# PhD Program

## Doctor of Philosophy in Mathematics

The TCU Doctor of Philosophy (Ph.D.) degree in mathematics is designed for students who desire to do original research in pure mathematics and who desire to teach at the college/university level. After each doctoral student acquires a strong mathematical foundation in algebra, real and complex analysis, and topology, that student concentrates on one of the research specialties of the TCU mathematics faculty. These fields of mathematics include real and complex analysis, harmonic analysis, functional analysis, algebraic geometry, differential geometry, number theory, topology, global analysis, representation theory, operator algebras, and K-theory. Award-winning professors, faculty expertise in a wide range of research areas, and abundant opportunities for one-to-one student-to-faculty interaction characterize our program.

**Program Details**

The Ph.D. program consists of coursework, examinations and research, as specified below. In accordance with TCU policy, a grade point average of at least 2.75 must be maintained with no more than two grades of C+ or lower being utilized in satisfying the degree requirement. Also, only five courses may be taken at the 50000-level; other classes must be at the 60000-level or higher.

Core Coursework: The following courses are required:

- MATH 50253 Abstract Algebra I
- MATH 60263 Abstract Algebra II
- MATH 50503 Real Analysis I
- MATH 60513 Real Analysis II
- MATH 60223 Applied Linear Algebra
- MATH 60313 Topology
- MATH 60323 Algebraic Topology I
- MATH 60413 Advanced Complex Analysis
- MATH 60523 Measure Theory

Any of these courses may be waived for more advanced students by departmental permission.

**Preliminary Examinations:** The program requires substantial training in algebra, real analysis, topology and complex analysis. The student must pass three of the following four preliminary written exams:

- Real Analysis Exam (based on MATH 50503 and 60513)
- Algebra Exam (based on MATH 30224, 50253 and 60263)
- Topology Exam (based on MATH 60313 and 60323)
- Complex Analysis Exam (based on MATH 50403 and 60413)

The student must pass the Real Analysis Exam, the Algebra Exam and either the Topology Exam or the Complex Analysis Exam. These exams are administered twice each year and must be passed by the end of the sixth semester. The department maintains exam syllabi and practice problems to help students prepare for the exams.

**Research-Specific Coursework:** After passing the preliminary examinations, the student decides on his or her direction of research and thesis adviser. Based on the recommendation of the department, the dean appoints an advisory committee of at least three members, including the thesis adviser as chair. Possible areas of research specialization include real analysis, complex analysis, functional analysis, algebraic geometry, differential geometry, number theory, topology, global analysis, K-theory, operator algebras and representation theory.

Students must also take at least one semester of three hours of MATH 80880 Graduate Student Seminar, providing training in the oral presentation of research-level mathematics.

**Qualifying Examination: **The student and advisory committee agree on a detailed plan of study to prepare the student for mathematical research. They decide on a syllabus of qualifying topics; after due preparation, the student takes the oral qualifying exam on these topics, administered by the advisory committee. In accordance with university requirements, the exam may only be taken after passing the preliminary examinations and not before the second semester of the second year. If the exam is not passed, at most one re-examination is allowed. The student advances to candidacy upon passing the qualifying examination.

**Dissertation:** Advancement to candidacy is the prerequisite to enrollment in dissertation research, consisting of an original research project directed by a graduate faculty member at TCU. Six hours of MATH 90980 Dissertation and six hours of MATH 90990 Dissertation are required. According to TCU policy, the time allowed to complete the dissertation is at most six years after advancement to candidacy. Also in accordance with TCU rules, the student must submit an Intent to Graduate form at the beginning of the last semester, for which there is a non-refundable fee. At the completion of the dissertation, a final oral examination is required, which is open to the public.

For more information, contact

Ken Richardson

Director of Graduate Studies in Mathematics

TCU Box 298900

Fort Worth, TX 76129

817-257-6128

k.richardson@tcu.edu