Choice of adviser 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?
 A: 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.  
A: 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
A: 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.
A: 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.
