Mentors and Advisers
Arriving graduate students are mentored by the Director of Graduate Studies and the Assistant Director of Graduate Studies. They are your primary contacts in the department. They advise on selecting courses to take and seminars to attend, help to resolve any problems with courses, exams, teaching, and practical matters of any kind. Students should not "swim on their own", and mentors help to take care of them until they select advisers.
Selecting an adviser is perhaps the most important decision during graduate studies. Even though you can change your adviser at any time, choosing or changing advisers too late in the program is the primary cause of failing to obtain a Ph.D for students who have passed their qualifying exams. It is a requirement that students select their advisers by the end of their fifth semester. It is strongly recommended that you choose your adviser no later than one semester after you passed all qualifying exams.
There are a number of strategies for selecting advisers. You should certainly do some web surfing and look at research of faculty you are considering. You may go to seminar talks or attend several classes taught by faculty working in your general area of interest. You may schedule appointments with several faculty members and ask them to give you sample problems or just exercises, or to simply talk to you about their research and projects they could offer you. You should however be prepared that some faculty may not agree to advise you since they have too many advisees or if you do not have necessary background; others may set up a "trial period" to see if it works well for both of you.