Your choice of dissertation advisor is the single most important decision you will make as a graduate student, and in fact it is one of the major decisions you will make in your mathematical career. You will therefore want to devote some thought to this choice.
Your advisor is primarily responsible for directing your thesis research, and very often will choose the subject of your dissertation. He or she is also the chair of your dissertation committee. If you need to be brought up to speed on certain topics related to your dissertation, your advisor will help you with that too. When you apply for a job, the letter that your advisor writes is the most important one. This relationship does not terminate when you get your degree, you may continue to rely on your advisor for advice and suggestions regarding the direction of your research, or help with mathematical problems.
You will want to keep the following questions in mind when you make this decision:
- Does this advisor have a flourishing research program?
Is it a subject that interests you? You will, after all, spend a lot of time thinking about it.
- Does this advisor have a lot of students? There are various pros and cons here. A lot of students may mean that the advisor has less time for you individually, but on the other hand it means that you will have other students interested in and informed about your specialty. A related question: does the advisor have extensive commitments elsewhere? (in another deparment or institution, say).
- Do you like the advisor’s teaching style? You will certainly want to take any special topics courses your advisor is offering, or any seminars your advisor is running.
- What is the advisor’s working style? Some allow their students considerable freedom, whereas others get very involved with their students’ research. Which works best for you?
- How long has it taken the advisor’s former students to finish?
- What is the advisor’s record in placing students?
Taking a reading course with a potential advisor is a very good way to see how the two of you interact. Typically the time to do this is after you have finished the Ph.D. exams in your general area. Since your advisor is the chair of your dissertation committee, you will form the committee at about the same time. These decisions are not irrevocable, but you will want to be reasonably sure of them.