**Doctor of Philosophy (Ph.D.) in Mathematics**

Students accepted to the Ph.D. program in Mathematics are divided in two categories:

1. Those who possess a Master’s degree in Mathematics.

2. Those who possess a Bachelor’s degree in an academic discipline, but do not possess a Master’s degree in Mathematics.

Students of the second category are encouraged to obtain a Master’s degree in Mathematics even though a Master’s degree is not a formal requirement for obtaining a Ph.D. in Mathematics.

**Requirements for the Ph.D. in Mathematics**

- A minimum of 39 credit hours of graduate course work is required for the Ph.D. degree. This course work must include Modern Algebra (Math 625, 626), Real Analysis (Math 753, 754), Complex Analysis (Math 655, 656), and General Topology (Math 501). Of these 39 credit hours, at least 30 must be in courses open only to graduate students. For the purpose of this requirement, Dynamical Systems (Math 564, 565) will be counted as courses open only to graduate students.
- A student must pass three comprehensive examinations. The list of subjects in which comprehensive examinations are administered is given below.
- A student must successfully complete the Prospectus Defense.
- The final step is a successful public defense of the student’s Ph.D. dissertation.
- Moreover, the Graduate School has strict rules about the Grade Point Average (GPA) that must be maintained by the student. Independently of the status of a student with respect to the rules of the Graduate School, the Department of Mathematics has the right to dismiss from its Ph.D. program any student whose combined GPA falls below 3.0 after two consecutive semesters.

**Examinations**

*Comprehensive examinations* are 3 hour written exams which take place each year in January and August. Each prospective doctoral candidate is required to pass comprehensive examinations in three areas: Real Analysis, Algebra, and a third area chosen by the candidate. It is strongly suggested that the choice of the area for the third exam should be done in consultation with the potential advisor of the candidate since the performance on this exam will provide a first assessment of the student’s readiness to do research in the intended area of specialization.

The Real Analysis Exam covers topics from Math 753/754 and the Algebra Exam covers topics from Math 625/626. For the third comprehensive exam, the candidate may request an exam on material from any 600-level two-semester sequence. Under usual circumstances, the exam is administered by the faculty member who is teaching the subject in the year of the examination. A student must average a grade of B or better in a two course sequence in order to take the comprehensive exam in that subject.

A committee will be appointed by the chair of the department for each subject to be examined in each comprehensive examination period. Whenever possible a committee will include the last two faculty who taught that material in the classroom. Each committee will be responsible for the setting and grading of their examination. Copies of previous examinations will be made available to students preparing for examinations.

- A student who is admitted on a part-time basis has 6 years, starting from the initial entrance, to pass the three comprehensive exams.
- A student who is first admitted into the M.A. or M.S. program and who is subsequently accepted as a Ph.D. student, has four years, starting from the initial entrance, to pass the three comprehensive exams.
- A student who is accepted into the Ph.D. program from the start, has three years to pass the three comprehensive exams.

Failure to complete the three comprehensive exams in the above time limits will result in dismissal from the Ph.D. program. Only two failures on comprehensive exams are allowed (whether in the same subject or not). A third failure on a comprehensive exam will result in dismissal from the Ph.D. program.

**Exceptions**

In an individual case, an exception to any of the above listed rules may be made by a majority vote of the tenured and tenure-track faculty.

**After the comprehensive exams**

After passing the comprehensive exams, the PhD candidate must find a faculty member of the Department of Mathematics who will agree to serve as the supervisor of the candidate’s dissertation. A Ph.D. advisory committee of five members of the university faculty will then be appointed by the chair for the department after consultation with the supervisor. The committee consists of the supervisor of the dissertation, three faculty members of the department, and a faculty member from another department of our university. Changes to the committee can be made with the approval of the candidate’s faculty supervisor and the chair.

After the Ph.D. advisory committee is formed, the next steps are the “Prospectus Defense” and the “Dissertation Defense.” According to university rules, there must be a minimum of four calendar months between these events.

*The Prospectus Defense: *During the Prospectus Defense, a candidate must present the prospectus of his/her proposed dissertation research for approval by the committee. At this time, the candidate must demonstrate a thorough understanding of the dissertation topic. The signatures of all committee members are required to pass the defense.

*The Dissertation Defense:* After successful completion of the Prospectus Defense, and upon mutual agreement between the supervisor and the candidate that the dissertation is complete, the candidate must undertake a final public defense of the dissertation. The dissertation must be delivered electronically or in print to all of the committee members at least 20 business days before the date of defense. A majority vote of the committee members is required in order to pass the defense.