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Thursday, November 5, 2015

**Abstract:** In this talk, I will describe several new results concerning the modularity of indefinite theta functions. From Zwegers' thesis, we know that special types of indefinite theta functions with prescribed signatures give rise to mock modular forms, which combined with important work of Andrews and others gives one road to understanding the mock theta functions of Ramanujan. Here, we will study several important examples of more general indefinite theta series inspired by physics and geometry and describe how to study the modularity properties of more complicated objects such as these, giving a glimpse into the general structure of indefinite theta functions. We will also study another class of indefinite theta functions, and we will discuss a new family of examples which give rise to quantum modular forms, and provide a family of canonical Maass waveforms whose Fourier coefficients are described by combinatorial functions with integer coefficients, placing the famous functions $\sigma$ and $\sigma^*$ of Andrews, Dyson, and Hickerson in a natural framework.