2021-22 Graduate Catalog

Mathematics, Ph.D.


The Ph.D. in mathematics 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 50253Abstract Algebra I


MATH 60263Abstract Algebra II


MATH 50503Real Analysis I


MATH 60503Real Analysis II


MATH 60513Multivariable Analysis


MATH 60223Applied Linear Algebra


MATH 60313Topology


MATH 60323Algebraic Topology I


MATH 60413Advanced Complex Analysis



MATH 80880Graduate Student Seminar


MATH 90980Dissertation


MATH 90990Dissertation


Any or all 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:

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 real analysis, complex analysis, functional analysis, algebraic geometry, differential geometry, number theory, topology, global analysis and K-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 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.


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.