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Ph.D. Program in Mathematics

Candidates for the Ph.D. degree must comply with all relevant Graduate School requirements. In addition, the specific steps toward obtaining a Ph.D. degree in Mathematics are as follows:


    This is accomplished through passage of preliminary (prelim) examinations and course work. The Department offers two-semester course sequences in six areas: Algebra, Analysis (real and complex), Applied Mathematics (functional analysis), Numerical Analysis, Probability, and Topology (algebraic and differential). Examinations covering these areas are administered twice per year (each August and January). Each exam covers one semester’s material. Incoming students are expected to pass one exam before February of their first year. Students are required to pass two exams by the end of August before their second year begins, and three exams before their third year begins. In addition, before passing into candidacy, a student needs to pass four semesters of prelim coursework, disjoint from the three semesters on whose content he or she passed exam segments. Click here for details.


    Within one year of passing his or her third prelim exam, a student must declare an area of specialization and identify an academic adviser, in consultation with whom he or she will select an advisory committee of three faculty members to oversee the candidacy exam, and a committee of four to oversee the dissertation. Students are guided in this process by the Graduate Adviser, who ensures that all steps conform to Graduate School requirements.


    By August of a student’s third year in the Ph.D. program, a student must successfully complete an oral candidacy exam in his or her chosen area of specialization. The topics of this exam are set by the advisory committee in consultation with the student. In order to pass the exam, a student must demonstrate to their advisory committee that he or she has (a) acquired adequate content knowledge in the area of specialization, (b) demonstrated the ability to interpret existing research literature and devise a program of original research, and (c) effectively communicate mathematics in English.


    In order to pass into candidacy, a student must have completed all preliminary exams and coursework, obtained the agreement of faculty members to serve on candidacy and dissertation committees, successfully passed an oral candidacy exam, written a statement of proposed research approved by the advisory committee, and complied with any other applicable Graduate School requirements.


    The dissertation is the most important part of the Ph.D. program. It consists of original research in mathematics performed by the student in consultation with an academic supervisor. A student is expected to complete the dissertation within three years of passing the candidacy exam.

A student wishing an exception to any of the regulations above must first consult with the Graduate Adviser and then, if circumstances warrant, make a formal appeal to the Administrative Subcommittee of the Graduate Studies Committee (ASGSC). The ASGSC is the final arbiter in all such matters.

M.A. Programs in Mathematics

Currently the Mathematics Department only admits Master’s students into its Actuarial program. Please visit Actuarial Studies for information on this program.

Course Offerings

The Department offers a broad assortment of graduate courses. Every year, the department offers a two-semester course sequence in each of the six areas covered in its preliminary examinations, namely algebra, analysis, methods of applied mathematics (including functional analysis), numerical analysis and computation, probability, and topology. Every few years, the department offers courses covering standard topics integral to its research areas, so that all students have an opportunity to take classes covering this material during their time in the Ph.D. program. These courses include algebraic topology, approximation theory, differential geometry, experimental design, geometric topology. harmonic analysis, mathematical finance, multivariate statistical analysis, number theory, numerical solution of differential equations, and ordinary and partial differential equations. . In addition, the department offers a wide assortment of advanced topics courses in response to faculty interest and student demand. Recent such offerings include:

  • Algebraic number theory
  • Algebraic geometry
  • Applied statistics
  • Approximation theory
  • Banach spaces
  • Calculus of variations
  • Commutative ring theory
  • Computational modeling
  • Dynamical systems
  • Ergodic theory
  • Group representations
  • Group theory
  • Harmonic analysis
  • Hilbert space methods in P.D.E.
  • Iterative methods for linear algebra
  • Knot theory
  • Lie groups and algebras
  • Low dimensional topology
  • Nonlinear elliptic P.D.E.
  • Markov processes
  • Mathematical modeling
  • Mathematical physics
  • Minimal surfaces
  • Multiscale problems
  • Numerical treatment of differential equations
  • Numerical analysis for PDEs
  • Numerical methods for conservation laws
  • Quantum mechanics
  • Quasiconformal mappings
  • Statistical mechanics
  • Stochastic processes
  • Ricci flow
  • Riemann surfaces
  • Rings and modules
  • Theory of algebraic groups
  • Topological quantum field theory
  • Wavelets

Finally, individualized reading courses (conference courses) are encouraged for those students wishing to study a particular faculty member’s area of expertise. Other departments on campus offer courses with various mathematical content such as operations research, optimization theory, optimal control theory, engineering mechanics, statistics, et cetera.

