Ken Ono

Affiliation: University of Wisconsin

Email: ono@math.wisc.edu

Three Lectures:

1. Dyson's challenge for the future: Ramanujan's mock theta functions
2. Harmonic Maass forms and modular forms: Leveraging
3. The Birch and Swinnerton-Dyer Conjecture, Heegner Points, and Maass forms

Abstract: My lectures will summarize various roles that harmonic weak Maass forms plays in various areas of number theory. The first lecture will about Ramanujan's mock theta functions and partitions. The second lecture will give an overview of the interplay between harmonic weak Maass form and classical modular forms. In particular, we shall leverage properties of two differential operators to obtain a theorem which detects the vanishing of Fourier coefficients of modular forms such as Delta. In particular, we reduce Lehmer's Conjecture on the nonvanishing of Ramanujan's tau-function to the alleged irrationality of a single simple elliptic type integral. In my third lecture I shall further develop the notion of rationality and transcendence to investigate the Birch and Swinnerton-Dyer Conjecture, the Gross-Zagier Theorem on derivatives of L-functions and heights of Heegner points.

Last update made Wed Feb 13 22:31:34 EST 2008.
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