This was my first week of classes at Budapest Semesters in Math!
The first three weeks are a shopping period, which means I could attend as many classes as I wanted in order to decide which four or five to take. Registration will happen during the third week. While math classes started this week, most of the humanities courses don't start until next week, so the past few days were all math for me. I attended seven classes this week: Spectral Theory, Extremal Combinatorics, Complex Analysis, Topology, Galois Theory, Functional Analysis, and Real Functions and Measures. Here are descriptions and my thoughts on each!
Inquiry Based Spectral Theory
This class isn't offered every semester, so I was really excited to see it on the schedule. The course is mostly about the intersection of linear algebra (a course I've NINJAed twice) and graph theory (my research area), so I'm very interested in the subject. I'm also excited because the course is inquiry based, or Moore method. This means that before each class period, all the students will do homework, and then in class we'll present our results and proofs to each other. After everything is presented and we comment on each other's work, we'll each formally write up the proofs and turn the write-ups in. Because I'm shy, the heavy focus on presenting and commenting on each other's work will be hard for me, but I know from experiences in Olin classes and a couple of independent study classes in high school that this format is really rewarding.
The first class period was a little weird because we didn't have any work done yet, but we started doing some linear algebra review and presenting it. Today was the second class period, and for today the professor, Miklos, had asked us to finish the linear algebra exercises and work on some graph theory proofs. I presented one proof and helped with another, which was exciting!
Extremal Combinatorics
So far, this is my coolest class in terms of material. Hungary is known for combinatorics, which is basically the math of counting (sometimes very difficult counting), and graph theory is a subfield of combinatorics. This course covers a lot of graph theory as well as some other combinatorics topics, but it also connects to a lot of other areas of math! For example, this week we were mostly focused on a couple of problems that, when we initially stated them, were posed as number theory problems. We translated them into graph theory problems, and then we solved them geometrically! Seeing all the connections is fantastic, and the fact that combinatorics is so prone to having such connections is one of the reasons I love it.
Also, one of the best quotes of the week, said by my Extremal professor, Ervin:
"If you are lazy like me, then you choose combinatorics because you do not have to study so much."
Complex Analysis
One of the first things the professor, Gergely, told us was, "You ask any mathematician if complex analysis is one of the most beautiful areas of math, and they will say yes." He also told us that by the end of the course we'd think of the real line as "rather boring."
So far, we've mostly done review of complex arithmetic and basic analysis and topology, but the first homework looks fun, and we should start doing more complex analysis-y things this coming week! I do expect this to be my easiest math class.
Topology
I had considered taking Topology, but it wasn't one of the courses about which I was particularly excited. I went on Wednesday mostly because I wasn't planning on taking anything else offered in the time slot and felt like I should try it out. I'm really glad I did because I love this course; I think it's been my favorite this week, and it's definitely the one in which I've learned the most. Topology is sometimes called rubbersheet geometry; it's like geometry but stretchier/more flexible. That description had made me think of topology as weird and sort of mind-boggling, but so far it seems pretty natural.
The reason I like this class so much probably has as much to do with the professor, Agnes, as it does with the material. She's the most organized of my professors, she uses a lot of examples, and she gives us mini-exercises to discuss in class, which has helped my understanding a lot.
Galois Theory
This is the one algebra class I'm taking. I really enjoyed the group and ring theory courses I took at Mathcamp in high school, but I haven't done much algebra since. The first day of class was an introduction to the course. The professor had us play around with some polynomials as motivation and then built up definitions to state one of the major theorems we'll prove in the course. On the second day, we started really working with polynomial rings, which are one of the objects we'll spend a lot of time studying. That class period started out feeling a little scattered, but it came together so beautifully in a result we proved at the end of class.
This is the one class in which the professor hasn't assigned any homework yet. I'm wavering between taking this class or Functional (or neither, maybe), and I probably won't make a decision until after doing the first assignment.
Functional Analysis
Several of my friends have taken functional analysis classes over the past couple of years and enjoyed them, so I decided to try it out. This week was a bit different than future weeks will be because normally, we'll do problem-solving in class on Thursdays and have lecture on Fridays, but this week we had lecture both days. Of the courses I'm still considering, this is the one that feels fastest for me. I'm not worried about keeping up, though, and the prof's lectures are really clear and organized. My decision between this class and Galois, if I decide to take five math classes, will probably have a lot to do with how much I enjoy working the problems for each class.
Along with Complex Analysis and Topology, this is the third class in which we've already covered the definition of an open set. Right now there's a lot of overlap in these three classes, but that's because they're all covering/reviewing basic analysis and point-set topology. They'll diverge into more different topics pretty soon. This class will go in the direction of infinite dimensional vector spaces. (Linear algebra tends to focus on finite dimensional vector spaces.) I would use a little bit of material from this class in Spectral Theory, too, which would be cool!
Real Functions and Measures
I went to the first day of class but not the second. I thought the material we covered on the first day was interesting. I'd never done anything with measure before. (Measure is a generalization of things like length and area, which are actually really hard to define well!) Just because of what math I have and haven't studied, though, this course would be a lot of work for me, and Spectral Theory is going to be about twice the work of a normal class because it's inquiry based. I decided that I'd rather take either Galois or Functional, so I didn't go to the second day of RFM.
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