Ph.D. Program in Mathematics

The Doctor of Philosophy (Ph.D.) is a research degree designed to prepare students to discover, integrate, and apply knowledge as well as to communicate and disseminate it.  Students are required to enroll in 9 credit-hours per semester.  

While the Ph.D. degree formally requires a minimum of 30 semester hours of advanced coursework, including a minimum of six dissertation hours, high-performing students normally require five to six years of full-time enrollment (9-12 semesters) to complete requirements of the Ph.D. degree.  It is quite exceptional (applies to less than 1% of students, and only in unusual circumstances) that a student is able to complete the requirements in less than 9 semesters.

In particular, completion of a research program leading to a dissertation worthy of the Ph.D. degree usually requires at least three years of work, taking the form both of conference courses with an agreed academic advisor (before formal passage to candidacy) and as dissertation hours (post-candidacy).

While the overall degree generally requires five to six years, the distribution of the coursework and dissertation components of the degree varies considerably.   Among other factors, it depends on the mathematical preparation of the student on entry.

The following list lays out the kinds of coursework required of all Ph.D. students:

• Required coursework: Prelim courses
• Elective coursework: Topics courses and graduate courses offered by other departments
• Conference courses (to establish an academic advisor, prepare for candidacy and embark on a research program)
• Dissertation hours (minimum 6 hours)

The prelim courses will be completed early in the degree (in 1-3 years); conference courses will begin in year 1 or 2 and continue until formal passage to candidacy (in years 2-4); elective coursework will begin in year 1 or 2 and continue throughout the degree; dissertation hours will begin after formal passage to candidacy and will continue for the remainder of the degree.

Note that even after they have demonstrated their broad competence, students are expected to deepen and further broaden their knowledge by taking “topics” courses, as detailed in Section V below.  Students enroll in 9 hours of coursework per semester, consisting of prelim and other graduate courses, conference courses, and dissertation hours.

The GSC specifies the coursework PhD students must complete, the qualifying examinations Ph.D. student must pass, the conditions under which Ph.D. students must retake all or part of an examination, and the procedures Ph.D. students must follow in developing a dissertation proposal. Each student seeking the Ph.D. must be admitted to candidacy on the recommendation of the GSC. Students may not register for the dissertation course until they are admitted to candidacy, and completion of course work does not in itself constitute admission. The student must register for at least six hours of dissertation courses in order to graduate.  A dissertation is required of every candidate.

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:

1.        Demonstration of broad competence

This is accomplished through passage of Preliminary (“prelim”) examinations and coursework.  The department offers twelve prelim courses, which are usually presented as six two-semester sequences.  The twelve courses are: 

• Algebra (parts I and II);
• Analysis (Real and Complex);
• Methods of Applied Mathematics (parts I and II; these courses cover functional analysis, harmonic analysis, and other analytic methods);
• Numerical Analysis (part I, covering linear-algebraic topics, and part II, covering differential equations); Probability (parts I and II);
• Topology (Algebraic and Differential).

In addition to the courses, exams are offered twice yearly (in August and January) in the twelve prelim areas.  Students can fulfill a part of their prelim requirement by passing a course with a grade of B or higher, or by passing the corresponding exam.  Students are required to pass at least 7 prelims in distinct areas, at least 3 of them by exam. Details of the prelim policy will be explained in section VII.

2.      Identification of an academic advisor

You should identify an area of specialization and an academic advisor with one year of passing the third prelim exam.  To do so, you should identify a potential advisor, (a member of the GSC—see https://catalog.utexas.edu/graduate/fields-of-study/natural-sciences/mathematics/) and take a Conference Course under the advisor’s supervision - and subsequent conference courses if both you and instructor consider it likely that the faculty member will become your academic advisor.  Taking conference courses need not be delayed till the completion of prelim requirements; often the second or third semester is an appropriate time for a first conference course. Once an academic advisor has been identified, and has agreed to serve in this role, you will work with the advisor to select an advisory committee of three faculty members to oversee the candidacy exam; and, after passing the candidacy exam, a committee of four to oversee the dissertation. 

