This award supports the research in the analytic theory of automorphic forms of Professor Isaac Efrat of the University of Maryland at College Park. Dr. Efrat has proposed to investigate a collection of problems in the harmonic analysis of graphs and surfaces that arise as quotients of spaces of rank one, modulo discrete groups. These problems stem from questions about the existence of discrete spectra for such groups, and the construc- tion of discrete eigenfunctions for them. Other aspects of Professor Efrat's project involve the study of certain arith- metically defined Eisenstein series attached to non-congruence subgroups. 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.
