Isabel Lugo had a nice post recently on advice for prospective graduate students. Although not all of her advice rang true to me (the first year of graduate school was pretty fun for most people I know), she makes an excellent point in comment 2 that I wanted to respond to.
Your mathematical interests will change during the first year in graduate school, because a lot of subjects “feel” different at the undergraduate level than at the graduate level, and there are some things you just don’t see as an undergraduate at all.
Personally, my favorite subjects in mathematics are:
- The representation theory of finite groups from Frobenius through Brauer
- Algebraic and Analytic Number Theory from Gauss through sometime in the late 1800s
- Quantum Topology from the Jones polynomial through the present
On the one hand you can tell from this that I like algebra more than I like geometry or analysis. This was something I was quite aware of as an undergraduate and beginning graduate student. However, all of these have something else important in common: they are/were all young subjects. With the exception of Euler’s prescient work on zeta functions, there’s not a whole lot of precursors prior to the beginnings I’ve stated above. You don’t see on my list anything like modern homotopy theory, ell-adic cohomology, the classification of simple groups project, or 20th century number theory. I went into graduate school thinking I wanted to do number theory because I love number theory up until about 1920. The lesson I should have taken from that is that I like younger topics in algebra, not that I want to do number theory. Older subjects feel different from younger subjects. You have more tools, bigger machines, but the most natural questions have already been answered and the field has moved on to harder things. But there was no way for me to know this as an undergraduate, because undergraduates don’t know enough to have been exposed to any material in an older subject.
There are lots of other key differences between fields: Do people do more theory building or problem solving? Is the topic dominated by a few giants’ research programs with other people following their lead or is it more each researcher having their own smaller programs? Is it on the intersection of multiple fields or more isolated from other fields of mathematics?
Obviously a first-year graduate student isn’t going to know the answers to these questions, but these are the sorts of questions that you need to ask yourself rather than just “which subject did I like as an undergraduate.”