\documentclass[letter, 11pt]{article}
\usepackage[top=3cm, bottom=3cm, left=2cm, right=2cm]{geometry} 
\usepackage[utf8]{inputenc}
\usepackage{textcomp}
\usepackage{graphicx} 
\usepackage{amsmath,amssymb,amsthm}  
\usepackage{bm}  
\usepackage[pdftex,bookmarks,colorlinks,breaklinks]{hyperref}  
\hypersetup{linkcolor=black,citecolor=black,filecolor=black,urlcolor=black}
\usepackage{memhfixc} 
\usepackage{pdfsync}  
\usepackage{fancyhdr}
\usepackage{mathtools}
\usepackage{wasysym}
\usepackage{tikz}
\usepackage{enumitem}
\usepackage{booktabs}
\usepackage{ccaption}
\usepackage{subcaption}
\pagestyle{fancy}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Prob}{\mathbb{P}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\E}{\mathbb{E}}
\newtheorem{definition}{Definition}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}{Corollary}
\newtheorem{remark}{Remark}
\title{Recurrence vs. Transience in $\Z^2$ and Higher Dimensions: \\ A Walk Through Probability}
\author{Laura Kurtz, Mentored by Dr. Fehrman}
\date{\today}
\renewcommand\qedsymbol{$\smiley$}
\renewcommand{\thefootnote}{\alph{footnote}}
\begin{document}
\maketitle
\begin{center}
    \textbf{Abstract} 

    Blah blah blah..... I'm going to do some cool math... here's my result.
\end{center}
\pagebreak
\tableofcontents
\pagebreak
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\section{Introduction}

Make them want to read your paper here. Are they already bored??
\section{here's another section}
\subsection{Etc.}
I'm starting to talk about math now...
\subsubsection{A theorem.}
\begin{theorem}
    Here is an important result.
\end{theorem}
\begin{proof}
    Here's how we got that important result.
\end{proof}
Note: If you want a regular box to end your proofs instead of the smiley face, you can delete the renew command qed symbol in the header.
\pagebreak
\bibliographystyle{abbrv}
\bibliography{references}  % need to put bibtex references in references.bib 
 \nocite{*}
\section{Acknowledgements}
 Some people helped you. Who are they?
\end{document}