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\begin{document}
\title{Oral Qualifying Examination Syllabus}
\author{Sujith Vijay \\\\ \hspace*{-0.32in}
Committee: J. Beck (Chair), R. Bumby, V. Gurvich, D. Zeilberger 
\hspace*{-0.28in}
}
\date{}

\maketitle
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\section{Combinatorial Number Theory} 

\noindent

\begin{itemise}

\item Bertrand's postulate; sum of reciprocals of primes; Mertens's 
theorem. \li
\item Brun's pure sieve; upper bound for the number of twin primes not 
exceeding 
$x$; convergence of the sum of reciprocals of twin primes. \li
\item Schnirelmann density; connections with additive bases; primes as a 
basis. \li
\item Squares as a basis; sums of three squares; Cauchy's polygonal number 
theorem. \li
\item Roth's theorem; Van der Waerden's theorem; Schur's theorem; 
sum-free sets. \li
\item Ordered and unordered partitions; Euler's pentagonal number theorem;
Rogers-Ramanujan identities; elementary estimates for the number of 
unordered partitions. \li
\item Erd\H{o}s-Fuchs theorem; Sidon sets; $B_h$-sequences. 

\end{itemise}

\section{Diophantine Approximations}

\noindent

\begin{itemise}

\item Continued fractions; best approximations; Hurwitz's theorem; 
quadratic irrationals; Pell's equation. \li
\item Lagrange and Markoff spectra; gaps in the Markoff spectrum; Hall's 
ray; Markoff's equation. \li
\item Irrationality of $e, \pi, \ln 2$ and $\zeta(3)$; transcendence of 
$e$ and $\pi$; Lindemann-Weierstrass theorem; Gelfond-Schneider theorem. \li
\item Approximability of algebraic numbers; Liouville's theorem and 
improvements. \li
\item Distribution modulo $1$; Weyl's criterion; uniform distribution of 
$\{n \alpha\}$ for irrational $\alpha$; uniform distribution of 
polynomials; normal numbers. 

\end{itemise}

\section{Combinatorial Game Theory}

\noindent

\begin{itemise}

\item Impartial games; standard and mis\`{e}re Nim; sums of games;
Sprague-Grundy theorem. \li
\item Surreal numbers; field structure and total ordering; positive, 
negative, zero and fuzzy games. \li
\item Positional games; pairing strategy; Erd\H{o}s-Selfridge theorem; 
potential functions. \li
\item The digraph of a game; Richardson's theorem; uniqueness of kernel in 
acyclic digraphs. \li

\end{itemise}

\end{document}
