Mathematical Sciences: Combinatorics and Number Theory

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This award supports the research in combinatorics and number theory of Professor Wen-Ching Li of The Pennsylvania State University. Dr. Li's project is centered on questions relating modular forms to graph theory. Among the questions she will investigate are: to realize certain Ramanujan graphs as quotients of trees associated to projective linear groups over p-adic fields; to continue her collaboration with Dr. F.K. Hwang of Bell Laboratories on generalized double loop networks; and to study the Galois structure of p-Sylow subgroups of ideal class groups. Non-Euclidean plane geometry began in the early nineteenth century as a mathematical curiosity, but by the end of that century, mathematicians had realized that many objects of fundamental importance are non-Euclidean in their basic nature. The detailed study of non-Euclidean plane geometries has given rise to several branches of modern mathematics, of which the study of modular and automorphic forms is one of the most active. This field is principally concerned with questions about the whole numbers, but in its use of geometry and analysis, it retains connection to its historical roots. The work of Professor Li is in this tradition, but extends beyond the traditional boundaries of the theory of modular forms in a particularly interdisciplinary thrust.