To qualify for the Ph.D. program, the student must first pass the qualifying examination. This examination serves to demonstrate that the student has mastered the foundational material necessary for more advanced study. The qualifying examination consists of three separate area exams, chosen from among algebra, real analysis, complex analysis, topology and differential equations. The student must choose at least one of algebra or topology, and at least one of either real or complex analysis.
The content of the exams are roughly the content of the corresponding year-long sequence, although not strictly limited to the material of the courses. The corresponding courses are MATH 5123, 5133 (Algebra), MATH 5503, 5513 (Real Analysis), MATH 5523, 5533 (Complex Analysis), MATH 5703, 5713 (Topology), MATH 5723, 5733 (Differential Topology) and MATH 5303, 5313 (Differential Equations). Syllabi for the exams, as well as copies of prior exams, are available in the department office.
The exam is given in January and August, before the beginning of the spring and fall semesters. In case of failure, students may retake the exam at most twice. An area exam passed in the first attempt usually will not need to be retaken in a second attempt, but this is a decision of the Graduate Committee.
The Qualifying Examination should be passed by the middle of the third year in the program. (By the middle of the second year for students entering with an M.S. in mathematics.)
After passing the qualifying examination, in order to become a Ph.D. candidate, the student will take the preliminary examination. This is an oral candidacy examination to ensure that the student is prepared to undertake research in the intended area of the dissertation.
The exam is conducted by the student's advisory committee. The content of the exam is determined by the chair of the committee in consultation with the student. It is expected, but not required, that the committee chair will become the student's thesis advisor upon passing the preliminary exam.
In the event of failure, the exam may be repeated once. At least three months must be given before the repeated attempt. In any event, the preliminary exam must be passed by the end of the fourth year in the program.
After the thesis is completed, the student is examined in a public oral defense of the thesis. The defense is conducted by the dissertation committee, whose chair is the major advisor, and includes at least two other mathematical sciences faculty members. The dissertation committee must approve the thesis and its defense for the student to obtain the degree of Ph.D.