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\centerline{\huge Oral Qualifying Exam Syllabus}
\bigskip
\centerline{\large Amit Priyadarshi}
\centerline{\large December 4, 2006}
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\centerline{\large Committee: Roger Nussbaum, Richard Bumby, Michael Vogelius, Richard Wheeden}


\begin{enumerate}[I.]
\item {\bf Major Topic: Fractal Sets and Dimensions}
\begin{enumerate}[1.]
\item {\bf Introduction}
\begin{itemize}
\item Definition of Hausdorff measure and Hausdorff dimension
\item Basic properties
\item Vitali covering theorem
\item Relationship between $n$-dimensional Hausdorff measure and Lebesgue measure
\item Calculation of Hausdorff dimension and measure for Cantor set
\end{itemize}
\item {\bf Self-similar sets and Hutchinson's theorem}
\begin{itemize}
\item Contractions and Similitudes
\item Invariant sets and self-similar sets
\item Open set condition
\item The Hutchinson's theorem
\end{itemize}
\item {\bf Apollonian packings}
\begin{itemize}
\item Lower and upper bounds for the Hausdorff dimension of the residual set for the Apollonian packing
\item The disk-packing constant
\end{itemize}
\item {\bf Basic density properties}
\begin{itemize}
\item Upper and lower densities
\item Regular and irregular sets
\item Elementary density bounds
\end{itemize}
\end{enumerate}

\item {\bf Minor Topic: Linear Operator Theory}
\begin{enumerate}[1.]
\item {\bf Bounded Linear operators on Hilbert spaces}
\begin{itemize}
\item Some examples of bounded linear operators with estimates of norms
\item Bounded linear functionals and the Riesz representation theorem
\item Self adjoint and normal operators
\item Compact operators
\end{itemize}
\item {\bf Spectral Theory}
\begin{itemize}
\item Spectrum and resolvent
\item Spectral radius
\item Functional Calculus
\item Spectral theorem for compact self adjoint operators
\item Spectral theorem for compact normal operators
\item Spectral theorem for bounded self adjoint operators
\end{itemize}
\item {\bf Trace class and Hilbert-Schmidt operators}
\item {\bf Trace and determinant}
\end{enumerate}
\end{enumerate}

\begin{thebibliography}{}
\bibitem[F]{F} Falconer, K.J., {\it The Geometry of Fractal Sets}, Cambridge University Press, 1985.
\bibitem[B]{B} Boyd, D.W., {\it The Disk-Packing Constant}, Aequationes Mathematicae 7, 182-193, 1971.
\bibitem[B]{B} Boyd, D.W., {\it The Residual Set Dimension of the Apollonian Packing}, Mathematika 20, 170-174, 1973.
\bibitem[GG]{GG} Gohberg, I., Goldberg, S., {\it Basic Operator Theory}, Birkh\"{a}user, 1981.
\bibitem[GGK]{GGK} Gohberg, I., Goldberg, S., Kaashoek, M.A., {\it Classes of Linear Operators} Vol.I, Birkh\"{a}user, 1990.
\end{thebibliography}
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