is more or less how I would describe my life this term. I almost never stop doing assignments. I really need to at least write a statement of purpose so that I’m ready to apply to grad schools, but I haven’t even had the chance to give that any thought lately. Before I give my impressions of courses so far, I want to ask for your opinions. Next Winter is my last term as an undergraduate, and I’m wondering which courses I should take (of the ridiculous number that are listed on my “Plan” page). I feel mainly interested in algebraic geometry and functional analysis, and I probably have a decent enough analysis background already (for an undergrad), so I was thinking about focusing on geometry, with something like this:
- Riemann surfaces
- Index theorems (I will have all the prerequisites for this, except algebraic topology)
- Commutative algebra or algebraic topology (which one?)
- 2 CO courses: nonlinear and combinatorial designs (to finish CO major requirements)
- Some lame non-math course (since I need 0.25 more non-math units… sigh)
(If you want to see the official course codes/titles, again, check the “Plan” page). Thoughts?
Here’s a summary of how I’m finding my courses this term. I’m also sitting in on PMATH 955, but I won’t talk about that here. I ordered them by ascending difficulty/time commitment.
PMATH 445 (Representations of Finite Groups): So far, this course has been really easy. I never thought I would use that word to describe a 4th year PMATH course. The assignments are doable in a couple of hours tops, and the material seems pretty standard. Initially I thought it was just because I had experience with representation theory from when I took Lie Groups, but it seems like almost everyone is finding the course easy going. I can’t complain since I honestly don’t think I would have the time to throw at this course if it were much more demanding. It’s my first class at 9:30am, and notes are posted online, so admittedly I haven’t been attending very regularly (things became hectic in the past week or two).
PMATH 900 (Valued Fields): This course was also easy going in the beginning. It’s a bit more technical now, but still rather palatable since all the objects involved are just fields and certain kinds of rings, and other stuff we all know and love. There’s no highly sophisticated machinery to deal with (I’ll save that for last). The first assignment was very reasonable, but not something I would want to attempt doing in a single day. Algebraic cleverness comes in short bursts for me.
PMATH 465 (Riemannian Geometry/”Diff Geo 2″): Just like representation theory, the material here is pretty tame, or maybe it just seems that way since (having been through 753, 763, 441, 365, and just about every other course) I’m used to seeing linear algebra everywhere. The concepts seem natural enough to me, although I feel kind of uncomfortable when I have to get my hands dirty and think about solutions of ODEs. I don’t really know the first thing about differential equations. Lectures aren’t hard to follow for the most part, but I just zone out when he starts grinding through horrific tensor calculations, mainly because I have to focus so much on TeXing them, which is always a pain. Assignments are long and routinely absorb my weekends, including the current one. 😦
CO 430 (Algebraic Enumeration): This is a serious course on enumeration. The operations on species are really cleverly chosen to make the generating functions behave as you’d expect, and they allow you to write down extremely concise formulas that capture some pretty nontrivial ideas. Even if I’m dealing with an equation involving 3 species, it can take me a few minutes to unwrap the definition and grok what it’s actually trying to say. Lectures aren’t the easiest to follow, and the assignments take me a while to solve and write up (he could probably make them significantly more difficult while staying within the margins of reason, though).
PMATH 822 (Operator Spaces): I don’t even know where to start. A lot of people who took Lie Groups with me last Winter would describe it as the hardest course they ever took. This course is at least 5 times more insane. In addition to the disfigured tensor products everywhere, there are now von Neumann algebras flying around and we never even defined them. The other students (who are mostly PhD students) seem way better than I am at analysis, and math in general. I’m not blaming myself for this, after all I’m just an undergrad; hopefully I will be able to reach a similar level in a couple of years. To be honest, it feels like the lectures are aimed at the PhD students in analysis, and my background in functional analysis just feels inadequate (I also have no measure theory background beyond PMATH 450). Keep in mind that this is coming from someone who generally wrote perfect assignments in Functional Analysis and meticulously checked every detail. It seems like we are also expected to have a lot of time to spend thinking about the material outside of class, which is understandable since grad students usually only take two courses. Unfortunately, I have to deal with five. I can’t even emphasize this enough; if you can’t do functional analysis upside-down blindfolded while reciting the alphabet backwards in your sleep UNDERWATER, you probably barely have a chance of grokking this material during the lecture. I started the first assignment fairly late, and proceeded to sink an enormous amount of time into it. I still didn’t completely finish. I should have known better; operator theory is notoriously subtle. I did feel like I could solve the remaining things with a couple more days, but even so, there was no room in my schedule to continue working on it past today. 😦