DOCTOR OF PHILOSOPHY IN MATHEMATICS

Program for the PhD degree

Prerequisites: B.A. or B.S., with a mathematics major or 24 semester hours of mathematics, including courses in elementary calculus of one and several variables, linear algebra, and at least 8 hours of departmentally approved advanced undergraduate courses in mathematics. Students deficient in any area of preparation may be required to take the necessary course work during the first year of graduate study. Scores on the Graduate Record Examinations are used as an aid in the overall evaluation of the applicant's graduate potential.

Application Process: An application package consists of a complete application form (available online at www.cse.tcu.edu), two sets of official transcripts from all college work, GRE scores, and three or more recommendation letters. A non-refundable $50 application fee in the form of a personal check (drawn on U.S. bank), money order, or cashier's check made payable to TCU is required with all applications. International student should contact the graduate director before applying and refer to www.cse.tcu.edu for further required forms. Mail all materials to:

College of Science & Engineering
Attn: Graduate Studies
TCU Box 298960
Fort Worth, TX 76129

Graduate program: The PhD program consists of 72 semester-hours of graduate study beyond the Baccalaureate or 39 semester-hours beyond the Master's degree, as specified below. In accordance with university 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. The program consists of the following:

I. Core Coursework (27 hours): : Coursework for the Ph.D. degree must include the following nine core courses:

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

An advanced student may be waived from any or all of these required courses by passing the corresponding Ph.D. preliminary exam(s) or otherwise showing proficiency in the material, as determined by the department.

II. Preliminary Examinations:The program requires substantial training in algebra, real analysis, topology, and complex analysis. The student must pass 3 of the following 4 preliminary written exams:

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

These exams are administered twice each year and must be passed by the end of the first semester of the third year.

III. Research-Specific Coursework: After passing the preliminary examinations, the student decides on his/her direction of research and thesis advisor. Based on the recommendation of the department, the dean appoints an advisory committee of at least three members, including the thesis 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.

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. Six hours of MATH 90980 Dissertation and six 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. At the completion of the dissertation, a final oral examination is required, which is open to the public.