The Math Department offers essentially four kinds of graduate courses. First, there are the courses associated with qualifying exams. These courses have regular homework assignments and exams, and culminate in an associated qualifying exam. These will almost certainly be your primary focus for your first year of study. A list of mathematics (graduate and undergraduate) courses can be found here.
The department also offers so-called “topics courses”. These are more specialized courses. The course content is up to the instructor, who may solicit input from interested students as to the topics covered. Topics courses may or may not have associated homework assignments or exams.
Next are the “reading courses.” These are custom courses that are designed by you and a faculty member. You spend the quarter reading selected text(s) and meeting regularly with the supervising faculty member to discuss the material. Taking a reading course with a professor is one of the critical stepping stones to finding an advisor. In addition to learning specialized material from an expert in one-on-one meetings, you get to know the professor on a more personal level. It can be hard to tell from a classroom setting whether or not someone will be a good advisor. Taking a reading course from someone is a good way to see what doing research with that person will be like.
Last are the “seminar courses.” These are regular meetings between students and faculty to discuss current research in a particular area. The regularity with which these courses are held varies widely. For some seminars, there are different speakers and topics every week. Sometimes, the speakers are invited guests of a faculty member. Some seminars have the goal of understanding a particular paper (or collection of papers). These will typically have a rotating schedule of participant-speakers.
Sample schedule of classes for the first two years of study for a PhD program and examples for Masters students can be found here.
Advice from Oded Yacobi, former graduate student:
There is one singular piece of advice I once got that completely changed my graduate school career. I remember it clearly. I was in my first year, and had just finished taking the topology qual. I was walking out of the testing room with Ross Richardson, at the time a second year student. We got to talking, and he asked me what my plans were for the summer and the following year. I was stumped. Up to that point I had not even pondered my existence post-quals.
Ross asked me about my interests and I responded vaguely that I enjoyed Prof. Garsia’s applied algebra class. In this class we studied the representation theory of finite groups and symmetric function theory. Ross, ever helpful, told me about other professors in our department with overlapping interests. And then he said, “Why don’t you ask to read with one of them for the summer?” Amazingly, the thought had not even occured to me.
Of course I knew that grad students do reading courses with professors. But I thought this was something that the “older” students do, like third and fourth years. I didn’t know that a newbie, such as myself, infinitely naive and unknowledgeable, can deign to waste professors’ time. And, anyhow, I still had one more qual to take. Shouldn’t I focus on that?
The answer is, of course, no. In fact, even before you finish all your quals, you can start reading with professors. The way our program is structured, with so much emphasis on the quals when we arrive, it’s easy to forget that you actually came here to do research. And while it’s perfectly natural (and necessary) to let quals take over your first year, that doesn’t need to happen again your second year. Sure, a third qual class in your second year might be the only thing with deadlines, but there is more to life than just that class. Too often, many students get completely bogged down studying for a qual, to the detriment of other facets of their grad career. Instead of spending the whole week on a problem set, try balancing that qual class with a reading course.
The value of a reading course (as opposed to a topics class) is several-fold. Most importantly, you get to know a professor well. Even if you don’t end up working with this professor, knowing him/her well will be helpful later on. Also, the one-on-one interaction forces you to be responsible for the material in a way that topics class just don’t. It’s nerve-racking presenting solutions to a professor by yourself, and you will understand the material inside and out as you prepare for your weekly meeting. Finally, you will get a feel for what research in this area is like. This will be invalueable as you navigate your grad career, regardless of what area you will finally settle on.
To summarize, grad school doesn’t have to be neatly divided into quals, reading courses, and research. These distinctions are rather blurry, and you will do yourself a great favor by beginning the latter two as quickly as possible.