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\begin{center}
	 { \Large \bf 
Topics for oral qualifying exam for David Radnell, June, 2000} 
\end{center}

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

The general theory of vertex operator algebras, simple examples, the
basic geometry of vertex operator algebras, determinant lines
over Riemann surfaces with parametrized boundaries and the diffeomorphism
group of $S^{1}$ as presented in \cite{FLM} (Chapters 1--8),
\cite{FHL}, \cite{H} and \cite{L}.  The specific topics are:

\begin{description}


\item The notions of vertex algebra and of vertex operator algebra,
and basic properties

\item Rationality, commutativity and associativity (various formulations,
including ``weak'' formulations, as presented in \cite{FLM},
\cite{FHL} and \cite{L})

\item Examples (using local Systems theory): Vertex algebras and vertex operator algebras based on
Heisenberg Lie algebras,  on the Virasoro
algebra and on even lattices. 



\item The moduli spaces of spheres with tubes, the sewing operation
and sewing equation, the geometric and formal solutions of the sewing
equation, 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, as presented

\item The notion of geometric vertex operator algebra and the isomorphism
theorem

\item Determinant lines of Fredholm operators, basic categorical
properties, basic canonical isomorphisms

\item Determinant lines over Riemann surfaces with parametrized
boundaries, the construction of the sewing canonical isomorphisms, the
associativity of the sewing canonical isomorphisms

\item The groups $GL_{\mbox{\rm res}}(H)$ and $\mbox{\rm Diff} S^{1}$, the
central extension of $GL_{\mbox{\rm res}}(H)$, two constructions of
the basic central extensions of $\mbox{\rm Diff} S^{1}$


\end{description}

\newpage

\begin{center}
{\bf Minor topic: Functional Analysis}
\end{center}
Basic Hilbert and Banach space theory. Banach algebras and elementary
spectral theory. Ideals of bounded operators. Fredholm operators. 

\begin{description}
\item {\bf Hilbert and Banach Spaces:} \cite{R},\cite{S},\cite{D} \\
Existence of orthogonal bases. Dual
space and adjoint operators. Riesz representation theorem. 
$C([a,b])$ and $C([a,b])^*$. Topologies (weak, weak* and operator).
Banach-Alaoglu, Hahn-Banach and Open Mapping theorems.
Embedding into $C(X)$.


\item {\bf Banach algebras and spectral theory} \cite{D},\cite{M},\cite{R} \\
Stone-Weierstrass Theorem. Example (Disc Algebra). 
Elementary properties of the spectrum and
spectral radius. Spectral mapping theorem. Gelfand
Representation for abelian Banach Algebras.

\item{\bf Operator Ideals:} \cite{M},\cite{BS} \\
Definition of, and relation between, finite
rank, compact, Hilbert-Schmidt and trace class operators. Example
using kernels.
Correspondence between operator ideals and certain sequence
spaces (Calkin Theory).

\item{\bf Fredholm operators:} \cite{D},\cite{M} \\
Fredholm Alternative. Relation to existence/uniqueness of differential equations. Atkinson Theorem.



\end{description}
\newpage
\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[L]{L} 
H. Li, Local systems of vertex operators, vertex superalgebras and
modules, J. Pure and Applied Algebra 109 (1996), 143-195.
\bibitem[D]{D}
R. G. Douglas, Banach Algebra Techniques in Operator Theory (second
edition), Graduate Texts in Mathematics 179, Springer, 1998

\bibitem[M]{M} 
G. J. Murphy, $C^*$-Algebras and Operator Theory, Academic Press, 1990

\bibitem[R]{R}
W. Rudin, Functional Analysis, McGraw Hill, 1973 

\bibitem[S]{S}
G. F. Simmons, Topology and Modern Analysis, McGraw Hill, 1963

\bibitem[BS]{BS}
B. Simon, Trace Ideals and their Application, Cambridge university
Press, 1979


\end{thebibliography}
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