I guess the first overdue announcement is that I passed my Real Analysis qualifying exam at the start of Fall quarter. This came as a very nice surprise as I am still confident that I had no business passing that test. The test is formatted into 4 subsections. I’ve described them loosely below, but there’s a heavy amount of overlap and hopping around of certain quesitons. Overall, the test covers undergraduate real analysis and the first 9-10 chapters of Folland.

  • Undergrad type problems
  • Problems pertaining to integration and measure theory
  • Problems pertaining to differentiation and functional analysis
  • Problems pertaining to Fourier analysis and $L^p$ theory

Each section has 4 problems and we are tasked to choose 2 to solve out of each section. I attempted 6 problems and had a few mistakes on some of them, so it really is a miracle that I passed. Either way, I’m very happy to have checked off one milestone towards my degree, and I’m still very fond of real analysis. I will probably make a separate post just about studying for qualifying exams later on. I still have one more (likely in Partial Differential Equations), so I’ll be looking forward to that.

Reflections on my courses

After taking Real Analysis and Algebra, whatever was left of my confidence led me to register for the Differential Equations sequence and the Applied Math sequence. Take that to mean whatever you’d like it to mean. This first quarter, my courses were Ordinary Differential equations, which focused on qualitative analysis of various types of linear and nonlinear systems of first order equations, as well as Probability Theory.

I very much enjoyed the content of ODE. The types of problems definitely felt closer to undergraduate analysis and it was nice to see a more qualitative approach to analyzing linear systems as opposed to the strangeness that is the undergraduate ODE experience. I do agree with Rota for the most part about this, so this class was a lot of fun. Even existence theorems were more interesting than I would have initially thought. I hope to continue looking into ODE theory at least a bit more as it seems we were just about to reach some nice results by Lyapunov. One thing I’m not satisfied with were the exercises given to us throughout the course. We used Brauer and Nohel’s The Qualitative Theory of Ordinary Differential Equations as well as Strogatz’s well known classic, Nonlinear Dynamics and Chaos. The exercises from Brauer were fine, but some from Strogatz definitely left something to be desired in terms of rigour. Some problems about bifurcations just leave too much to numerical approximation and heuristic observation for me to feel good about. I’m also not sure how to approach such problems in a satisfying way without passing to perhaps some topological argument (which I would have no knowledge on).

As for probability theory, it was quite a trainwreck. I’m not sure if the issue was with my effort in the class, my lack of ever having taken an undergraduate probability theory course or the fact that the instructor was very new, but it just felt like throughout the course, all I saw were very distinct trees with a complete absence of a forest to situate myself in. I was actually very eager to see how measure theory lies as a foundation for probability theory, but it felt strangely hamfisted in the way it was incorporated. It almost feels like probability theory is just a superficial mask atop measure theory. I know it’s definitely not that once you reach independence, but I was just not getting any kind of motivation towards any of the large results we covered.

Regardless of what happened in my courses, it feels a lot better to be in person. There are so many little moments of chatter between classmates that just don’t happen remotely. I really appreciate moments like that because it builds that feeling of community that I was missing a lot last year.

Now, I don’t exactly recall what I wrote last episode about note taking and overall effort in my courses, but I’m sure my past self would still be disappointed in my present performance.

I think it’s unreasonable to TeX notes for all my courses. That’s it. It takes way too much time, especially given how particular I am about formatting. I was barely able to keep up with TeXing for ODEs and I was at a loss of how I was supposed to incorporate things like bifurcation diagrams and such into my notes.

My classmate R.W. does a much more reasonable method of taking notes on a physical notebook and then rewriting neatly onto his tablet later on and even with this, he felt he was not able to keep up with his two classes. Thus, I’m now debating what the appropriate way to proceed is. I’ll definitely be making another post about this, but to summarize my idea for next quarter, I want to spend less time taking and reading notes, and more time solving problems. I’ll need to get the essential definitions and results down to memory, but the real learning must come from solving problems.

Reflections on TAing in person for the first time

It was very exciting TAing in person for the first time and it turned out to be one of the things I looked most forward to each week this quarter. I seriously had a blast.

By some grace of God, I landed 4 sections of Math7A under the exact same primary instructor, so I did have an easier time than my peers when it came to preparing and grading.

To briefly summarize my approach, my discussions were fixed as a quiz based discussion. That is, the quiz pertains to material they’ve learned recently, and my job is to spend 35 minutes doing something to help them prepare, and then administering the quiz in the last 15 minutes.

I decided to treat discussion as problem solving sessions to prepare my students. I’d start by taking any questions on homework and lecture, going over those, and then writing some example problems that reflect the quiz and having the students work on them for as long as I could spare, and then finally going over them fully and then giving out the quiz.

It was immediately apparent that I did not have time to do all of that, so I soon scrapped fielding questions. This may seem to be a strange thing to scrap, but students can email me whenever they want for questions, and there are always office hours, which I always offer in a surplus.

It also became apparent that most students struggled to solve even the most standard problems. Thus, I would leave my fully written out solutions to the example problems on the board while they took the quiz. Even then, some students still had quite a bit of trouble with the quizzes. It was very strange, but I do empathize a bit.

Things to keep in mind for future TAing:

  1. I need to prepare something like a syllabus for the first day. Too many students had no idea when my office hours were even to the final week of the quarter. I should also try not to change my office hour schedule mid-quarter.
  2. I should be more upfront about resources I’m providing. I spent a good amount of time writing up solutions to quizzes and posting helpful links and resources, but I don’t think most students knew about them. I need to keep reminding them each week to check things out. Especially when it literally points out what the final exam will look like.
  3. Example problems should be more carefully crafted and fully worked out to avoid any unintentional complications.
  4. Continue being careful about “math phrases”. Although much better than my undergrad years, I still say words like “clear” or “easy” too often.
  5. I should try to be more attentive to students that are struggling. It’s easy to slip into the comfort of talking to the students closest to the front or those that are most eager to speak up, but this neglects those that likely need more attention.

Goals I’m working towards

The exciting news of the quarter is that I found a faculty I really enjoy talking to. Dr. Yat Tin Chow has been incredibly kind to me and has offered to take me up on a reading course. He even offered to be my PhD advisor, so that’s even more incredible.

The particular work I’ll be diving into is in Optimal Control Theory with the eventual goal of learning about Mean Field Game Theory.

I hope to have lots of chats with him about our work as well as various other things. I want to learn how to be a mathematician from him.

My other goal is to brush up on and continue to learn more real analysis. I’m still very fond of the field and the problems within. It’s a lot of fun to get into the nitty gritty measure theoretic arguments, so I want to solve more problems and look into other textbooks from the ones we used last year.

Last is that there’s an interesting project being done right now by a mathematical philosopher, Silvia de Toffoli, about the fallibility of rigour and proofs for mathematical justification. Other than following her journey, I’m hoping to read some of her already published works and absorb some wisdom and ideas from them.

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