## Ph.D. in Pure Mathematics

**Departmental Requirements for Ph.D. in Mathematics:****Pure Mathematics**

The Ph.D. degree with emphasis in pure mathematics is intended to prepare a student for a career in university and industrial research and college instruction. Recipients of the Ph.D degree in pure mathematics are expected to have significant breadth across the core areas of mathematics as well as skills to undertake independent and original research in a particular area.

**Credit requirements:** A total of 90 hours above the B.S. degree or 60 hours above the M.S. is required, including 15-24 credit hours for the Ph.D. dissertation.

**Core requirements:** All candidates for the Ph.D. in pure mathematics are required to complete 24 hours of core courses from the following list. A minimum GPA of 3.0 must be maintained in core courses.

- Real Analysis I and II (MATH 5143 and 5153)
- Complex Analysis I and II (MATH 5283 and 5293)
- Algebra I and II (MATH 5613 and 5623)
- Geometric Topology (MATH 5313) and Algebraic Topology I (MATH 6323)

**Additional courses:** In addition to the core course requirements, every plan of study must contain at least 12 hours of graduate courses in the mathematical sciences (mathematics, statistics, or computer science).

Comprehensive exams: A Ph.D. student in pure mathematics must select three of the four areas from the core requirements listed above and for each must pass a comprehensive exam covering material in that area. The mathematics comprehensive exams are administered in January, May/June, and August. See the separate document covering timetables and requirements. Students who have passed two comprehensive exams may, with the approval of the student's advisor, substitute a minor thesis (as described below) for one of the comprehensive exams.

Minor thesis: The minor thesis must be a creative, written work of research or exposition on a topic of mathematical content. Further requirements are given in a separate document.

Language requirement: Candidates for the Ph.D. in pure mathematics must pass an examination demonstrating reading knowledge of one foreign language, usually French, German, or Russian, before they take the qualifying exam. Other languages may be substituted subject to the recommendation of the student's committee and the approval of the Graduate Committee, including a working ability with a computer language or symbolic computation meeting the satisfaction of the student's committee.

Qualifying exam: The student must pass an oral qualifying exam over the area of specialization for the Ph.D. dissertation. This exam covers material on a reading list presented to the student by the advisory committee. Its purpose is to test the student's readiness to begin dissertation work.

Dissertation proposal: An outline of the proposed dissertation research must be presented to the student's advisory committee for approval.

Dissertation: A dissertation must be written according to Graduate College guidelines. The dissertation consists of an original research contribution in pure or applied mathematics. It must be prepared according to the style required of doctoral dissertations and defended.

Graduate College requirements: All requirements listed in the university catalog must be satisfied.