Courses
I took four courses this semester.
MATH 389: Advanced Analysis. This was a topics course on C*-algebras, and it was the coolest class I've taken at Williams.
We spent about half of the semester building up the necessary background, covering metric and topological spaces before diving into some theory about Lebesgue measure and Hilbert spaces. Once we had that foundation, we did the main structure theorems for C*-algebras, including the Gelfand representation theorem and continuous functional calculus. After that, we did some representation theory, using the GNS construction to link pure states to irreducible representations and ultimately proving the general structure theorem. We wrapped up the semester by briefly touching on C*-dynamics, KMS states, and a tiny bit of Tomita-Takesaki theory.
What made this class especially enjoyable was the combination of challenging, engaging material and a remarkably low-stress grading environment. The content was advanced enough that studying never felt like a chore, while the grading was generous enough that I had almost no anxiety about doing badly (I finished with a 98%, and the cutoff for an A was 85%). We had only six problem sets, an in-person midterm, an oral final, and a written final, which is a very low amount of graded work for a math class here. The lectures were excellent, and because they were early in the morning, attendance was often sparse. In fact, it was the first math course I've taken at Williams in which I never dreaded the homework. When I spent an absurd amount of time preparing for the final exam for no reason, I did it because I enjoyed the material. Looking back, I wish I had spent more time working through Bratteli and Robinson rather than relying primarily on the professor's lecture notes. I also would have liked the course to dive a bit further into Tomita-Takesaki theory toward the end of the semester. But overall I'm very happy with how this course went.
MATH 413: Algebraic Geometry. This was the math course I spent the least time on, averaging about 10 hours per month outside of class, with an additional ~10 hours during months with exams. Despite a seemingly heavy workload (including weekly problem sets, weekly quizzes, weekly forum posts, two written midterms, an oral midterm, an oral final, and a final project) the course was very manageable. The lectures were pretty good and super easy to follow.
We covered Chapters 1-4 and 8 of Ideals, Varieties, and Algorithms. Much of the material overlapped with stuff I was already confortable with, so most of the stuff I needed to learn were related to algorithms for computing things in algebraic geometry. Unfortunately I do not find things like Buchberger's algorithm and radical membership test interesting. Fortunately I didn't have to learn the algorithms super well: even with frequent mistakes on the algorithmic questions, I finished with a 97%. The highlight of the course was the final project, for which I wrote a short expository paper titled Projective Toric Surfaces via Convex Lattice Polygons.
MATH 419: Tiling Theory. This was probably the easiest math course I took in terms of technical difficulty, though it ended up being one of the classes I struggled with the most. We spent much of the semester working through The Tiling Book, a recreational math text written by the professor.
While I can see the appeal, a lot of the material felt somewhat ad hoc to me, as if interesting facts were being introduced simply because they happened to exist. Maybe that's an unfair criticism, but I never found myself particularly invested in whether the gyrobifastigium tiles the plane.
Ironically, despite finding most of the concepts fairly accessible, I performed exceptionally badly on both exams. There wasn't much substantive material to be tested since much of the course felt like a collection of random facts presented without proofs, so across the exams there were perhaps only five core ideas in total. I still managed to do everything wrong. Fortunately, the grading curve was generous enough that my final grade survived my mistakes.
- THEA 103: Acting Fundamentals. This class was an experience. We met once a week for three hours, but there was never much structure and I fell asleep in every class I attended. The professor would just improvise random activities and monologue for the entire class. One day I arrived late and everyone was sitting silently. When someone finally asked what we were doing, the professor said that we were meditating. I probably should not say much more since I got an A and have no complaints about that. Altogether, I spent maybe five hours outside of class on coursework for the entire semester.
I also TA'd for Tiling Theory while taking the class. Most of my responsibilities involved grading homework, which was more time consuming than I expected since many of the problems required checking detailed drawings. Spending so much time grading definitely made me less enthusiastic about doing the assignments myself. The work was fine overall, but it was not especially exciting.
Other
I don't really know what I did this semester outside of class. I think I spent about 20 hours a week looking at pictures of koalas on instagram and a couple hours a week watching sports.
The main thing I remember is hating how much time I spent filling out applications (for the summer and for next school year). I got pretty good results and I am happy with how everything turned out, but I strongly dislike writing application materials. If given the choice, I would happily trade all of that time back for more math.
Most of the math I did outside of classes was related to my research project with Professor Yang. I'm pretty excited about our results.
In the last few weeks of class, I mostly studied advanced analysis while taking long walks to the Clark. I would bring a list of definitions/theorems/problems and try to recall stuff as I walked, which seemed to work well since I did well on my exams. Around the same time, I also sat in on a few of the final meetings of the Galois theory course, even though I wasn’t enrolled. They covered some interesting topics, including homological algebra.
Looking forward: Over the summer, I'll be doing research at the University of Maryland. Next school year, I will be at the University of Oxford.