This award supports the research in the analytic number theory of automorphic and modular forms of Professor Dorian Goldfeld of Columbia University. Dr. Goldfeld plans to investigate the growth of the coefficients of meromorphic modular functions and hopes to derive asymptotic formulas for these in terms of Kloosterman sums. A related project of Professor Goldfeld is to relate the arithmetic of a modular elliptic curve to the minimal hyperbolic distance between the zeros of the Hecke cusp form of weight two associated to this curve. 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.
