Analysis and Algebra

Going into BSM, I knew I wanted to take at least one analysis class and at least one algebra class. My analysis ended up being Complex Analysis, and my algebra was Galois Theory. Here are my thoughts on each:

Complex Analysis
The professor, Gergely, started this class by telling us that complex is the most beautiful field of mathematics. After a semester course, it's definitely the most beautiful field I've studied. Real analysis can be pretty tedious and messy, and there's very little of that in complex. It's much cleaner, and problems that look real but ugly can sometimes be done quite elegantly with complex tools, which is awesome. Two of the handful of theorems I learned this semester that blew me away were from this class (Liouville's Theorem and the Maximum Modulus Principle).

I would call four of my math classes this semester lecture courses, but this was the most lecture-y of them. There was rarely discussion, and Gergely didn't set aside any class time as problem sessions or office hours. He was always willing to go over anything we asked about though, even if it took up a significant portion of class time. The class started out rather slow because our first day and a half was mostly complex arithmetic, just to make sure we were comfortable with it from algebraic and geometric perspectives. Throughout the course, Gergely was extremely detail-oriented and precise in his proofs (and expected that sort of exactness from our work), and while later that was helpful, at the beginning it dragged out material that was familiar to all of us. The course sped up as it went, though, especially once we reached Cauchy's Theorems. After that, everything started falling into place. The second half of the class felt like theorem after theorem after theorem.

We had weekly problem sets with two to four problems. Not all problems had equal weight, and Gergely always marked how many points each problem was worth. Even within a given number of points, though, it felt like problem difficulty varied a lot. Especially in the last month of the course, lots of the problems were pretty tricky, so we worked together often. Gergely tended to respond to email quickly, though, which was helpful, and he was also very good at giving hints that pushed us in the right direction without giving everything away.

I was pleased with my work in Complex. I usually started working on the homework early enough and put in an appropriate amount of work, though sometimes I should have spent more time writing up my work. I was happy with how often I worked with other people, and because several of us were in both Complex and Topology, we had a study group for both that was productive and a lot of fun.

In general, I really liked Complex (the course and the material), and the class confirmed for me that I like analysis in general. Much more than algebra, it turns out.

Galois Theory
Galois was my hardest course this semester. I took it because I wanted to take an algebra class, and I had enjoyed studying fields (the main mathematical objects in Galois Theory) in the past. It turns out that the course relied more on group theory than ring theory, though, and my group theory was really rusty, so I had to review that and get a better sense for groups as the course went. Despite the difficulty, I loved and completely understood Galois correspondence, which is the key result in the course. Near the end of the class, we covered some really fun things, like working with finite fields (which are so much easier to work with than infinite ones) and showing with Galois Theory that it's impossible to trisect an angle or double a cube.

Prof. Domokos lectured on Tuesdays, and then we had problem sessions on Thursdays. He would give us a sheet with ten to fifteen problems, and we'd go through the first few during the problem session, and then he would assign three of the problems as homework, due a week later. For some reason in this class more than others it felt like the material on the homework lagged behind the class, but I'm not sure how that could have been fixed. Like most of the professors he would go over homework problems that had caused several people issues, but for more individual homework corrections he not only wrote notes on the homework but talked to each of us, which was really helpful.

I either wish I hadn't taken Galois or had put more time into it. I certainly learned, and I like the material, but I would have gotten far more out of the course if I'd put in a couple more hours a week. The time I put into Galois was about the same as my other non-Spectral math classes, but the material covered in lecture and the homework problems were more distinct from each other in Galois than in any of my other classes. What we did in class tended to be going through the proofs of the relevant theorems, but the homework was much more computational. The problems used what we covered in class time, but understanding what went on in lecture and knowing how to do the homework were completely different, and I should have adjusted for that.

Through this class as well as parts of Topology and Spectral Theory, I figured out that while parts of algebra are definitely cool, I'm really not an algebraist. I far prefer discrete math and analysis.