Related papers: Renormalization and quantum modular forms, part I:…

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,…

Ramanujan studied the analytic properties of many $q$-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious $q$-series fit into the theory of…

In the theory of harmonic Maass forms and mock modular forms, mock theta functions are distinguished examples which arose from $q$-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular…

Andrews-Dyson-Hickerson, Cohen build a striking relation between q-hypergeometric series, real quadratic fields, and Maass forms. Thanks to the works of Lewis-Zagier and Zwegers we have a complete understanding on the part of these…

Andrews recently introduced k-marked Durfee symbols, which are a generalization of partitions that are connected to moments of Dyson's rank statistic. He used these connections to find identities relating their generating functions as well…

In this paper, we use theta integrals to give a different construction of mock Maass forms studied by Sander Zwegers. With this method, we construct new real-analytic modular forms, whose Fourier coefficients are logarithms of algebraic…

Ramanujan derived a sequence of even weight $2n$ quasimodular forms $U_{2n}(q)$ from derivatives of Jacobi's weight $3/2$ theta function. Using the generating function for this sequence, one can construct sequences of quasimodular forms of…

In this Ph.D. thesis, written under the direction of D.B. Zagier and R.W. Bruggeman, we study the mock theta functions, that were introduced by Ramanujan. We show how they can be interpreted in the theory of (real-analytic) modular forms.…

Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves $E/\mathbb{Q}$. We…

We analyze the mock modular behavior of $\bar{P}_\omega(q)$, a partition function introduced by Andrews, Dixit, Schultz, and Yee. This function arose in a study of smallest parts functions related to classical third order mock theta…

By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish,…

The modular forms are revisited from a geometric and an algebraic point of view leading to a geometric interpretation of the weak Maass forms connecting them to the Ramanujan Mock Theta functions and to the cusp forms generated from the…

The study of arithmetic properties of coefficients of modular forms $f(\tau) = \sum a(n)q^n$ has a rich history, including deep results regarding congruences in arithmetic progressions. Recently, work of C.-S. Radu, S. Ahlgren, B. Kim, N.…

In this paper, we answer a question of Li, Ngo, and Rhoades concerning a set of $q$-series related to the $q$-hypergeometric series $\sigma$ from Ramanajun's lost notebook. Our results parallel a theorem of Cohen which says that $\sigma$,…

Using results from Ramanujan's lost notebook, Zudilin recently gave an insightful proof of a radial limit result of Folsom, Ono, and Rhoades for mock theta functions. Here we see that the author's previous work on the dual nature of…

Ramanujan's original definition of mock theta functions from 1920 involves their asymptotic behaviors at roots of unity on the boundary of the disk of convergence $|q|<1$. More recently this topic has been related by several authors,…

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in…

Ramanujan introduced mock theta functions in his last letter to G.H.Hardy. He provided examples and various relations between them. G.N.Watson found transformations for the third order mock theta functions $f(q)$ and $\omega$(q). Zwegers in…

In this paper we add to the literature on the combinatorial nature of the mock theta functions, a collection of curious $q$-hypergeometric series introduced by Ramanujan in his last letter to Hardy in 1920, which we now know to be important…

The mock theta conjectures are ten identities involving Ramanujan's fifth-order mock theta functions. The conjectures were proven by Hickerson in 1988 using q-series methods. Using methods from the theory of harmonic Maass forms,…