Colloquia, Lecture Series, Seminars

The Department supports several venues by which leading national and international mathematicians visit for various lengths of time and give presentations about their research interests. These venues include a weekly colloquium and weekly seminars in analysis and partial differential equations, applied and computational mathematics, Banach spaces, differential and algebraic geometry, dynamical systems, mathematical finance, mathematical physics, number theory, and topology. Please see our Seminar Calendar for more details. Additionally, the Department supports a Distinguished Women in Mathematics Lecture Series. The Department frequently invites distinguished lecturers for longer visits during which they may give a series of talks and interact with local faculty and students. These visits are often supported by NSF Research and Training Grants within the Department, in the areas of Analysis/Applied Mathematics, Geometry, and Topology.

Computer Facilities

The Department of Mathematics maintains a state-of-the-art computer network to facilitate research and departmental administration. This is predominantly a UNIX-based system consisting of Linux PC's and servers, with a few Windows XP workstations and a number of Apple Macintosh systems. Every graduate student office contains at least one Linux PC. Within the department, there are five computer labs available for general use, including one 40-seat instructional laboratory for its undergraduate mathematics program. The Department also operates a 46 node cluster for research computing, locally developed mathematical software, and our internationally recognized Mathematical Physics Electronic Journal (MPEJ) and preprint archive (mp_arc). The most important element of the departmental computer operation is the ready availability of innovative mathematical and instructional software and free computer resources that create an environment conducive to experimentation and exploration by faculty and students alike.

The Texas Advanced Computing Center (TACC) is a research center at UT reporting to the Office of the Vice President for Research. TACC provides advanced computing resources & services to enable computationally-intensive research and conducts research & development to enhance the capabilities of these resources. TACC offers a number of high performance computing facilities including a world leading 500-teraflops system. The Computational Visualization Center is available nearby for visualization of scientific and computational data, as well as virtual reality simulations.

Financial Support

It is the intent of the Mathematics Department to provide five years (ten fall/spring semesters) of financial support to all Ph.D. students who are making satisfactory progress towards their degrees. We fully expect to be able to furnish such support, except possibly in cases of severe financial crisis. A sixth year may be available when circumstances warrant. We also provide financial aid to some Actuarial Master's degree students, when sufficient funds are available.

Financial support takes three forms: (1) Teaching Assistantships and Assistant Instructorships; (2) Graduate Research Assistantships; and (3) Fellowships. All support is contingent upon meeting the scholastic requirements for eligibility established by the Graduate School, providing satisfactory service to the Department, and complying with all applicable University policies. Additionally, please note that final decisions regarding reappointment also are dependent upon available resources.

Note: International students must speak fluent English and pass an oral English assessment exam administered by the University before they can be supported as Teaching Assistants or Assistant Instructors.

Teaching Assistantships and Assistant Instructorships

The most common form of financial support, especially for beginning students, is appointment as a Teaching Assistant (TA). TA duties vary from course to course. A typical assignment as a Calculus TA would involve attending the instructor’s lectures (three days per week), conducting discussion sessions (two days per week), holding office hours, and helping to proctor and sometimes grade exams. (Precise information on workload policies may be found here.) Advanced students are sometimes appointed as Assistant Instructors (AIs).

Graduate Research Assistantships

Many of our faculty hold active NSF or NSA grants that contain funding for graduate students working in their research specialty. Advanced students (those who have chosen a research supervisor) are supported as Graduate Research Assistants (GRAs) by these grants, without teaching duties, as funding permits.


The Department awards fellowship funds to academically deserving students. Like GRAs, fellowship recipients do not have teaching duties. The Department awards approximately eight semesters of fellowship support each year, drawn from various bequests and endowments (see below). Beginning students receive summer support whenever possible, to facilitate their studying for preliminary examinations. Additionally, several semesters of support are available annually from the Department’s NSF Research and Training Grant (RTGs) for students working in the area of Geometry/Topology.

Fellowship bequests and endowments:

  • Edward Louis and Alice Laidman Dodd Fellowship
  • Arthur LeFevre, Sr., Scholarship in Mathematics
  • Regents Endowed Graduate Fellowships in Mathematics
  • David Bruton, Jr. Graduate Fellowships in Mathematics
  • Professor and Mrs. Hubert S. Wall Endowed Presidential Fellowship
  • Charles Rubert Scholarship
  • John L. and Anne Crawford Endowed Presidential Scholarship