The comprehensive examination is the most advanced posed by MU. The Graduate School requires all doctoral candidates to pass their comprehensive exams no later than 7 months before defending their PhD thesis. This page gives guidelines for Mathematics Doctoral Students to compete their Comprehensive Exam.
The various Graduate School forms and additional information are available from the Graduate School
The comprehensive exam is conducted by a committee. It should have a minimum of 3 math members (including advisor) and one outside member. Often this committee is the same as the doctoral committee declared on the D-2 form, but this is not a requirement. The membership of the comprehensive committee does not need to be declared on a form in advance. At the completion of the comprehensive exam, the results are recorded on the D-3 form.
At least 3 of the math members of the committee (separately) arrange with the student to study a specific topic for a few months, and then the student reports to the committee member on the topic. (The outside member is usually not included in this part.)
The Topic: The topic can be an important paper in the area of research or theorems in a book at the level of an Advanced Graduate class. If you have questions about the appropriateness of a topic please consult the DGS. The report to the committee member is considered a component of the comprehensive exam.
The Report: Typically the report to each committee member would be an oral presentation on the board, with the committee member asking questions to test understanding. The committee member should also be available to answer occasional questions while the student is preparing for the exam, or possibly to modify the topic and/or expectations, but this is primarily independent study. It is allowable to submit written reports instead of oral presentations by prior arrangement with the committee.
Timing and Coordination: All of the individual reports to the committee members should be within a one-month period, and that period should be contained within an academic semester (summer semester is ok). Normally when students arrange for a committee and topics, they would also discuss the month when the reports will be completed. Usually people go 3-6 months from planning to completion, but the Comprehensives can be completed in a single semester. After a student’s completes each of the reports to the math members of the committee, the entire committee (including the outside person) confers (email is ok) before each member signs the D-3 form.
For each topic of the exam, the student and committee member should fill out and sign the math department Report on the Comprehensive Exam Form. These forms should be submitted by the student to the director of graduate studies before the DGS signs the D-3 form and sends it to the graduate school.
Students are encouraged to do one of their comprehensive exam topics on something that is NOT closely related to their thesis research, this is a good opportunity to study an advanced topic in another area. This is not a requirement, and some students and advisors think that it's better to focus all energy on an area close to the thesis area.