Friday, January 23, 2015

Combinatorics and Topology

Extremal Combinatorics and Topology were my math classes at BSM that had the most fun homework and were generally the lowest stress. Here's a bit more about each:

Extremal Combinatorics
I was at the Joint Mathematics Meetings this past week, and I had the opportunity to talk to a few students who are headed to Budapest this spring. When they asked for advice on what courses to take, I told them they needed to take at least one combinatorics or graph theory course. Over the past century, combinatorics in general and graph theory in particular have been really strong in Hungary, so Budapest is arguably the best place in the world to take combinatorics.

My combinatorics class was Extremal Combinatorics, which was one of the advanced graph theory courses as well as one of the courses I was most excited to take at BSM. Throughout the semester, Extremal homework was consistently my favorite. The problems, even when they were very difficult, just tended to be fun. The homework lined up well with what was covered in class (more so in this course than in any of my others), so the material as a whole was interesting and enjoyable. We covered a lot of topics, and despite the class feeling very nonlinear at times, everything tied together in surprising and beautiful ways.

The professor was Ervin Gyori, and the most difficult part of the class was keeping up with Ervin's nonlinearity, not just in topics but in proofs. The course was advanced enough that he rarely did a full, formal proof, opting for sketches (usually detailed, but still sketches) instead. For example, we did dozens of induction proofs, and I can only think of a couple of times when Ervin actually included a base case. We all knew it worked, and so it was assumed. (Of course, we had to be more formal in our own proofs.) Sometimes Ervin would go through the basic idea of a proof with a picture, and then go back and add in a little more formality. This was the class in which I talked the most to other people about proofs from class because a lot of details ended up as lines or dots on a diagram and not as written words, so we would check with each other to make sure we had caught everything. Despite that, I never felt seriously lost. There were times when I looked at my friends during break and said, "I didn't get the second half of that proof," but it was always easy to get caught up by asking other students or Ervin for help.

Homework was assigned on Thursday and was due the next Thursday, and on the Tuesday in between, Ervin set aside the second half of class as an office hour. We usually had questions about the homework by that point, and he would give hints. Sometimes he would go over past homework if he thought there had been a common problem. If there was time left, he would do fun lectures on topics related to the class but that we didn't need to know.

Extremal was low-stress while still having interesting and challenging problems, and it was definitely my most fun class. It was cool to go to combinatorics and graph theory talks at the Joint Meetings this year and see from how many more terms and techniques were familiar because of how much I learned in Extremal.

Topology
Topology was one of the classes for which I turned in paperwork to the Course Substitution and Transfer Board at Olin last spring, but when BSM actually started, I didn't think I would end up taking Topology. During shopping period, though, it didn't conflict with any of my other classes, so I went, and it was so much fun that I ended up staying in the course.

The professor, Agnes, spent a lot of time on examples in this course, which was generally pretty helpful. There are a lot of things in topology that don't behave in a way that's particularly intuitive, so we spent a lot of time working with concrete sets and topologies. If anything, there were sometimes too many examples. (When you get into a double digit number of examples, sometimes you lose track of what you were originally doing.)

Agnes didn't formally set aside time as an office hour or problem session, but she sometimes posted non-homework problems that we worked through in class. She always took questions about homework if we had them, and doing so many examples (in a class discussion form, generally) meant that it wasn't straight lecture.

Homework was due on Fridays and was always reasonable, particularly after we asked questions on Wednesday. There were eleven other students in Topology, and I had at least one other class with more than half of them, which meant that I often ended up working on Topology along with work from another class with other people.

Topology was really useful for Complex and came up in Spectral as well, it helped me remember a lot of my rusty real analysis, and it made a lot of the topological graph theory I've read or heard about over the past few years make even more sense. Topology is also pretty fun to think about in its own right, so I'm really glad I ended up taking the class!

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