Quoted from "Guidelines for Qualifying Examinations"
The second exam is an oral, 80-minute exam, given by a committee of four faculty members. It will cover two topics chosen by the student in consultation with a prospective dissertation advisor. It will normally be taken by the beginning of the second semester of the student's third year; any delays past then will have to be approved each semester by the program director, and failure to either take the exam or obtain such permission will result in the student being dropped from the Ph.D. program.
The student, or faculty giving the second exam, will write a syllabus (or approve a previously written syllabus) for each topic on the exam. A copy of the two individual topic syllabi, along with the membership of the examining committee, must be filed for approval with the program director well in advance of the scheduled date of the exam. The members of the examining committee should normally be chosen from faculty who work in areas related to the proposed topics; a prospective dissertation advisor should be on this committee. It is expected that the student will do some reading independent of course work for this examination.
Ideally, the two topics should have some relationship to each other and to a proposed area of dissertation research, but not be so closely related as to be essentially one topic. However, we recognize that a student who decides late on an area of specialization may present two unrelated topics. The examiners may continue as an advisory committee for the student's dissertation. They are also likely to be on the student's dissertation defense committee if they are available.
It will continue to be possible for the student to change advisors if so desired, even after this exam is taken. In this case, the program director might consider also appointing a new advisory committee.
A student who fails this exam may attempt it a second time within a one year period, potentially with a different proposed dissertation advisor and committee. Students who fail a second time, or whose delay in retaking the exam is excessive (as determined by the program director) will not be allowed to continue in the Ph.D. program.
An archive of topics and syllabi is provided in the link at the top of the page. It should be strongly emphasized that these are only examples. Each syllabus is specific to a student, the student's committee, the aims of the student's studies, and the time at which the exam is given. Research in mathematics evolves constantly, and suitable syllabi at one time may not be appropriate a year or two later. The examples presented here should help the process be understood better, and also should help students and faculty members create suitable syllabi for future oral exams.
Four members! The Graduate School has been enforcing the rule that committees for this exam must have at least four members. To avoid difficulties, students should assemble committees in agreement with that regulation.