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Not sure how to tag this one so feel free to edit and add tags.

When I initially started graduate school my choice for an area of study was quite nebulous. I had only figured out enough to know that I wanted to do some work involving a lot of category theory. So when I applied to schools I figured I could find some interesting topic to work on since almost anything involves category theory. Now I'm a bit more mature and I have a much better perspective and a much better idea of what I would like to do. My question then is how do I go about finding an adviser working in an area I would like to specialize in and if this person is not at my current school what are my possible courses of action?

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I strongly recommend finding a good advisor (someone who you get along with, who has compatible understanding of how hands-on the advisor will be, who will keep you funded, who can get you a postdoc, who actually wants a student etc.) over choosing a particular subfield. There's too little correlation between what you learn about a subject as a starting graduate student and what it's like to do research in that field for you to make decisions on that basis. That said, it's reasonable to narrow your search to broad fields (say "I want to do algebra") because that still allows a huge range of subfields to work in. As long as your tastes are reasonably broad (and if they're not you should work on broadening them) you and your advisor should be able to find something that interests you both for you to work on no matter what your advisor's particular speciality.

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Also, I'd be very careful about being sure one knows what one wants to work on. It may seem like that now, but a lot of people do a lot of bouncing around in grad school, keep an open mind, and remember that almost everyone drifts a little bit in their area of specialty. Having an advisor who doesn't do exactly what you want to, even if you do your thesis on their kind of stuff, doesn't mean you'll have to do that forever.

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  • $\begingroup$ Are analysis and stochastic equations your only options? $\endgroup$
    – Ben Webster ♦
    Commented Dec 1, 2009 at 17:25
  • $\begingroup$ @davidk01: Probably for every subfield of mathematics X, there is a subfield Y such that starting to work in X will make it harder to move to Y. True, some subfields are more "central," but that doesn't necessarily make them a better choice for you. If you love stochastic equations, then go for it! $\endgroup$
    – John Goodrick
    Commented Dec 1, 2009 at 19:08
  • $\begingroup$ I think the point was that he (? I'm assuming that's not short for Davida) doesn't love stochastic equations. It mostly just hard to believe that those are the only practical options. On ther other hand, you could always take a reading course if there's a professor you like who does stochastic equations, and see how you feel at the end. $\endgroup$
    – Ben Webster ♦
    Commented Dec 1, 2009 at 21:58
  • $\begingroup$ @davidk01- Ah, I see. This is where threading would be handy. I didn't mean to suggest that you hadn't thought about, just that you are reasonably likely to overestimate your ability to know these thing, and that having a good advisor, one who gives you good advice and looks out for you, as opposed to having one who does exactly what you think you want to, may be more important than you think. $\endgroup$
    – Ben Webster ♦
    Commented Dec 2, 2009 at 0:25
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One thing to keep in mind is that if there is no one at your school, you can often either have a local advisor and an advisor elsewhere, just an advisor elsewhere, or have a local advisor who will sign things while you're really talking to someone else.

As far as finding an advisor at your school, go to talks! Professors (at least here at U. Penn) give talks fairly regularly about what they're working on, and if they aren't, you can often email them to ask if they can talk to you about what they do, and then if what they do is like what you're interested in, you can talk to them about being your advisor. Another way to do it is to talk to professors who've taught classes that you've taken, and try to do a reading course with them.

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  • $\begingroup$ I am quite curious about how the local-elsewhere advisor setup could work. If your advisor is quite far away from where you are, then how can the supervising process work? $\endgroup$
    – user709
    Commented Dec 1, 2009 at 16:19
  • $\begingroup$ I'm not doing it, so I can't go into much detail. Everyone I know who has done it has their elsewhere advisor at a nearby school (in particular, the main example I'm thinking of was in Boston, and so were both of his advisors, just at different schools). As for how resistant profs may be, I would assume it varies wildly from prof to prof. $\endgroup$
    – Charles Siegel
    Commented Dec 1, 2009 at 16:43
  • $\begingroup$ It also helps to have two schools in very close proximity; I'm guessing the example Charlie is thinking of involves people at Harvard and MIT, which are about two miles apart. $\endgroup$
    – Michael Lugo
    Commented Dec 1, 2009 at 17:04
  • $\begingroup$ These days, with the telephone, email, and skype, you can definitely work with a remote advisor, if you don't need constant guidance. My recommendation is to start by establishing email contact and asking for informal guidance ("what should I read if I want to learn...") from someone appropriate. If you develop a good rapport, then you can ask if that person will be your advisor. The most successful example that I know of this is Curt McMullen, who was a graduate student at Harvard, but somehow became a student of Dennis Sullivan, who had joint positions at IHES and CUNY at the time. $\endgroup$
    – Deane Yang
    Commented Dec 1, 2009 at 17:05
  • $\begingroup$ I'll note: the MIT and Harvard situation is different from any other pair of schools. As far as I can tell, no distinction is made for advisor pairing between the two schools. I know a couple of people who had remote advisors (most, though not all, as a result of the advisor moving to another school), where it worked fine, but that doesn't mean I would recommend it. $\endgroup$
    – Ben Webster ♦
    Commented Dec 1, 2009 at 17:25
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Here is some advice that my department wrote some years ago for our undergraduate math majors that might be relevant. It takes a rather extreme stance, but this was done purposely to provide counterpoint to the advice usually given to Ph.D. students in math:

Choosing a Ph.D. program or advisor

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  • $\begingroup$ The website you link to was unresponsive when I tried to follow the link. $\endgroup$
    – Alison Miller
    Commented Dec 3, 2009 at 3:48
  • $\begingroup$ Yes, unfortunately it appears to be down. $\endgroup$
    – Deane Yang
    Commented Dec 3, 2009 at 15:06

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