Then we moved away from simply the real numbers and the Lebesgue outer measure and onto more general spaces and measures, and what properties these fields must have. We even got examples!
I love analysis. I'm not sure what I'm going to do in another week when the short courses and research seminar go away and it's just us stuggling over a problem we may or may not understand. The structure and rigor of analysis is one of the main things keeping me sane right now, and I would be completely lost if it was just lectures on algebraic topology (and whatever else Dr. Farley wants to talk about) every day.
We'll cross that bridge when we come to it next week. But first, a weekend break, antoher week of classes and seminar, and another weekend. I wonder if it will fly by as quickly as the last week seemed to.
