This page gives a detailed overview of the following programs offered in Mathematics. There are four main categories of graduate degrees that are offered in Mathematics:

- Ph.D. in Mathematics (for students in the Department of Mathematics)
- M.S. in Mathematics (for students in the Department of Mathematics)
- Co-major Ph.D. in Mathematics + Outside Area (for all graduate students)
- Graduate minor in Mathematics (for graduate students from outside departments)

See below for more information.

## Ph.D. in Mathematics

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 500 or 600 level mathematics courses excluding Math 590, 591, 592, 599 and 699. There is also a 1 credit seminar requirement which is satisfied by taking both Math 591 and Math 592.

Included in the 42 credits of formal courses is a 0 credit cognate study requirement. A cognate course is defined to be a course which is (i) acceptable for graduate credit, (ii) taught in another department (a course cross-listed with Mathematics can count toward the cognate requirement if taught by a faculty member whose primary appointment is not in the Department of Mathematics, or if approved by the Graduate Committee), and (iii) relevant to the major. The course work for the cognate study requirement must be approved by the student’s POS committee. Students are encouraged to consider a minor in another department to meet the cognate study requirement.

The student is also required to take at least 3 credits of Math 699, 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:

- At least 6 core courses.
- At least 3 core courses must be in Algebra, Analysis or Discrete Mathematics.
- The core courses must include
**Math 504**and**Math 515**.

Core Courses (for more details see Course Catalog)

- Algebra:
**Math 504:**Abstract Algebra I**Math 505:**Abstract Algebra II**Math 510:**Linear Algebra

- Analysis:
**Math 515:**Real Analysis I**Math 516:**Real Analysis II**Math 511:**Complex Analysis

- Discrete Mathematics:
**Math 607:**Graph Theory**Math 606:**Enumerative Combinatorics**Math 566:**Discrete Optimization

- Applied Mathematics:
**Math 519:**Methods of Applied Mathematics I**Math 520:**Methods of Applied Mathematics II**Math 565:**Continuous Optimization

- Numerical Analysis:
**Math 561:**Numerical Analysis I**Math 562:**Numerical Analysis II**Math 517:**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:

- Retaking the course for credit and earning a B or better.
- Retaking the final examination of the course and earning a B or better (permission of the instructor giving the final is required).
- 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:

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

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

- Abstract Algebra,
**Math 504,**Algebra - Linear Algebra,
**Math 510**, Algebra - Real Analysis,
**Math 515**, Analysis - Complex Analysis,
**Math 511**, Analysis - Graph Theory,
**Math 607**, Discrete Mathematics - Enumerative Combinatorics,
**Math 606,**Discrete Mathematics - Applied Mathematics I,
**Math 519**, Applied Mathematics - Applied Mathematics II,
**Math 520**, Applied Mathematics - Numerical Analysis,
**Math 561**, Numerical Analysis - Numerical Linear Algebra,
**Math 562**, 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 Program of Study (POS) committee, and has an approved Program of Study (POS) 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 Program of Study (POS) committee, equivalent supervised experience in oral mathematics communication may be substituted for teaching. In that case the Program of Study (POS) committee must specify in writing what the equivalent experience will be.

## M.S. in Mathematics

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 500-600 level mathematics courses excluding Math 590, 591, 592, 599 and 699, 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 591 and Math 592.

Each student must elect one of the following options:

**Thesis option:**the thesis program required 6 credits of Math 699, 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 599 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:

- At least 4 core courses.
- At least 2 core courses in Algebra (
**Math 504**,**Math 505**,**Math 510**) or at least 2 core courses in Analysis (**Math 515, Math 516, Math 511**).

### Note about cognate study

Although cognate study (as described in above) is not required at the M.S. level, it is strongly recommended. When cognate study is in the form of a minor, it typically consists of 6-9 credits in a department other than Mathematics. These credits must be acceptable to the representative of the minor department on the student’s POS (Program of Study) committee.

### 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.

## Co-Major in Mathematics and Outside Area (Interdisciplinary Ph.D.)

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 Program of Study (POS) 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.

- Co-major Ph.D. students are required to earn at least 24 credits in 500-600 level mathematics courses other than Math 590, 591, 592, 599 and 699. They are required to take a total of four courses from the mathematics core including at least one one-year sequence (Math 504-505 or Math 515-516). They are also required to pass two of the qualifying examinations.
- 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.

## Graduate Minor in Mathematics (for graduate students from other departments)

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 590, 591, 592, 599 and 699, and of which at least 6 must be in 500-600 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 590, 591, 592, 599 and 699, at the 400-level or above.

The Mathematics Graduate Student Handbook contains the full details of program rules.