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. 😦

Some advice from someone who’s been there, done that:

Your posts are dominated by discussion of courses, which is reasonable given that the course notes section is a main draw, but something that you will move away from in graduate school. The centerpiece of graduate school, of course, is the thesis, which represents new work, and is not a subset of any existing courses (although the intersection may be of considerable size). Even before the thesis, you’ll need to start reading papers, and engage in 1-on-1 discussion with experts in your field, and these experiences differ considerably from reading textbooks and attending lectures or seminars respectively, although 800/900 level courses might come close. The point is, you need to start transitioning away from the view of mathematics as a collection of packaged university courses, and instead view mathematics as an actively evolving subject, in which you personally can change the future direction of your chosen field of study.

I would like to emphasize how delightful a privilege it is for mathematicians to be able to personally affect the development of their own subject. This is not an automatic attribute of any arbitrary academic subject area, and we should not take it for granted that we have this ability! For example, at the other extreme, very few historians are themselves historical figures. Likewise, not every literature professor is a literary writer, not every law professor is a lawyer, and so on. But pretty much every research-active math professor is a mathematician. It is an intrinsic property of mathematics that to study mathematics is to do mathematics. Your job in graduate school is to develop your own abilities as a mathematician. While background knowledge is a big part of that process, it is just the first step, and one that you have already largely completed.

The next step is to train yourself to form connections between different areas of mathematics. Such connections give new insights and new theorems, and those who understand any given link have a huge advantage and a huge head-start over those who don’t. This step is something that most people need help from their Ph.D advisors to accomplish, and it’s certainly not one that I can effect through a web page. Not everyone succeeds in making this leap. As long as you are ready, which you are, the earlier you start thinking about this stage, the better.

For all these reasons, you really should not be considering any courses in isolation at this juncture. Evaluate each course according to your own goals (which I don’t know), keeping in mind that the course itself is only a stepping stone. Learning N courses takes O(N) work. Understanding all possible connections between them takes O(N^2) work. The implied constant in the first O(N) is larger, but eventually asymptotic behavior dominates. That said, if I were you, which I’m not, I would definitely take both algebraic topology and commutative algebra. Both are essential core subjects.

There is much more I can say, but not all of it can be said in public. Feel free to email me if you wish to learn more. This applies for any readers of your blog, not just yourself.

Last and least, it’s no surprise that finite groups are easy if you’ve done Lie groups. A finite group is a zero-dimensional Lie group.

Hey Mike, I have really enjoyed reading your blog. Although I am not taking any pure math courses (so my courses are much easier), I’m taking six courses this term, five of which are crosslisted as graduate-level courses. I feel your pain. I just finished 5 midterms but I’m not even half way done. I guess some of us like to put ourselves through _hell_, but at the same time, don’t forget to take a step back and think about what you’ve learned. Keep working towards your goal – grad school and research in pure mathematics. You’ve chosen a very difficult path, and you have our best wishes.