1. Take at least 60 hours beyond the bachelor’s degree - 45 hours, excluding MATH 7300.

There must be 30 hours of mathematics courses numbered 6000 or above, excluding MATH 7300. It is required by the University that the dissertation hours and 30 hours of course work be completed after admission to the doctoral program. The 60 hours will include the following courses.

• One course in each of Real Analysis (MATH 6700), Complex Analysis (MATH 6760), Topology (MATH 6210), and Algebra (MATH 6300).
• Three two-semester graduate sequences, including at least two of
  • Real Analysis (MATH 6700-6710)
  • Algebra (MATH 6300-6310)
  • Topology (MATH 6210-6240), MATH 6250 may be substituted for MATH 6240.
• These three two-semester graduate sequences are the standard option, if no exception is sought as described below.
• An approved course in applied mathematics or probability/statistics.

2. Take three comprehensive examinations.

• A student must pass three comprehensive examinations, including at least two of Analysis, Topology, and Algebra. These three examinations constitute the standard option, if no exception is sought as described below.
• In general, a student is strongly encouraged to take an examination as soon as it is offered once the student has completed the corresponding two-semester sequence. However, the student must work with their advisor to determine the student’s readiness for each examination such that they are completed within the maximum allowable timeline: success on at least one examination at 2.5 years into the program and success on all three examinations at 3.5 years. Any alteration to the timeline must be approved by the Graduate Committee.
• A student must successfully complete a given examination by the second attempt and will be notified in writing of the results of each comprehensive examination within two weeks of its administration.

3. Exceptions to the two-semester sequence or comprehensive examination.

A student may, with the approval of the advisor and the Graduate Committee:

• Replace one of the two-semester sequences above with two courses (preferable a sequence) at the graduate level in the student’s planned area of specialty. (The courses may include MATH 6990 Independent Study)
• Replace one of the three comprehensive examinations above with one in the student’s planned area of specialty.

An exception to one of these sequences may include, but are not limited to, graph theory, collegiate mathematics education, applied mathematics, or statistics.

Substitution proposals for either must be submitted by the advisor in writing to the Graduate Committee at least three months prior to the plan proceeding. The proposal must include a clear description of and rationale for the submission, a timeline for completion, and a list of faculty (minimum of two) who have agreed to be responsible for any courses or exam (construction and grading) under consideration. Any proposals for a substitute comprehensive examination must also include a syllabus for the alternative comprehensive examination.

4. Demonstrate competency in research tools.

Candidates must demonstrate proficiency in mathematical typesetting, presentations, and mathematical literature search tools. In addition, in consultation with the candidate’s advisor, candidates must choose one of the following tools: foreign language or mathematical research software.

5. Complete a teaching practicum including teaching an approved undergraduate mathematics course.

6. Complete a dissertation that is a significant new contribution to mathematics.

And defend the dissertation before the student’s doctoral committee. This requires at least 15 hours of the following course:

• MATH 7300 - Doctoral Dissertation Credits: 1 to 15 hours

7. The following courses may not be included in the required 60 hours.

• MATH 6110 - Mathematical Applications Credits: 3 hours
• MATH 6150 - Intermediate Analysis Credits: 3 hours
• MATH 6160 - Survey of Algebra Credits: 3 hours
• MATH 6170 - Survey of Discrete Mathematics Credits: 3 hours
• STAT 6120 - Data Analysis Credits: 3 hours