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Sander Zwegers showed that Ramanujan's mock theta functions are$q$-hypergeometric series, whose $q$-expansion coefficients are half of theFourier coefficients of a non-holomorphic modular form. George Andrews, HenriCohen, Freeman Dyson, and Dean Hickerson found a pair of $q$-hypergeometricseries each of which contains half of the Fourier coefficients of Maasswaveform 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 aseries which contains half of them. This construction unifies examplesassociated with mock theta functions and examples associated with Maasswaveforms. Thus confirming a conviction of Freeman Dyson. This construction isnatural in the context of Don Zagier's quantum modular forms. Detaileddiscussion of the role quantum modular forms play in this construction isgiven.  New examples associated to Maass waveforms are given in Part I. Part IIcontains new examples associated with mock theta functions, and classicalmodular forms. Part II contains an extensive survey of the ``renormalization''construction. A large number of examples and open questions which sharesimilarities to the main examples, but remain mysterious, are given.

Renormalization and quantum modular forms, part II: Mock theta functions