\documentclass[12pt]{letter}

\begin{document}
\thispagestyle{empty}
\setlength{\topmargin}{-1.5cm}
\setlength{\oddsidemargin}{.5cm}
\setlength{\evensidemargin}{.5cm}
\large
\begin{center}Oral Exam Syllabus \\ \large Benjamin Bunting \\ June 24, 2005 \end{center}

\normalsize
Committee: Michael Vogelius, Stephen Greenfield, Zheng-Chao Han and Michael Kiessling
\renewcommand{\labelenumi}{\Roman{enumi}.}
\renewcommand{\labelenumii}{\roman{enumii}.} 
\renewcommand{\labelenumiii}{\alph{enumiii}.} 

\begin{enumerate}
{\bf \item Partial Differential Equations}
\begin{enumerate}
{\bf \item Laplace Equation}
\begin{enumerate}
\item Derivation of the fundamental solution
\item Mean value formula and maximum principles
\item Analyticity of harmonic functions via gradient estimates
\item Existence Technique 1: The Perron method and Harnack's inequality
\item Existence Technique 2: Discretization of the Laplace equation
\item Dirichlet's Principle
\end{enumerate}
{\bf \item Heat Equation}
\begin{enumerate}
\item Derivation of the fundamental solution
\item Maximum principles
\item Energy methods
\end{enumerate}
{\bf \item Wave Equation}
\begin{enumerate}
\item Solution by spherical means
\item Uniqueness and finite propagation speed via energy methods
\end{enumerate}
{\bf \item Sobolev Spaces}
\begin{enumerate}
\item Definitions: $H^s, W^{k,p}$
\item Density of $C^\infty$ in $W^{k,p}$
\item Extensions, traces, compactness
\item Sobolev inequalities
\item Fourier transform methods
\end{enumerate}
{\bf \item Second Order Elliptic Equations}
\begin{enumerate}
 \item Definition of elliptic equations, weak solutions
\item Existence of weak solutions via Lax-Milgram theorem, energy
estimates, Fredholm alternatives
\item Maximum principles weak and strong, Hopf lemma
\item Regularity of solutions in $H^k$
\item Existence via Schauder estimates in $C^\alpha$
\item Consequences of Riesz-Thorin
\item Eigenvalues of symmetric elliptic operators
\end{enumerate}
\end{enumerate}
{\bf \item Functional Analysis}
\begin{enumerate}
{\bf \item Hilbert Spaces}
\begin{enumerate}
\item Riesz representation and consequences
\item Orthonormal bases
\item Compact operators on a Hilbert space
\item Diagonalization of compact self-adjoint operators
\item Application: Sturm-Liouville systems
\end{enumerate}
{\bf \item Banach Spaces}
\begin{enumerate}
\item Linear functionals
\item Hahn-Banach theorem
\item Applications: Banach limits, Runge's theorem
\item Baire category, open mapping, closed graph theorems
\item The principle of uniform boundedness
\end{enumerate}
{\bf \item Weak Topologies}
\begin{enumerate}
\item Alaoglu's theorem, consequences of reflexivity
\item Schauder, Markov-Kakutani, Ryll-Nardzewski fixed point theorems
\item Application: Haar measure on a compact group
\end{enumerate}
\end{enumerate}
\end{enumerate}

{\bf References}
\renewcommand{\labelenumi}{\roman{enumi}.}
\begin{enumerate}
\item Evans, Lawrence C., {\it Partial Differential Equations}
\item Folland, Gerald B., {\it Introduction to Partial Differential Equations. Second Edition}
\item Han, Qing and Lin, Fanghua, {\it Elliptic Partial Differential Equations}
\item Conway, John B., {\it A Course in Functional Analysis}
\item Folland, Gerald B., {\it Real Analysis}
\end{enumerate}

\thispagestyle{empty}
\end{document}
