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\title{Oral Exam Syllabus}
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\section{Lie Groups}
\begin{itemize}
\item Definition of a Lie group; examples (including classical Lie groups)
\item Lie algebras and their relation to Lie groups
\subitem Exponential mapping
\subitem Adjoint and co-adjoint representation

\item Representations of compact, connected Lie groups
\subitem Peter-Weyl theorem
\subitem Maximal Tori: existence, uniqueness up to conjugation, Weyl
\subitem covering theorem, examples for classical groups.
\subitem Weyl group; action on maximal torus and its Lie algebra
\subitem Complexification; roots; positive roots; dominant alcove 
\subitem Dynkin diagrams
\subitem Weight spaces, dominant weights
\subitem Highest weight theorem
\item Formulae
\subitem Weyl integration formula
\subitem Weyl character formula
\subitem Dimension formula

\item Homogeneous vector bundles
\subitem Induced representations
\subitem Frobenius reciprocity

\item Borel-Weil theorem

\end{itemize}

\section{Functional Analysis}

\begin{itemize}
\item Banach spaces 
\subitem Examples ($L^p$ spaces, sequence spaces, direct sums, quotients)
\subitem Linear functionals: duals, reflexive spaces, Hahn-Banach theorems
\subitem Baire category theorem, Open Mapping theorem, Closed Graph 
\subitem theorem, Banach-Steinhaus (uniform boundedness) theorem
\subitem Hilbert spaces (polarisation, adjoints, Riesz lemma)
\item Topological devices
\subitem Nets
\subitem Compactness (Tychonoff theorem, Urysohn's lemma, Stone-Weierstrass 
\subitem theorem)
\subitem Banach-Alaoglu theorem

\item Bounded operator theory
\subitem Adjoints
\subitem Spectrum
\subitem Compact operators
\subitem Fredholm alternative
\subitem Spectral decomposition of compact, self-adjoint operators

\item Differential operators and spectral theory
\subitem Schwarz space
\subitem Fourier transform
\subitem Distributions
\subitem Sobolev spaces

\end{itemize}

\begin{thebibliography}{99}

\bibitem{brocker} Br\"ocker, T. and tom Dieck, T., {\it Representations of Compact Lie Groups}
\bibitem{kolk} Duistermaat, J., and Kolk, J., {\it Lie Groups}
\bibitem{simonreed} Reed, M. and Simon, B., {\it Functional Analysis} 

\end{thebibliography}



  









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