This award supports the research of Professor A. Pizer to work on the connections between modular forms, quaternion algebras, and Ramanujan graphs. He intends to investigate the extremal properties of Ramanujan graphs constructed via modular forms. Modular forms arose out of Non-Euclidean geometry in the middle of the nineteenth century. Combinatorics, on the other hand, represents a systematization of the first mathematical activity, counting. The recently discovered connections between these two areas demonstrates once again how modern mathematics progresses by connecting together seemingly diverse fields. This research is potentially important in communications and information technology.
