Rob Rhoades

Affiliation: Stanford University

Email: rob.rhoades@gmail.com

Title Of Talk: Perspectives on Mock and False theta functions

Abstract: The mock theta functions are $q$-series which exist for $|q|<1$ and have pseudo-modular properties. We understand that the partial theta functions are ``shadows" of the mock theta functions in the region $|q|>1$ with no modular properties. We explain four ways to view this relationship.

  1. asymptotic expansions (Lawerence-Zagier, Hikami, $\dots$)
  2. Poincare series and "expansions of zero" (Rademacher, Lehner, Knopp, R, $\dots$)
  3. $q$-series (Zwegers, Zagier, Hikami, Folsom-Ono-R, $\dots$)
  4. the Mordell integral (Mordell, Ramanujan, Zwegers, Chern-R, $\dots$)
The mock theta functions often have a dense set of singularities on $|q|=1$. Despite this, we will explain how these constructions allow the mock theta function to ``leak" through the region $|q|=1$ into the region $|q|>1$. This phenomenon leads us to quantum modular forms. It is remarkable that Ramanujan's own study of the Mordell integral shows how the partial theta functions have a pseudo-modular transformation on the region $|q|=1$.

WARNING: This page contains MATH-JAX

Last update made Wed Oct 24 15:31:12 EDT 2012.
Please report problems to: fgarvan@ufl.edu