\documentclass[12pt]{article}
\pagestyle{empty}
\parindent=0in

\oddsidemargin=0in
\evensidemargin=0in
\marginparsep=0in
\marginparwidth=0in

\topmargin=0in
\headheight=0in
\headsep=0in
\footskip=0in

\textwidth=6.5in
\textheight=9in

\begin{document}
Syllabus -- Kai Medville
\begin{enumerate}
  \item Functional Analysis
     \begin{enumerate}
       \item Hilbert spaces 
	\begin{enumerate}
	\item Riesz lemma
        \item orthonormal basis
	\end{enumerate}
       \item Banach spaces
	 \begin{enumerate}
       		\item Hahn-Banach theorem
       		\item Baire category theorem, principle of uniform boundedness,
open mapping theorem, closed graph theorem
	 \end{enumerate}
       \item weak topologies, weak-$*$ topology, Banach-Alaoglu
theorem
       \item Fredholm alternative and the spectral theory of compact operators
     \end{enumerate}
  \item Sobolev Spaces
     \begin{enumerate}
       \item density of smooth functions in Sobolev space
       \item global extensions
       \item trace theorem
       \item Sobolev inequalities (Morrey,
Gagliardo-Nirenberg-Sobolev)
       \item compact imbedding
       \item Poincare inequality
     \end{enumerate} 
  \item Laplace's Equation
     \begin{enumerate}
       \item the fundamental solution
       \item Poisson's equation, $\Delta u=f$ in $\Omega$, $\Omega$ a bounded domain, $f$ H\"{o}lder continuous
       \item mean value formula and maximum principle for subharmonic functions
       \item Harnack's inequality
       \item Green's function: for ball and half-space
       \item single and double layer potentials
     \end{enumerate}
  \item Second Order Elliptic Equations
     \begin{enumerate}
       \item strong and weak maximum principle
       \item uniqueness of Dirichlet and Neumann boundary value
problems
       \item definition of weak solutions
       \item existence: Lax-Milgram, energy estimates, Fredholm
alternative
       \item regularity of solutions
         \begin{enumerate}
            \item interior and boundary regularity, smooth coefficients
            \item H\"{o}lder continuity for bounded coefficients
         \end{enumerate}
      \end{enumerate}
\end{enumerate}
References
\begin{enumerate}
	\item Partial Differential Equations by L.C.Evans
	\item Elliptic Partial Differential Equationa of Second Order by Gilbarg and Trudinger
	\item Maximum Principle by Protter and Weinberger
	\item Functional Analysis I by Reed and Simon
	\item Lectures on Partial Differential Equations by G.B. Folland
\end{enumerate}
\end{document}

