# Mathematics Ph.D. Program Requirements

The Ph.D. in mathematics at TCU consists of coursework, examinations, and research, as specified below. In accordance with University policy, a GPA of at least 2.75 must be maintained with no more than two passing grades of "C+" or lower being utilized in satisfying the degree requirement.

**Core Coursework (27 hours): **

Coursework for the Ph.D. must include the following nine core courses:

- MATH 50253 Abstract Algebra I (3 hours)
- MATH 60263 Abstract Algebra II (3 hours)
- MATH 50503 Real Analysis I (3 hours)
- MATH 60503 Real Analysis II (3 hours)
- MATH 60513 Multivariable Analysis (3 hours)
- MATH 60223 Applied Linear Algebra (3 hours)
- MATH 60313 Topology (3 hours)
- MATH 60323 Algebraic Topology I (3 hours)
- MATH 60413 Advanced Complex Analysis (3 hours)

Any or all of these courses may be waived for more advanced students by departmental permission. Additional courses may be taken in addition to these core classes as the student progresses through the program.

**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 MATH 60513, approximately)
- Algebra Exam (based on MATH 50253, MATH 60263, and MATH 60223, approximately)
- Topology (based on MATH 60313 and MATH 60323)
- Complex Analysis Exam (based on MATH 50403 and MATH 60413)

More specific information about the syllabi for the exams is given at https://faculty.tcu.edu/richardson/Prelims/ .

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.

**Research-Specific Coursework **

After passing the preliminary examinations, the student decides on their direction of research and dissertation advisor. Based on the recommendation of the department, the dean appoints a Ph.D. Advisory Committee of at least four members, including the dissertation advisor as chair.

Possible areas of research specialization include Number Theory, Differential Geometry, Global Analysis, K-Theory, Operator Algebras, Algebraic Geometry, Harmonic Analysis, p-adic Groups, and Algebraic Topology.

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 can 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.

**Admission to Candidacy **

The student advances to candidacy after passing the qualifying examination.

**Research **

The student performs research in his/her area of specialization, which leads to a dissertation, if successful.

**Dissertation (at least 12 hours) **

Admission to candidacy is the prerequisite to enrollment in dissertation research, consisting of an original research project directed by a graduate faculty member at TCU. 6 hours of MATH 90980 Dissertation and 6 hours of MATH 90990 Dissertation are required. According to University policy, the time allowed to complete the dissertation is at most six years after advancement to candidacy. Also, in accordance with University 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. For the student to be eligible for the degree, the Ph.D. Advisory Committee must approve the dissertation upon its completion. A final oral examination is required and is open to the public.