Bill Martin's goal is to find mathematical research projects that lie between beautiful and powerful mathematical theory, on the one hand, and pressing technological applications, on the other. This effort requires one to keep abreast of both mathematical developments and applications in computer science and engineering. 

Professor Martin's mathematical research is in the area of algebraic combinatorics, where tools from linear and abstract algebra are applied to problems in discrete math. An association scheme is a collection of graphs, which give rise to a highly structured matrix algebra whose eigenspaces reveal information about these graphs and their substructures. The vertices of the graphs might, for example, be the set of all binary n-tuples in which case we have a tool for the study of error-correcting codes. In this and numerous other cases, by embedding unstructured configurations into well-structured ambient spaces, we obtain algebraic leverage over what are otherwise messy applied problems. Martin and co-authors have applied the theory of association schemes to the study of experimental designs, finite geometries, highly regular graphs, error-correcting codes, (t,m,s)-nets, and structures appearing in quantum information theory. 

Martin's current research activities are split across four areas.  With his collaborators, he is carrying out research in quantum information,  obtaining results on quantum random walks, quantum games, quantum error-correcting codes, and mutually unbiased bases. With Professor Berk Sunar and co-authors, Martin has investigated homomorphic encryption schemes, random number generators, and other ideas in cryptography.  Finally, and centrally, he also uses algebraic and combinatorial techniques to develop association scheme theory itself. In addition to these main activities, Professor Martin is interested in K-12 education, contributing to math clubs, competitions, summer camps, and high school curricular development.