Spectral Theory

I've already written a fair bit about Inquiry Based Spectral Theory already, but it still deserves a post of its own. It's definitely the BSM class I'll miss the most.

In general, spectral theory is about using linear algebra to study mathematical structures. In Spectral, we focused on spectral graph theory. My usual description of the course was "throwing eigenstuff at graphs." I love graph theory, and I'm now NINJAing Linearity I for the third time, so I was comfortable using linear algebra (or I thought I was; I'm much more so now!). But we didn't just use linear algebra and graph theory. Over the course of the class, we did a lot of group theory and analysis as well as some set theory, topology, and functional analysis; so in terms of seeing how important a general  mathematics background is, Spectral was really valuable.

I think Spectral will be the kind of class that sticks out when I think back on my four years of college. I've taken plenty of project-based and do-learn classes, and I've NINJAed for discovery-based math classes, but none of those was inquiry based in quite the way Spectral was. This wasn't UOCD with occasional lectures and too many readings or Linearity I with problem sets full of exercises and profs and NINJAs around during problem solving sessions. We learned by doing, but the doing happened mostly between class periods and was thus unsupervised, and it took the form of proving mathematical statements.

We met for three hours every Wednesday evening. Each weekend, the professor, Miklos, sent out a sheet with a few definitions and then many statements to prove. We would work through the proofs, and in class on Wednesday someone would present each proof to the class. If we had mistakes in our proofs, that helped us find them, and we also all improved in our presentation of mathematics. Then we had a formal write-up of all of our proofs due Sunday, by which point we would already have the next week's sheet. I said earlier that there were no readings and no lectures, which makes it sound like we were redoing everything from scratch, but the sheets were organized in such a way as to push us in the right direction (and Miklos was always willing to give hints).

I put a lot of work into Spectral, and I wish I had put in even more time. My average number of Spectral work hours a week was twice that of any of my other classes, and I still usually hadn't finished the sheets completely before class on Wednesday. Because of the format of Spectral, it was more like research than any of my other classes, and I think the ways I learned to work through Spectral problems will improve how I do math research, especially research with other people. Presenting work in class was somewhere between teaching and giving a conference presentation, and while I've done a little lecturing, I'd never had to present a proof to a class before. It's different from working through problems, so the experience of presenting was also really good.

There were eight of us in the class, and we were pretty different in mathematical background. Those of us with less background worked together a lot, often spending the afternoon leading up to class going over each other's proofs in the club room downstairs at BSM and trying to make some last bit of progress. Even when we were really stuck (both before and in class), it was fun. We spent a lot of time together, so we got to know each other well. In class, we joked with each other, and Miklos in particular told terrible jokes. Three hours was a long time to go without a break, so halfway through we would take a short break and then someone would talk for five to ten minutes on a non-math subject. (The day Miklos suggested this, I volunteered to go first, so with very little prep time I talked about roller coasters.) The way we talked about non-math things we loved was surprisingly similar to how we presented math.

I'll really miss Spectral, both the subject matter and the people. The book I got at the Joint Meetings this year was on expanders and Cayley graphs, which we started to cover in Spectral. As a class we had a couple of final meetups outside of class time. We all went over to Miklos' flat after class one Wednesday and ate foie gras and were there for several hours, just eating and talking, and we all left the BSM farewell party early to get frozen yogurt together. Spectral was a huge amount of work, but I'm so glad I chose to take it.