Speaker: Professor Ken Ono (University of Virginia)

Title: Variants of Lehmer’s  conjecture

Abstract: Generating functions are central devices and tools in arithmetic geometry, combinatorics, number theory, physics, and representation theory. Amazingly, such generating functions often turn out to be examples of modular forms. Despite many deep developments in the arithmetic geometric and analytic aspects (e.g. Deligne’s proof of the Weil Conjectures, the development of Galois representations, Birch and Swinnerton-Dyer Conjecture, to name a few) of modular form theory, some of the seminal questions about these generating functions remain open. Perhaps the most prominent of these is Lehmer’s Conjecture on the nonvanishing of Ramanujan’s tau-function. In joint work with J. Balakrishnan, W. Craig, and W.-L. Tsai, the speaker has obtained the first results that establish that many integers are never modular form coefficients. This lecture is geared to a very general audience and begins with a historical survey of illustrating the importance of modularity.