You are guided in this process by the Graduate Advisor, who ensures that all steps conform to Graduate School requirements.

3.     Oral candidacy exam

Having identified an area of specialization and an academic advisor, you must successfully complete an oral candidacy exam in the 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, you must demonstrate to their advisory committee:

• acquired adequate content knowledge in the area of specialization,
• the ability to interpret existing research literature and devise a program of original research, and
• the ability to effectively communicate mathematics in English.

The candidacy exam takes the form of a lecture presentation by the student, followed by questions from the committee.  While some candidacy lectures present original research by the students, this is not a requirement; presentation of existing material relevant to the area of specialization and the student’s planned research may also be appropriate, as determined by the academic advisor and candidacy committee.

Students become eligible for oral candidacy exams after passing 5 distinct prelims, at least 3 by exam. Students are expected to complete their candidacy exam by August of their third year.

 4.    Formal admission to Ph.D. candidacy

To advance to doctoral candidacy, you must have:

• passed the oral candidacy exam (as well as the prelim eligibility requirement as described above);
• obtained the agreement of faculty members to serve on the dissertation committee (including one external member, meaning a qualified faculty member, either from another university, or from UT Austin but not a member of the Mathematics GSC); and
• submitted a formal candidacy application to the Graduate School. The formal application includes a statement of proposed research which must be approved by the student’s academic advisor, chair of the GSC, Graduate Advisor, and Graduate School. Students must comply with any other applicable Graduate School requirements.

5.     Completion of Prelim requirements

If you have any remaining prelim requirements, these must be completed within a year of formal admission to Ph.D. candidacy.  The requirements are 7 prelim courses and exams in distinct areas, passed via courses or exams, but at least 3 of them by exam.

6.    Completion of dissertation

The research written up in the dissertation is the most important part of the Ph.D. program.  This, fundamentally, is what a Ph.D. is about.  It consists of original research in mathematics performed by the student with regular input from his or her academic advisor.  You are expected to complete the dissertation within three years of passing the candidacy exam.  While writing the dissertation, students must be continuously registered in the dissertation course during the Fall and Spring semesters.

7.   Exceptions

A student wishing an exception to be made to any of the regulations above must first consult with the Graduate Advisor, 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

The Mathematics Department provides expert training in actuarial mathematics, both at the Bachelors and Masters levels.   The MA program with a focus in Actuarial Mathematics is a professional program focused on providing the academic preparation for entry-level actuarial jobs in the United States. 

The program is funded by the State of Texas and strongly supported financially by actuarial employers throughout the United States, who hope to recruit qualified graduates of the program.

The program is intended for students needing to learn the academic content of SOA and CAS Exams IFM, LTAM, STAM, 3F, 3L and 4.  Possible schedules for graduate students vary enormously, depending on exam status upon arrival, but the program often takes 4 semesters to complete.

The MA degree requires completion of at least 33 semester-credit-hours of coursework (11 three-credit-hour courses) to include M 389U, M 389V, M 389W, M 389J, and M 389P.   Up to nine hours of upper-division undergraduate coursework may be used to satisfy program requirements, with no more than six of those nine in a single subject.  At least 18 semester hours must be completed in Mathematics coursework, and at least six hours must be completed in supporting work, or coursework offered outside of Mathematics.  Graduate program requirements vary by individual and are determined based upon on each student’s exam status on arrival.

Some students in this program are employed as Teaching Assistants (TAs).  These students are required to take the 3-hour course 398T: Supervised Teaching in Mathematics (these hours do not contribute to the 33 required).

Further information about this program can be found at https://sites.cns.utexas.edu/actuarial-science/ma-actuarial-focus.

