%% 7 January 2002

\documentclass{article}
\begin{document}

\section{Primary Topic- Functional Analysis}

\subsection{Topological Vector Spaces}
\begin{enumerate}
\item{Types of Topological Vector Spaces}
\item{Metric spaces and seminorms}
\item{examples}
\end{enumerate}

\subsection{Completeness}
\begin{enumerate}
\item{Banach-Steinhaus}
\item{Open Mapping theorem}
\item{Closed Graph theorem}
\end{enumerate}

\subsection{Duality}
\begin{enumerate}
\item{Hahn-Banach theorem}
\item{weak and weak * topologies}
\item{examples of duals}
\item{Compact and integral operators}
\item{Fredholm alternatives}
\end{enumerate}

%% Perhaps compact operators should moved down with SA, sym, uni, normal
%% and their spectral properties discussed in the spectral section.


\subsection{Applications}
\begin{enumerate}
\item{Convergence theorems}
\item{Stone-Weierstrass}
\item{Haar measure}
\end{enumerate}

\subsection{Distributions}
\begin{enumerate}
\item{topology of the test functions}
\item{local representation of distributions}
\item{PV(1/x), $\delta$ function}
\end{enumerate}

\subsection{Fourier Transform}
\begin{enumerate}
\item{rapidly decreasing functions and tempered distributions}
\item{Parseval/Plancherel/Inversion on $L^2$}
\item{range of Fourier transform}
\item{Sobolev lemma}
\end{enumerate}

\subsection{Application to Differential Equations}
\begin{enumerate}
\item{Fundamental solutions}
\item{Sobolev spaces}
\item{Elliptic equations and characteristic polynomials}
\item{regularity theorems}
\item{solution via Fourier transform}
\end{enumerate}

%% Add spectral properties to SA, unitary, ?normal?

\subsection{Bounded operators and Spectral Theorem}
\begin{enumerate}
\item{self-adjoint, unitary, normal operators}
\item{definition and basic properties of Spectrum}
\item{Spectral Theorem (Functional Calculus)}
\end{enumerate}

\subsection{Unbounded operators}
\begin{enumerate}
\item{densely defined operators}
\item{closed, symmetric, and examples}
\item{Spectral Theorem (Functional Calculus)}
\end{enumerate}


\section{Secondary Topic- Differential Geometry}

\subsection{Manifolds}
\begin{enumerate}
\item{Submanifolds, submersions, immersions}
\item{homeomorphisms, diffeomorphisms}
\item{Implicit and Inverse Function theorems}
\end{enumerate}

\subsection{Tangent Space}
\begin{enumerate}
\item{Tangent space, derivation theorem}
\item{Vector fields, flows, and the FT ODE}
\item{Lie Bracket and Derivative}
\end{enumerate}

\subsection{Bundles}
\begin{enumerate}
\item{bundles, fibers, sections}
\item{pull back bundle}
\item{Principle G bundle}
\item{frame and associated bundle}
\end{enumerate}

\subsection{exterior algebra}
\begin{enumerate}
\item{cotangent bundle, exterior algebra and derivative}
\item{closed, exact, de Rham cohomology}
\item{homotopy invariance}
\item{insertion, Cartan's formula}
\item{orientation, orientation of compact manifolds}
\item{integration, Stoke's theorem}
\end{enumerate}

\subsection{Covariant derivatives and curvature}
\begin{enumerate}
\item{distributions, vertical, Frobenius' Theorem}
\item{Connections, covariant derivative, connection 1-form}
\item{existence theorem}
\item{parallel transport and lifts}
\item{Curvature}
\item{Bianchi identity} %% if this doesn't need a metric
\item{Metrics, Levi-Civita connection}
\item{Geodesics}
\end{enumerate}

\subsection{exp}
\begin{enumerate}
\item{exp, completeness, Hopf-Rinow}
\end{enumerate}

\subsection{Basic Symplectic Geometry}
\begin{enumerate}
\item{Symplectic forms and manifolds, symplectomorphisms}
\item{Canonical symplectic structure of cotangent bundle}
\item{Darboux's theorem, including linear version and Moser's trick}
\end{enumerate}

\section{References}
\begin{enumerate}
\item{Rudin, W \it{Functional Analysis}}
\item{Reed, M and Simon, B, \it{Functional Analysis}}
\item{Bott, R and Tu, L, \it{Differential Forms in Algebraic Topology}}
\item{Bredon, G, \it{Topology and Geometry}} 
\end{enumerate}

\end{document}

