In 1956, Alder conjectured an integer partition inequality which generalized Euler’s partition identity, the first Rogers-Ramanujan identity, and a partition identity of Schur. Alder’s conjecture, proved in part by Andrews in 1971, followed by Yee in 2008, and finally completed by Alfes, Jameson, and Lemke Oliver in 2010 states that...
Following the work of Asai, Kaneko, and Ninomiya for Faber polynomials associated to the modular group, and Bannai, Kojima, and Miezaki's partial proof for the case of the Fricke group of level 2, we show that the zeros of certain modular functions for some low-level genus zero groups associated to...
In this dissertation, we consider two problems in number theory, both relating to modular forms. First we consider when a given modular form can be expressed as a quotient in Dedekind's $\eta$ function. Rouse and Webb \cite{RW} have determined the integers $N \leq 500$ such that the graded ring of...
In 2013, Lemke Oliver created a list of all eta-quotients which are theta functions. Then in 2016, Folsom, Garthwaite, Kang, Swisher, and Treneer utilized this list of ``eta-theta'' functions along with Zwegers's construction of mock theta functions to create a set of mock modular forms which are also quantum modular...