Course Offerings

The Department offers a broad assortment of graduate courses, of the following types:

• Prelim courses (comprehensive materials)
• Topics courses (specialized materials)
• Conference (i.e., individual reading) courses
• Actuarial courses

1.    Prelim courses

Every year, the department offers a two-semester course sequence in each of the six areas covered in its preliminary examinations, namely Algebra (with part I covering groups, rings and module, part II fields and Galois theory);  Analysis (Real and Complex); Methods of Applied Mathematics (with part I focused on functional analysis and part II on Fourier analysis and methods useful for PDE); Numerical Analysis (with part I about linear algebraic methods, part II about numerical approaches to differential equations); Probability (with part I covering foundations of measure-theoretic probability and discrete-time random processes, part II on continuous-time random processes); and Topology (part I Algebraic Topology, part II Differential Topology).  The syllabi are largely standard, and are available at https://www.ma.utexas.edu/academics/courses/course-syllabi.

2.   Topics courses

In addition, the department offers graduate “topics” courses (typically 14-16 per year).  Many topics courses cover standard material, integral to the research areas represented in the department.  For instance, a two-semester sequence on Partial Differential Equations is offered most years, while a number of courses in geometry and topology (for example, Algebraic Geometry, Riemannian Geometry) are repeated every few years so that all students have an opportunity to them.  Further topics courses cover advanced and cutting-edge material, in response to faculty interest and student demand.

Recent such offerings include:

3.    Conference courses

Conference courses, that is, reading courses for one or a small number of students working with a supervisor, are encouraged as part of the processes of identifying an area of specialization and academic advisor, training and embarking on research in that area, and preparing for candidacy. (After candidacy, they are supplanted by dissertation hours.)

Students are discouraged from taking conference courses purely for the purpose of learning material if this is not likely to lead to the establishment of an academic supervising relationship.

4.   Offerings from other departments

Other departments on campus offer courses with various mathematical content, such as computer science, operations research, optimization theory, optimal control theory, engineering mechanics, statistics, etc. Students may select such courses when they are relevant to their area of specialization, but are advised to consider taking them on a credit/no credit basis.

5.   Actuarial courses

Courses in the following topics are regularly offered:

• Introduction to financial mathematics for actuaries
• Theory of interest
• Probabilistic models with actuarial applications
• Actuarial contingent payments I
• Actuarial contingent payments II
• Actuarial case studies
• Actuarial statistical estimates

Seminars & other Lectures

Seminars and other lectures are advertised on the departmental Seminar Calendar.  Regular seminar series include:

• Analysis
• Geometry
• Topology
• Groups and Dynamics
• Mathematical Physics
• Geometry and String Theory
• Mathematical Finance
• Numerical Analysis

Students are also welcome to attend seminars elsewhere in UT, for instance at the Oden Institute for Computational Science, Engineering and Mathematics and in the Physics Department.

These seminars feature researchers ranging from leading figures in the national and international mathematical scene to postdocs and graduate students with exciting results to report.

There is also a regular colloquium, that is, a series of lectures aimed at a general mathematical audience rather than at specialists in a particular field.

There are also junior seminars, run by and for Ph.D. students, including

• Junior Analysis
• Junior Topology
• Junior Geometry
• Junior Geometry and String Theory
• Sophex (a friendly, introductory seminar series run by and for first-year Ph.D. students)

Ph.D. students are strongly urged to participate and present regularly at junior seminars.  Learning to present mathematics effectively is an important skill, but one that takes practice. Junior seminars provide a friendly environment in which to acquire this skill, as well as to hear about interesting topics and learn about what your peers are working on.

Students are also encouraged to participate in the ‘senior’ seminars, asking questions without worrying about whether they are too ‘elementary’.  Speakers are generally very open to questions from students learning about their field.   You will probably find that senior seminars become gradually more useful as you develop from a beginner in a field to an expert in your own right. Once you have a core of knowledge in a particular field, seminars are a very effective way of adding to that core and hearing about recent developments.

Other student-run seminars include:

• Learning seminars (recent topics include algebraic geometry, mapping class groups)
• STEM inequity seminar

The department supports a long-running Distinguished Women in Mathematics Lecture Series; since 2020 it has been complemented by the Distinguished Mathematicians of Color Series.  Both provide opportunities for Ph.D. students to meet the speakers and learn about their experiences

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.