Doctor of Philosophy in Mathematics (Ph.D.) - University of Houston
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Doctor of Philosophy in Mathematics (Ph.D.)

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.

Priority Deadline: Ph.D. with Teaching Assistantship (both US and International): December 1st (Fall)
Each fall semester we have approximately 15—20 TA positions. Review of applications will start shortly after December 1st and offers will be made on a continuous basis until all TA positions are filled. We usually give the first round of offers before December 25th, the second round of offers in January and the third round of offers in February.

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, Fall followed by Spring. Alternatively, the candidate must be in residence and take a full load in consecutive Spring, Summer, and Fall terms.
  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.
  8. NSM Deadlines & Academic Calendar: This link provides deadlines for the submission of Dissertations.
  9. *The Graduate Record Examination (GRE) is waived for the Ph.D program within the Department of Mathematics.
  10. International students can not exclusively register for online courses.


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).
  • Course Selection Requests: Please contact the Director for Instructional Support and Coordination <> for more information.


Teaching Opportunities for Ph.D. Students:

Since teaching is extremely important for PhD students, the department has a program to enhance students' teaching skills. PhD students, prior to finishing the PhD degree, may be allowed to teach regular courses. In this case, some senior professors will be mentors/advisors to help assist students with this task. 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.

All PhD applicants who submit their complete application before the appropriate deadline are automatically considered for Teaching Assistantship.

Please contact the Director for Instructional Support and Coordination for more information about course selection requests.


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 usually offered twice a year: at the end of the Fall and Spring semesters.

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 topics: Probability, Algebra, Topology group, Analysis group.
4. The total number of attempts for all Preliminary Examinations is five.

Examination topic groups:
MATH 6302: Modern Algebra
MATH 6342: Topology
MATH 7352: Riemannian Geometry
MATH 6320;6321: Theory of Functions of a Real Variable
MATH 6322: Theory of functions of a Complex Variable
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
Sample exam
Sample exam
Numerical Analysis

Additional problems from past preliminary exams:

Applicable Analysis
Modern Algebra
Real Variables
Complex Variables
Probability Sample Problems
Statistics Sample Problems
Optimization Sample Problems
Numerical Analysis Sample Problems

All preliminary exams are based on the content of the corresponding course. Please contact the instructor who taught the corresponding course most recently to obtain the up-to-date information.