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 \title {\bf Syllabus for  Oral Qualifying
Exam}
\author{Yuan Yuan}
\date{}
\begin{document}
\maketitle
\par \noindent
{\it Committee:} Sagun Chanillo, Stephen Greenfield, Xiaojun
 Huang, Feng Luo.
\bigskip

\noindent I.\ {\bf Several Complex Variables}

\smallskip

1. Cauchy integral formula and its applications:
\smallskip

  $\bullet$ Cauchy integral formula in polydiscs, Cauchy
estimates;

  $\bullet$ Bochner-Martinelli formula;

  $\bullet$ Hartogs extension Theorem;

  $\bullet$ Bochner's extension Theorem.\\

2. Subharmonicity and convexity:
\smallskip

 $\bullet$ Properties of subharmonic functions and plurisubharmonic functions;

 $\bullet$ Domain of holomorphy, Continuity Principle;

 $\bullet$ Pseudoconvexity, Levi pseudoconvexity;

 $\bullet$ Oka's Theorem.\\

3. $L^2$ theory for $\bar{\partial}$ on pseudoconvex domains:
\smallskip

$\bullet$ Morrey-Kohn-Hormander Theorem;

$\bullet$ $L^2$ Existence Theorem for $\bar{\partial}$ operator;

$\bullet$ $\bar{\partial}$-Neumann problem;

$\bullet$ The Levi problem.\\

4. Kodaira's Embedding Theorem:
\smallskip

$\bullet$ Kodaira-Bochner formula;

$\bullet$ Vanishing theorem;

$\bullet$ Hodge manifolds and embedding;

%\bigskip
\newpage
\noindent II.\ {\bf Riemannian Geometry}


\smallskip


1. Riemannian metrics and connections
\smallskip


2. Curvature
\smallskip


3. Jacobi fields
\smallskip

4. Hopf-Rinow Theorem and Hadamard Theorem

\bigskip
\begin{thebibliography}{99}

\bibitem[C]{C} Do Carmo, Riemannian Geometry.

\bibitem[CS]{CS} So-chin Chen, Mei-chi Shaw, Partial Differential Equations in Several Complex
Variables.

\bibitem[M-K]{M-K} Morrow, Kodaira, Complex Manifolds.

\bibitem[Kr]{Kr} Steven G. Krantz, Function Theory of Several
Complex Variables.



\end{thebibliography}
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