Mathematics Graduate Program

For the Ph.D. program, a minimum of 72 acceptable credits is required, with at least 36 of these earned in residence. (See the GCH for details regarding credit requirements.) At least 42 credits must be in formal courses (not research); 18 of the 42 must be in the core courses listed below. In addition, at least 36 must be in 5000 or 6000 level mathematics courses excluding Math 5900, 5910, 5920, 5990 and 6990. There is also a 1 credit seminar requirement which is satisfied by taking both Math 5910 and Math 5920.

The student is also required to take at least 3 credits of Math 6990, Research in Mathematics.

In addition to the course work, the Ph.D. student must pass four written qualifying examinations and an oral preliminary examination, prepare a dissertation, and pass an oral final examination, which is usually limited to the defense of the dissertation. These requirements are described in subsequent

Core course requirements

The core course requirements for the Ph.D. in Mathematics are:

1. At least 6 core courses.
2. At least 3 core courses must be in Algebra, Analysis or Discrete Mathematics.
3. The core courses must include Math 5040 and Math 5150.

Core Courses (for more details see Course Catalog)

• Algebra:
  • Math 5040: Abstract Algebra I
  • Math 5050: Abstract Algebra II
  • Math 5100: Linear Algebra
• Analysis:
  • Math 5150: Real Analysis I
  • Math 5160: Real Analysis II
  • Math 5110: Complex Analysis
• Discrete Mathematics:
  • Math 5670: Graph Theory
  • Math 5680: Enumerative Combinatorics
  • Math 5660: Discrete Optimization
• Applied Mathematics:
  • Math 5190: Methods of Applied Mathematics I
  • Math 5200: Methods of Applied Mathematics II
  • Math 5650: Continuous Optimization
• Numerical Analysis:
  • Math 5610: Numerical Analysis I
  • Math 5620: Numerical Analysis II
  • Math 5170: Finite Difference Methods

A grade of B or better must be earned in each core course used to satisfy the requirements in this section. A deficiency may be made up any one of the following methods:

1. Retaking the course for credit and earning a B or better.
2. Retaking the final examination of the course and earning a B or better (permission of the instructor giving the final is required).
3. A pass on the associated qualifying examination, if such examination exists. Students are strongly encouraged to consult their advisor prior to deciding which core courses to take, since certain core course combinations may not be suitable for certain areas of research.

Qualifying examinations

A Ph.D. student in the Mathematics Program must pass four written Qualifying Examinations, including:

1. at least two in Algebra and/or Analysis, and
2. at least two in one area.

The 10 qualifying examinations are (listed as examination name, associated course #, area):

• Abstract Algebra, Math 5040, Algebra
• Linear Algebra, Math 5100, Algebra
• Real Analysis, Math 5150, Analysis
• Complex Analysis, Math 5110, Analysis
• Graph Theory, Math 6070, Discrete Mathematics
• Enumerative Combinatorics, Math 6060, Discrete Mathematics
• Applied Mathematics I, Math 5190, Applied Mathematics
• Applied Mathematics II, Math 5200, Applied Mathematics
• Numerical Analysis, Math 5610, Numerical Analysis
• Numerical Linear Algebra, Math 5620, Numerical Analysis

A student granted full admission to the Ph.D. program is expected to pass four qualifying examinations (subject to the rules above) within the first two calendar years in the program.

Oral preliminary exam

The oral preliminary examination of a Ph.D. student tests a student’s knowledge of the major, minor and supporting fields of their research area. The examination is taken after a student has passed four written qualifying examinations, satisfied the graduate English requirement (if required), formed a Academic Plan (AP) committee, and has an approved Academic Plan (AP) form.

Final oral examination

The final examination of a Ph.D. candidate is oral, and is usually limited to a defense of the dissertation.

Graduate English requirements

Graduate students whose native language is not English must meet the Graduate College English Requirement.

Teaching requirement

Each Ph.D. student is required to have one year of supervised teaching. However, if approved by the student’s Academic Plan (AP) committee, equivalent supervised experience in oral mathematics communication may be substituted for teaching. In that case the Academic Plan (AP) committee must specify in writing what the equivalent experience will be.

An M.S. student is required to a attain a minimum of 30 acceptable credits, with at least 22 of these earned in residence. The total of 30 credits must include 21 hours of 5000-6000 level mathematics courses excluding Math 5900, 5910, 5920, 5990 and 6990, and these 21 hours must include at least 12 hours of core courses subject to the conditions (see below). There is also a 1 credit seminar requirement which is satisfied by taking both Math 5910 and Math 5920.

Each student must elect one of the following options:

• Thesis option: the thesis program required 6 credits of Math 6990, which may be included in the 30 credits. The Masters thesis must include original mathematical work.
• Non-thesis option: non-thesis program requires a creative component that entails reading, synthesizing, and presenting mathematical content from approved journal articles. Three credits of Math 5990 may be included in the 30 credits for the creative component.

Core course requirements

The core course requirements for the M.S. in Mathematics are:

1. At least 4 core courses.
2. At least 2 core courses in Algebra (Math 5040, Math 5050, Math 5100) or at least 2 core courses in Analysis (Math 5150, Math 5160, Math 5110).

Final oral examination

The final examination of an M.S. candidate is oral and comprehensive. It normally consists of a defense of the thesis or creative component and an examination of the candidate’s knowledge of the topics covered in the program of study.

Well-qualified students are encouraged to consider a Ph.D. program having a co-major in Mathematics and some other appropriate program. Such programs are to be initiated by a written proposal from the student to the Mathematics Department Graduate Committee. The proposal must contain an outline of how all requirements are to be met. Authorization by the Graduate Committee to embark on a co-major program will be based on this proposal, and on the academic history of the student. The Academic Plan (AP) committee is to be directed by co-chairmen, one from each of the co-major departments. The dissertation must have significant content in both fields. Co-major programs are subject to the following minimum standards.

1. Co-major Ph.D. students are required to earn at least 24 credits in 5000-6000 level mathematics courses other than Math 5900, 5910, 5920, 5990 and 6990. They are required to take a total of four courses from the mathematics core including at least one one-year sequence (Math 5040-5050 or Math 5150-5160). They are also required to pass two of the qualifying examinations.
2. Co-major Ph.D. students are required to have two years of professional experience including at least one year of supervised teaching. The other year may be supervised research as a research assistant or associate.

Ph.D. students who declare a minor in Applied Mathematics are required to have at least 12 credits in Mathematics courses which are acceptable for non-major graduate credit, excluding Math 5900, 5910, 5920, 5990 and 6990, and of which at least 6 must be in 5000-6000 level Mathematics courses.

M.S. students who declare a minor in Applied Mathematics are required to have at least 6 credits in Mathematics courses that are acceptable for non-major graduate credit, excluding Math 5900, 5910, 5920, 5990 and 6990, at the 4000-level or above.