Renormalization and quantum modular forms, part I: Maass wave forms

Sander Zwegers showed that Ramanujan's mock theta functions are $q$-hypergeometric series, whose $q$-expansion coefficients are half of the Fourier coefficients of a non-holomorphic modular form. George Andrews, Henri Cohen, Freeman Dyson, and Dean Hickerson found a pair of $q$-hypergeometric series each of which contains half of the Fourier coefficients of Maass waveform of eigenvalue $1/4$. This series of papers shows that a $q$-series construction, called ``renormalization'', yields the other half of the Fourier coefficients from a series which contains half of them. This construction unifies examples associated with mock theta functions and examples associated with Maass waveforms. Thus confirming a conviction of Freeman Dyson. This construction is natural in the context of Don Zagier's quantum modular forms. Detailed discussion of the role quantum modular forms play in this construction is given. New examples associated to Maass waveforms are given in Part I. Part II contains new examples associated with mock theta functions, and classical modular forms. Part II contains an extensive survey of the ``renormalization'' construction. A large number of examples and open questions which share similarities to the main examples, but remain mysterious, are given.