Stratton Hall, 302A
William J. Martin
- Professor, Mathematical Sciences
- Affiliated with:
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 ands 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 Professor Berk Sunar, Martin is currently investigating homomorphic encryption, exploring techniques for efficient implementation of existing schemes as well as developing entirely new schemes. With his collaborators, he is carrying out research on mutually unbiased bases. He also works with co-authors in Korea on completely regular error-correcting codes and their connection to distance-regular graphs. Finally, 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.
- Algebraic Combinatorics
- Association Schemes
- Digital Communications
- Error-correcting Codes
- BA, State University of New York Potsdam, 1986
- MA, State University of New York Potsdam, 1986
- PhD, University of Waterloo, 1992
- “Commutative association schemes” (with Hajime Tanaka). European Journal of Combinatorics 30, no. 6 (2009), pp. 1497-1525.
- “There are finitely many Q-polynomial association schemes with given first multiplicity at least three” (with Jason Williford). European Journal of Combinatorics 30, no. 3 (2009), pp. 698-704.
- “Width and dual width of subsets in polynomial association schemes” (with A. E. Brouwer, C. D. Godsil and J. Koolen). Journal of Combinatorial Theory, Series A 102 (2003), 255-271.
- Journal preprint access (with subscription) ``Resilient functions: Just how resilient are the they?'' (with Berk Sunar). Error-Correcting Codes, Finite Geometries and Cryptography Contemporary Mathematics, Vol. 523 Aiden A. Bruen, and David L. Wehlau, editors.
- "A provably secure true random number generator with built-in tolerance to active attacks.'' (with B. Sunar and D.R. Stinson) IEEE Transactions on Computers, 56, no. 1 (2007), 109-119.