# Ph.D. Mathematics Program Guidelines

## Program Guidelines

Described below are the Department of Mathematical Sciences requirements for the Ph.D. degree in Mathematics. These departmental requirements supplement those set out by The Graduate School. There are no foreign language requirements or qualifying exam for a Ph.D. in Mathematics.

## Course Requirements

- A minimum of 60 credit hours are required
- A minimum of 18 credit hours must be dissertation credits (M 690)
- The Ph.D. student's Program of Study listing their intended coursework must be approved by all committee members.

Typically, a Ph.D. student takes 18 credits of mathematics in courses numbered 500 or higher to prepare for their comprehensive examination. Students are encouraged to begin some form of doctoral reading or research (either informally or in the form of M 689 credits) with a committee member by their second year of study.

## Ph.D. Mathematics Written Comprehensive Exam

The Ph.D. comprehensive examination in Mathematics consists of both a written and an oral component. The candidate must pass the written comprehensive exam before taking the oral comprehensive exam.

The written part of the Ph.D. examination in mathematics consists of three tests, chosen from the following eight areas listed below. Students entering the Mathematics Ph.D. program from the Mathematics M.S. program who pass both written components of the M.S. Comprehensive Exam at the Ph.D. level need only take two Ph.D. written comprehensive exams from the following list below. Students entering the Ph.D. program from another institution may also sit for the written M.S. Comprehensive Exam immediately upon entrance to the Ph.D. program. If the student scores a Ph.D. pass on both components of the exam in one attempt, then the student may use a Ph.D. pass on both components to satisfy one of the three components of the written Ph.D. comprehensive exam.

- M 547 - M 551 Measure Theory and Complex Analysis
- M 511 - M 512 General Topology and Geometry and Algebraic Topology
- M 595 - 596 Dynamical Systems
- M 584 - 585 Functional Analysis
- M 581 - 582 Numerical Solution of Partial Differential Equations
- M 544 - 545 Partial Differential Equations
- M 560 - 561 Applied Mathematics
- M 547 - 586 Probability

#### The following guidelines apply to the Ph.D. examination:

- All candidates must take and pass the test on Measure Theory and Complex Analysis.
- Students should choose their remaining tests in consultation with their advisor. One purpose of this is to ensure sufficient breadth in the choice of tests. The student's choice of tests must be approved by the student's Graduate Committee.
- At most one "nonstandard" component (not from 2-8) may be taken with the approval of the candidate's committee.
- Exams are given every August on dates determined by the department.
- All students must attempt at least one exam in August before the beginning of their second year, and must attempt three exams by August preceding their third year.
- Students have a final opportunity to complete the exams in January of their third year (after completing five semesters of coursework). Failure to pass at this time results in termination in the graduate program at the end of the third academic year.
- A student is permitted to take a maximum of three exams each exam period, and may repeat a failed exam only once, and only at the discretion of the student's supervisory committee.

## Ph.D. Mathematics Oral Comprehensive Exam

After passing the Ph.D. written comprehensive exam the candidate must pass an oral comprehensive exam within 1-2 years from the date of passing the written exam, as determined by the supervisory committee. The exam date will be agreed upon by the candidate’s committee. Normally the oral comprehensive exam is a thesis topic proposal where the candidate's ability to conduct research on the proposal is assessed. When this is not the case, the candidate will be informed about the nature of the oral comprehensive exam by the supervisory committee. The candidate has at most two attempts to pass the oral comprehensive examination.

## Dissertation Requirements and Final Defense

Please see later sections in this document for details on Dissertation Requirements and Final Defense.

## More Information

For more information, refer to The Department of Mathematics Graduate Handbook.

## Online Library

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