Discrete Mathematics

Discrete mathematics is a rapidly evolving area of mathematics with strong connections to theoretical computer science and operations research, as well as to core areas of pure mathematics, especially algebra and number theory. At ASU, graduate students can study Graph Theory, Ordered Sets, Ramsey Theory, Discrete Optimization, Algorithms & Computational Complexity, and Algebraic Combinatorics.

Prerequisites

Graduate students beginning the program should have a solid background in general undergraduate mathematics. They should be comfortable with both reading and creating mathematical proofs. Besides the department's minimum requirements, courses in real analysis, abstract algebra and probability are useful. Within Discrete Mathematics, students benefit from having taken courses such as Combinatorics, Graph Theory and Linear Programming.

Course Work and Examinations

Students with little or no background in Discrete Mathematics can start with the courses MAT 415 Introduction to Combinatorics and MAT 416 Introduction to Graph Theory. This 400 level sequence prepares those Masters and Ph.D. students who choose to take the Qualifier Examination in Discrete Mathematics. The sequence is also useful to students from other areas of Mathematics and Computer Science. MAT 419 Linear Programming is another elementary course in Discrete Mathematics that may be of particular interest to computational math students and computer scientists. Although it is not a prerequisite, it can serve as an introduction to MAT 518.

The department will offer a one year core sequence in one of the areas of Combinatorial Enumeration (MAT 514-515), Graph Theory (MAT 516-517), and Combinatorial Optimization (MAT 518-519) each year. These 500 level courses will be taught at the level of the Comprehensive Examination in Discrete Mathematics. All Discrete Mathematics Ph.D. students should plan to take these three sequences.

M.A.

Master's students in Discrete Mathematics should take the MAT 415-416 Qualifier sequence above, one 500 level core Comprehensive sequence above, and one other course in Discrete Mathematics (usually Linear Programming or a Topics course). They should either take one Qualifier Examination (typically in Discrete Mathematics, Algebra, or Real Analysis) and write a thesis in Discrete Mathematics, or take two Examinations (typically from the list of Qualifiers in Discrete Mathematics, Algebra, Real Analysis, and Statistics, and the Comprehensive in Discrete Mathematics).

Ph.D.

Ph.D. students in Discrete Mathematics usually take two Qualifiers from the list of Discrete Mathematics, Algebra, and Real Analysis; however other combinations are possible. The Comprehensive Examination in Discrete Mathematics will be based on one of the three core sequences in Graph Theory, Combinatorial Optimization or Combinatorial Enumeration.
Ph.D. students also must pass the Oral Examination, which is based on their individual written dissertation prospectus. The core sequences, advanced special topics courses, the Discrete Mathematics Seminar, and especially independent research courses under the direction of an
advisor are critical to preparing for it. During the Oral Examination, students may expect to answer questions on their research project as well as on literature relevant to it.

The final and most important step in a student's Ph.D. program is to write a thesis based on original research. This work should be publishable in respected research journals, usually as two to three medium length papers or possibly as one very long paper.

Faculty

For more information on specific aspects and content of this program, contact one of the Discrete Mathematics faculty:

http://math.asu.edu/research/discrete.html