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% You and your thesis

\newcommand{\myname}{Larry Rolen}
\newcommand{\mytitle}{Maass Forms and Quantum Modular Forms}
\newcommand{\mydegree}{Ph.D., Emory University, 2013}
\newcommand{\thisyear}{2013}
\newcommand{\mydepartment}{Mathematics}
%%% can be 'Master of Science' or 'Doctor of Philosophy'
\newcommand{\thisdegree}{Doctor of Philosophy}
%%% type of thesis, can be 'thesis' (for M.S.) or 'dissertation' (for Ph.D.)
\newcommand{\typeofthesis}{dissertation}

% Your committee
\newcommand{\myadvisor}{Ken Ono, Ph.D.}
% other committee members should be in alphabetical order3
\newcommand{\committeeone}{David Borthwick, Ph.D.}
\newcommand{\committeetwo}{David Zureick-Brown, Ph.D.}

%%% Acknowledgments
\newcommand{\myacknowledgments}{
I would like to thank my family who have been extremely supportive of my entire education, and without whom this thesis would not exist. I also dedicate this thesis to my lovely fianc\'ee Anna. Finally, I would like to thank my advisor Ken Ono who has guided me throughout my Ph.D. and especially on the topics contained in this thesis. 
}

%%% Abstract
\newcommand{\myabstract}{
This thesis describes several new results in the theory of harmonic Maass forms and related objects. Maass forms have recently led to a flood of applications throughout number theory and combinatorics in recent years, especially following their development by the work of Bruinier and Funke [10] the modern understanding Ramanujan's mock theta functions due to Zwegers [36,37].
The first of three main theorems discussed in this thesis concerns the integrality properties of \emph{singular moduli}. These are well-known to be algebraic integers, and they play a beautiful role in complex multiplication and explicit class field theory for imaginary quadratic fields. One can also study ``singular moduli'' for special non-holomorphic functions, which are algebraic but are not necessarily algebraic integers. Here we will explain the phenomenon of integrality properties and provide a sharp bound on denominators of symmetric functions in singular moduli.
The second main theme of the thesis concerns Zagier's recent definition of a \emph{quantum modular form}. Since their definition in 2010 by Zagier, quantum modular forms have been connected to numerous different topics such as strongly unimodal sequences, ranks, cranks, and  asymptotics for mock theta functions. Motivated by Zagier's example of the quantum modularity of Kontsevich's ``strange'' function $F(q)$, we revisit work of Andrews, Jim{\'e}nez-Urroz, and Ono to construct a natural vector-valued quantum modular form whose components.
The final chapter of this thesis is devoted to a study of asymptotics of mock theta functions near roots of unity. In his famous deathbed letter, Ramanujan introduced the notion of a {\it mock theta function}, and he offered some alleged examples. The theory of mock theta functions has been brought to fruition using the framework of harmonic Maass forms, thanks to Zwegers [36,37].  Despite this understanding, little attention
has been given to Ramanujan's original definition. Here we prove that Ramanujan's examples do indeed satisfy his original definition.
}

% If you want this, uncomment the lines at the end of preamble.tex as well
%%% Dedication
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%Dedication goes here.
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