\documentstyle[12pt]{article}
\textwidth 6.1in
\evensidemargin 0.1in
\oddsidemargin 0.1in
\setlength{\parskip}{2pt}
\newcommand{\SECT}[1]
		{\vspace{.25cm}
		$\!\!\!\!\!\!\!\!\!\!\!${\bf {#1}}
		\vspace{.15cm}}

\begin{document}
\bibliographystyle{alpha}
\pagestyle{empty}

\begin{center}
	 { \Large \bf
Topics for oral qualifying exam for Zhixiang Wang\\
\vskip .4em
Spring, 2003}
\end{center}

\begin{center}
{\bf Major topic: Vertex operator algebras}
\end{center}

The general theory of vertex operator algebras, basic examples, the
basic geometry of vertex operator algebras as presented in \cite{FLM}
(Chapters 1--8), \cite{FHL}, \cite{H} and \cite{LL}.  The specific
topics are:

\begin{enumerate}
\item Definitions and properties.
\begin{enumerate}
\item Formal calculus.
\item The notions of vertex algebra and of vertex operator algebra,
and basic properties.
\item Rationality, commutativity and associativity (various formulations,
including ``weak'' formulations.
\end{enumerate}

\item Examples of vertex (operator) algebras.
\begin{enumerate}
\item Vertex (operator) algebras based on Heisenberg Lie algebras.
\item Vertex (operator) algebras based on affine Lie algebras.
\item Vertex (operator) algebras on the Virasoro algebra.
\item Vertex (operator) algebras on even lattices.
\end{enumerate}

\item Representations of vertex (operator) algebras.
\begin{enumerate}
\item The notion of module and basic properties.
\item The relations between this notion and the topics listed
above, including the construction of modules from mutually local
operators.
\item The notions of intertwining operator and fusion rule, basic properties.
\item Examples of modules for the vertex operator algebras listed above.
\end{enumerate}

\item The geometry of vertex operator algebras.
\begin{enumerate}
\item The moduli spaces of spheres with tubes, the sewing operation,
examples for the sewing operation.
\item The geometric interpretations of vertex operators and
vacua, the geometric meanings of commutativity, associativity,
skew-symmetry, $L(-1)$-derivative property, vacuum property and
creation property.
\item The notion of geometric vertex operator algebra and the
statement of the isomorphism theorem.
\item Determinant lines of Fredholm operators, determinant lines over
Riemann surfaces with parametrized boundaries, relation to the
central charges of vertex operator algebras.
\end{enumerate}
\end{enumerate}

\begin{center}
{\bf Minor topic: Riemann geometry}
\end{center}

Basic Riemannian geometry as presented in the first six chapters of
\cite{P}.  The specific topics are:
%
\begin{enumerate}
\item Riemannian metrics
%
\begin{enumerate}
\item Riemannian manifolds and maps.
\item Groups and Riemannian manifolds.
\item Local representations of metrics.
\item Doubly warped products.
\end{enumerate}
\item Curvature
\begin{enumerate}
\item Connections and curvature.
\item The fundamental curvature equations and the equations of
Riemannian geometry.
\end{enumerate}
\item Examples
\begin{enumerate}
\item Warped products.
\item Hyperbolic space.
\item Left-invariant metrics.
\item Complex projective space.
\end{enumerate}
\item Hypersurfaces
\begin{enumerate}
\item The Gauss map
\item Existence of hypersurfaces.
\item The Gauss-Bonnet theorem.
\end{enumerate}
\goodbreak
\item Geodesics and distance
\begin{enumerate}
\item The connection along curves and geodesics.
\item The metric structure of a Riemannian manifold.
\item The exponential map.
\item Short geodesics are segments.
\item Local geometry in constant curvature.
\item Completeness.
\item Characterization of segments
\item Metric characterization of maps.
\end{enumerate}
\item Sectional curvature comparison
\begin{enumerate}
\item Basic comparison estimates.
\item Riemannian covering maps.
\item Positive sectional curvature.
\end{enumerate}
\end{enumerate}

\begin{thebibliography}{FLM2}
\bibitem[FHL]{FHL}
I. Frenkel, Y.-Z. Huang and J. Lepowsky, On Axiomatic Approaches
to Vertex Operator Algebras and Modules, Memoirs Amer. Math. Soc. 104
(1993).

\bibitem[FLM]{FLM}
I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras
and the Monster, Academic Press, 1988.


\bibitem[H]{H}
Y.-Z. Huang, Two-dimensional conformal geometry and vertex operator
algebras, Progress in Math., Vol. 148, Birkh\"{a}user Boston, 1997.


\bibitem[LL]{LL}
J. Lepowsky and H. Li, Introduction to vertex operator algebras and
their representations, to appear.

\bibitem[P]{P}
P. Petersen, Riemannian geometry, Graduate Text in Mathematics,
Springer-Verlag, New York, 1997.

\end{thebibliography}
\end{document}
