Quantum modular forms, mock modular forms, and partial theta functions

Defined by Zagier in 2010, quantum modular forms have been the subject of an explosion of recent research. Many of these results are aimed at discovering examples of these functions, which are defined on the rational numbers and have "nice" modularity properties. Though the subject is in its early stages, numerous results (including Zagier's original examples) show these objects naturally arising from many areas of mathematics as limits of other modular-like functions. One such family of examples is due to Folsom, Ono, and Rhoades, who connected these new objects to partial theta functions (introduced by Rogers in 1917) and mock modular forms (about which there is a rich theory, whose origins date back to Ramanujan in 1920).;In this thesis, we build off of the work of Folsom, Ono, and Rhoades by providing an infinite family of quantum modular forms of arbitrary positive half-integral weight. Further, this family of quantum modular forms "glues" mock modular forms to partial theta functions and is constructed from a so-called "universal" mock theta function by extending a method of Eichler and Zagier (originally defined for holomorphic Jacobi forms) into a non-holomorphic setting.;In addition to the infinite family, we explore the weight 1/2 and 3/2 functions in more depth. For both of these weights, we are able to explicitly write down the quantum modular form, as well as the corresponding "errors to modularity," which can be shown to be Mordell integrals of specific theta functions and, as a consequence, are real-analytic functions.;Finally, we turn our attention to the partial theta functions associated with these low weight examples. Berndt and Kim provide asymptotic expansions for a certain class of partial theta functions as q approaches 1 radially within the unit disk. Here, we extend this work to not only obtain asymptotic expansions for this class of functions as q approaches any root of unity, but also for a certain class of derivatives of these functions. These derivatives of partial theta functions play a key role in the partial theta formulation of the infinite family of quantum modular forms described above.;Through the main theorems of this work, we gain further insight into mock modular forms, the role of partial theta functions in the theory of modular forms, and the newly defined quantum modular forms.