The study of discrete mathematics is available at a number of universities throughout the country. Discrete mathematics doctoral programs are most commonly offered as a mathematics Ph.D. with a track in discrete mathematics.

## Information About Ph.D. Programs in Discrete Mathematics

Discrete mathematics Ph.D. programs can typically be completed over five years. The first two years are spent taking advanced coursework and qualifying examinations before advancing to Ph.D. candidacy, which is when students can begin working on their dissertations. The courses a student takes in these programs may inform their dissertation decision. Five common courses are discussed below.

### Graph Theory

This course provides students with a foundation in basic graph theory as well as a more advanced understanding of some common topics associated with this subject. Topics that could be covered include graph decomposition, paths and connectedness, Menger's theorem, Hall's matching condition, and edge connectivity.

### Algebra I & II

These programs typically require that students take both algebra I and II, depending on their mathematical background. An algebra I course focuses on topics like canonical form, multilinear algebra, Witt theorem, basic group theory, and alternating groups. Algebra II focuses on polynomial rings, finite and infinite groups, Chinese remainder theorem, and homomorphism, among other topics.

### Numerical Analysis

In this course, which can be offered as a two-part series, students learn about a number of different topics, like floating point computations, ordinary and partial differential equations, computational geometry, and linear and quadratic programming. In addition, students in numerical analysis courses may also learn about different methods and problem-solving techniques specific to the fields of engineering and science. A knowledge of MATLAB and programming languages may be required for these courses.

### Commutative Algebra

This course is often offered as an elective for students who want to gain an introduction to the principles and tools of commutative algebra. Topics that students learn about include localization, ring theory, chain conditions, and Noetherian property. In addition, this course may provide students with a look at the relationships between commutative algebra and algebraic geometry.

### Mathematical Statistics

Discrete mathematics programs may include introductory as well as more advanced coursework in this subject, and courses generally begin by covering some of the common theories used in mathematical statistics, like the theory of point estimation. Additionally, students learn about topics like sufficiency and completeness, parametric estimations, hypothesis testing, and optimality theory.

## Admission Standards for Ph.D. Programs in Discrete Mathematics

When applying to a Ph.D. program in discrete mathematics, students will need to submit undergraduate and/or graduate transcripts, GRE scores, letters of recommendation, a completed application form, and a statement of purpose. These programs often also require that students submit results from the GRE subject test in mathematics. Ph.D. programs in this field expect that applicants will have a sufficient background in mathematics to be prepared for graduate coursework, though it is not explicitly required that students have obtained a bachelor's degree in mathematics or a closely related field.

To summarize, it is possible to find Ph.D. programs in which students could study discrete mathematics. This field of study is typically offered as a track within a more generalized mathematics Ph.D. program.