Note: Faculty and graduate students can access past topic proposals here if using a department computer, or by entering the department username / password combination.
During the second year of graduate study, students are required to give a topic presentation in some field of mathematics. The goal of the topic is to help students bridge the gap between the material covered in the first year courses and some of the frontiers of research. It is, in addition, an opportunity for students to begin to engage with mathematics in a more sophisticated and active manner. Often, until this point, students have seen mathematics presented in highly refined and complete form, and their task has been to absorb it as thoroughly as possible. By and large, the path has been laid out quite clearly, and the student's task has simply been to follow it. In the topic, the student is expected to begin to take a more active role; the task is not simply to master a body of material (although it includes that), but also to understand the problems, questions and examples which have led to the development of the theory, and to begin to be able to formulate questions which might lead to productive future developments. In other words, he or she must learn not only answers to questions, but also what makes the questions interesting.
The student should have two advisors during the course of the topic, one of whom generally takes a primary role. At least one of the advisors should be a senior faculty member in the mathematics department. At the beginning of the topic, the student and advisor(s) should reach a preliminary agreement about the area to be addressed in the topic, and the kind of work to be undertaken in connection with it. Often, there will be a list of books and papers to be read and discussed with the advisor; in other cases, the advisor may run a seminar in which the student is expected to take an active role. The agreement about the scope and direction of the topic may change over the course of time, but both student and advisor(s) should be clear in their understanding of this. There are two requirements for the successful conclusion of a topic: a topic proposal and an oral presentation.
The first requirement is that the student write a brief topic proposal, generally about 5 pages in length. The goal in the proposal is to give an overview of the area in question which is accessible to a reader who is mathematically literate, but not a specialist in the area. The proposal should attempt not only to describe a set of results, but also to explain the source of interest in the subject, and what fertile areas for further development there might be. While the topic is intended to help the student reach the point of doing research, it is not desirable that the focus of the topic be narrowed excessively so as to ensure that the student is able to work on a problem at its conclusion. It is more important to gain a firm and broad foundation in a significant area of mathematical inquiry at this stage, even if this means that additional work may be necessary after the topic before the student can begin to engage seriously with a problem. The proposal should aim to convince the reader that such a foundation has been put in place. Obviously, there may be disagreements as to whether a topic is broad enough; there should be enough communication between the student, the advisor(s) and the graduate committee over the course of the year to ensure that these disagreements are resolved early rather than late. Proposals are to be turned in to Laurie Wail by February 15 at the latest. It is, however, advisable to turn in the proposal earlier if at all possible, and to begin work on the proposal well before the deadline. The proposal will be reviewed by the graduate committee and, if the committee is satisfied, distributed to the faculty. The faculty as a whole then has two weeks in which to raise questions or objections. If no objections arise, the proposal is considered approved.
The second and major requirement is an oral presentation to the advisors (and perhaps others), which may be scheduled after the proposal has been approved. The presentation itself is generally expected to last about an hour, but questions from the advisors may expand the time required to as much as two hours. Questions may be asked by the examiners either during or after the presentation; the energy and depth of the questioning will vary according to the inclinations of the examiners. The student may be asked to do some follow-up work in order to pass, or to give a retake. The topic should be presented by the end of the last week of classes in the winter quarter, though exceptions may be made in unusual circumstances, if the advisors and graduate committee agree. If a retake is required, the second presentation must take place by the end of the last week of classes of the spring quarter.