A friend’s advisor is rumored to have said, “An advisor/advisee relationship is like a marriage…only more permanent.” From that perspective, I did a lot of “dating” before I settled on an advisor. I did some “online dating” before I got to graduate school by checking out the web pages of professors at the schools I applied to, particularly in the areas I thought I might specialize in. I looked at their publication lists, titles of their advisee’s theses, and what their advisees did after graduate school. Next came “speed dates” at various graduate schools’ open houses, where I talked with those professors and tried to determine if we’d be a good fit. It was clear in some cases that we were not. In other cases, asking graduate students about a professor let me know that I either might not want to work with the professor or would not be able to work with the professor (for instance, if they were retiring, moving, or not taking advisees).
Once I got to graduate school, it was time for some short-term relationships: I took classes with professors I might want as my advisor. Most students in my program arranged in their first or second year to do an independent study for a quarter with a potential advisor, occasionally finding out that they did not work well together. For my qualifying exam in the area I wanted to specialize in, I chose a potential advisor to be on my committee, as that gave me an excuse to meet with them and talk about that area. I went to seminars and conferences organized by my potential advisor. I talked to their current students and asked if they would still choose this advisor if they had the chance to do things over. I also asked for advice on how to work with this professor. One student told me that it had taken her two years to figure out that her advisor expected her to stop him if he asked her to do an unreasonable amount of work.
After passing my qualifying exams, I went to a couple professors and asked them, if I were to be their advisee, what kinds of problems would I work on. They gave me papers and books to read on a variety of topics and we set up additional meetings so I could tell them if any of these subjects interested me or ask them more questions. At this stage, most of the students in my department would settle on one professor to keep meeting with and voila! They had a thesis advisor.
Of course, this isn’t the only way to go about things, or even the best way, and even if you do all this, it doesn’t necessarily mean that finding an advisor will go smoothly for you. If you are set on a certain area, you will have a limited number of choices for advisor once you get to graduate school. If you fall in love with a particular thesis topic, you may have only one or even no professor at your institution to work with. Even after starting work with a professor, you may find that your approaches to math and research clash so much that it makes working together difficult. Or you may find yourself in a similar situation to mine: while I was reading papers with one professor, I took a completely unrelated class on a whim and found it so interesting that I asked the professor teaching the course for more information. Before I knew it I was reading papers and meeting weekly with that professor, while the papers from the professor I had thought would be my advisor languished on my desk. For any number of reasons, you may have tough choices to make: do you stick with the topic and the professor you had previously settled on? Do you pursue work in a completely new area, where your background isn’t as strong? Do you switch to a different professor within your area and begin work on a new thesis problem? Do you transfer to a new institution where there is someone you think you could work with, and start over? Any of these could be the right choice: I’m sure that at any institution you could find people who took each of these approaches and would say they made the right decision. Whatever you do, hopefully in the end you will have found an area, a problem within that area, and a professor who works on that problem that are all a good (enough) match for you.