April 15 approaches, and seniors with math degrees are deciding where to begin their graduate study. Isabel Lugo at God Plays Dice has a great post addressed to prospective graduate students at Penn:

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. (This statement about “feeling” is incredibly difficult to make precise, but two examples are probability and number theory. Probability is usually taught in a “naive” way to undergrads and in a measure-theoretic way to grad students; number theory as taught to undergrads pretty much exclusively concerns itself with reasoning that takes place in the integers, whereas at higher levels it uses Big Fancy Algebraic Machinery. In addition, it may turn out that you think you are interested in X but in reality you had a particularly good teacher of X as an undergrad which colored your perception of that field.)

Read the whole thing: I agree with every word, except that Lugo says you should go to Penn, whereas in fact you should come to Wisconsin.

If you’re wrestling with this decision right now, the most important thing I can tell you is that this decision doesn’t matter as much as you think. More precisely: it matters a lot, but the ways in which it will matter are almost entirely unpredictable, so you might as well not worry about them. Four or five years from now you’ll very likely look back and say “How wise I was to have chosen graduate program X!” — which is just what you’d have said about graduate program Y, had you decided to go there.

Some things that matter more than you might think:

*How much do you like the city where you’d be living for the next several years? *Living in a place you hate can be depressing, and being depressed means not doing good mathematics.
*How strong are the graduate students at the program, how long do they typically take to finish, and what kind of jobs do they get when they do? *Make sure the program is well set-up to meet the kind of ambitions you have.
*How’s the atmosphere?* Some math departments are really fun. Some are kind of dead. I don’t know any way to figure out which is which without physically visiting the place and walking around.

Some things that matter less than you might think:

*How strong is the university “overall,” e.g., how highly rated is it in US News and World Report? *In math we only care how good the math department is; membership in the Ivy League, for instance, doesn’t necessarily mean much.
*How strong is the group of faculty in my research area?* As Lugo explained, you probably don’t really know what your research area is going to be, even if you think you do. Lots of undergrads have had some exposure to number theory and differential geometry; many fewer have seen algebraic geometry, or representation theory, or PDE, or homological algebra, or symplectic geometry, or homotopy theory, or …. Don’t be afraid to go to a place where you don’t have any idea who you might work with. If the institution is a good fit for you, it’ll work out.
*How much am I going to get paid?* In some circumstances, obviously (e.g. supporting a small child) this really does matter. But if you’ve got a B.A. in math, you’re already taking a huge pay cut by studying for a Ph.D. instead of becoming an I-banker. If you can go to a school that you like better and where you think you’ll do better math at the cost of making a few thousand dollars in annual stipend, you should do it — if you end up getting a fancier postdoc as a result, you’ll break even in a year.

I, too, found this decision really difficult. For a long time I was pretty sure I wanted to go to Berkeley. But a friend of mine who was going to M.I.T. found a nice house in Somerville, and needed an extra person to live there, and I’d heard finding a house in Berkeley was really hard. So I went to Harvard instead. Is that a dopey way to choose a graduate program? Maybe. But it worked out fine for me. And it’ll work out fine for you, too!

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