Ph.D. Program - University of Houston
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Ph.D. Program

Requirements Outline

The Ph.D. degree is a research degree and the principal requirement is that a student writes an original research thesis. The thesis is produced under the supervision of a faculty member and is examined by a committee of three departmental faculty and an outside expert. To qualify to write a thesis, a candidate for a Ph.D. in mathematics first must pass three Preliminary Examinations.

It is recommended that Ph.D. candidates discuss possible research opportunities with the Director of Graduate Studies and/or faculty members soon after they enter the Ph.D. Program. Entering students should outline an appropriate sequence of courses to learn the essential material for pursuing their research interests.

After a student has passed the Preliminary Examinations they must choose an advisor from the Mathematics Department faculty. A candidate's thesis usually is developed and written with the guidance of this advisor who will later chair the thesis defense committee.

The time required to obtain a Ph.D. degree varies a lot. The department does not support graduate students as Teaching Assistants for more than five academic years.

Ph.D. Degree Requirements

The requirements that must be satisfied for a candidate to receive a Ph.D. include:

  1. The candidate must pass Preliminary Examinations.
  2. The candidate must obtain a grade of B or better in at least 24 semester credit hours of courses in the Mathematics Ph.D. program. Students should take doctoral research classes MATH 8x98 (where “x” is the number of credit hours) while conducting thesis research. Students must register for. the course MATH 8x99 “Doctoral Dissertation” in the semester when they intend to graduate
  3. After passing all three Preliminary Examinations the candidate is subject to Annual Performance Review (APR). The APR evaluates research progress of the candidate. The APR is conducted in oral or written form by a committee consisting of at least two faculty members of the Mathematics Department. The APR committee is chaired by the candidate’s advisor. Candidates failing the APR are subject to termination from the Ph.D. program.
  4. The candidate must be in residence, and take 9 semester credit hours of courses, in two consecutive long semesters.
  5. The candidate must write a Doctoral Dissertation with the guidance of an advisor who is a regular faculty member of the Mathematics Department.
  6. The candidate must defend their Dissertation in a public examination by a thesis committee consisting of at least 4 members, three of whom are faculty members in the Mathematics  Department and at least one member outside UH Mathematics Department.
  7. A copy of the dissertation, approved by all the members of the thesis committee is transmitted to, and approved by, the office of the Dean of the College of Natural Science and Mathematics.


Course Selection:

  • Information about courses may be found at this link.
  • Students can discuss advisor selection process with the Director of Graduate Studies.
  • The above is only an outline of the primary requirements for the degree. The Director of Graduate Studies and others can provide more detailed information about conditions. The college and the university may have further requirements as listed at College and websites.
  • PhD students can take topics classes at Rice University, UT Health, UTMB, or Baylor College of Medicine. Students must submit the Inter-Institutional Course Registration Form to the Graduate Director for approval. Taking an outside class must be essential for the completion of graduate degree. Thus, students must obtain a prior approval of their PhD avisor (signature on the form).


Teaching Opportunities for Ph.D. Students:

Enhance Teaching Experience

Since teaching skill is important for PhD students, the department develops a program to improve students' teaching skill.

For any PhD student before finishing the PhD degree, the department may arrange him/her to teach regular courses. In this case, some senior professors will be mentors to improve the student's teaching skill.
If any student is interested in this program, the student should make request to the Director of Graduate Studies by the end of the fall semester about one and half year before graduation.
As a condition, a student should have experiences of teaching Calculus recitation class with reasonable teaching evaluation. For an international student, by Texas law, the student must pass the English SPEAK test or its equivalence.


Preliminary Examinations:

Background and Purpose

The Preliminary Examination is the final step in assessing the student’s ability and appropriate mathematical background to undertake a program of supervised research and study leading to a Ph.D. in Mathematics. Students who have completed their Master's degree in Mathematics may often be ready to take the Preliminary Examination without further course study.

Preliminary Examinations are three-hour, closed book written examinations that are given in each of the topics listed below. The questions in the examination emphasize problem solving skills and mathematical ability as opposed to rote memorization.



Preliminary Examinations are offered three times a year in May, August and January.

Students who receive support from the Department of Mathematics are expected to pass the Preliminary Examination according to the rules below. For non-supported students, the University rules apply.

All students are supposed to pass three Preliminary Examinations before the beginning of their third year in the Ph.D. program.


Administration of the Examination

The following rules apply:

  1. Students must pass three Preliminary Examinations from the different topic groups listed below.
  2. At least one out of the three Preliminary Examinations must be a core sequence. Core sequences are:
• MATH 6320;6321: Theory of Functions of a Real Variable
• MATH 6360;6361: Applicable Analysis
• MATH 6370;6371: Numerical Analysis
3. At least one of the three Preliminary Examinations must be from the following topic groups: Algebra, Topology, Analysis.
4. The student is subject to termination from the Ph.D. program in case of failing the same Preliminary Examination twice.
5. The total number of attempts for all Preliminary Examinations is five.

Examination topic groups:
MATH 6302: Modern Algebra
MATH 6342: Topology
MATH 63xx: Geometry
MATH 6320;6321: Theory of Functions of a Real Variable
MATH 6322: Theory of functions of a Complex Variable
Applied Mathematics
MATH 6360;6361: Applicable Analysis
Probability and Statistics
MATH 6382: Probability
MATH 6383: Mathematical Statistics
Computational Mathematics
MATH 6366: Optimization and Variational Methods
MATH 6370;6371: Numerical Analysis


Review information for the preliminary written examinations:

Applicable Analysis
Modern Algebra
Real Variables
Complex Variables
Probability & Statistics
Numerical Analysis

Additional problems from past preliminary exams:

Applicable Analysis
Modern Algebra       
Real Variables
Complex Variables
Probability & Statistics    Sample Problems
Optimization Sample Problems
Numerical Analysis Sample Problems
Refer to syllabus, schedule, and sample examinations for guidance in preparing for the examinations.