During their first year, most Ph.D. students concentrate on coursework and the written qualifying exam. After completing the written exam (usually by the beginning of the second year), students change their focus to (1) identifying potential research advisors and research topics, and (2) selecting qualifying exam topics and a qualifying exam committee.
Students are often unsure how to go about this. There is no one way to find an advisor and a thesis topic. Read the summary of faculty research areas. Ask advanced students how they found their advisors. Talk to your faculty mentor or the graduate program director. Most faculty are very happy to talk to students and to discuss possible areas for thesis research.
Don't assume that a faculty member who is not currently supervising students is not available. Some faculty members are not currently supervising students by choice, but others are not advising students simply because our student population is small relative to the size of the faculty.
The Faculty Research Perspectives Seminar held in the spring semester presents the research activity of the faculty on an introductory level, and may help you identify your interests. Department tea time is a good time to make informal contacts with faculty and initiate these discussions.
When talking to a prospective advisor, ask them about potential thesis areas that you could pursue with them, and the background required to work in these areas. Some advisors have specific topics ready for students to begin working on, others expect the students to take more responsibility for identifying the topic.
Don't expect that a faculty member will agree to be your thesis advisor during your first discussion. It is common for the prospective advisor to suggest some reading for you to start with, and for you to have a few technical discussions as part of the process of deciding to work together. One useful way to explore a potential research area or advisor is to take an independent reading course with the potential advisor. This can be done during the academic year or over the summer.
The oral qualifying exam is intended in part to help students make the transition from course work to research. The student forms a committee (including potential thesis advisors) and chooses two areas (which should encompass the student's possible thesis areas). The process of preparing for the exam should help the student build the necessary background for thesis research, and build relationships with potential advisors.
Students should aim to have their orals topics and committee set by June of their second year (but should not panic if they don't reach this goal.) Students are expected to finish the oral qualifying by the beginning of the spring semester of their third year. (Fourth year students who have not passed their oral exam are normally ineligible for support.) By the time they complete the exam, most students know who their advisor will be (or have a pretty good idea).