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

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

$\bullet$ Morrey-Kohn-Hormander Theorem;

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

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

$\bullet$ The Levi problem;

$\bullet$ Subelliptic estimates for $\bar{\partial}$-Neumann
operator.\\

4. Tangential Cauchy-Riemann Complex:
\smallskip

$\bullet$ Definition of CR manifold;

$\bullet$ Subelliptic estimates for the Tangential Cauchy-Riemann
Complex.\\

\bigskip

\noindent II.\ {\bf Partial Differential Equations}


\smallskip


1. Fourier analysis and Sobolev spaces:
\smallskip

 $\bullet$ Fourier transform;

 $\bullet$ Definition of Sobolev space;

 $\bullet$ Density Theorem of smooth functions into Sobolev space;

 $\bullet$ Sobolev embedding Theorem;

 $\bullet$ Rellich embedding Theorem.\\

2. Second order elliptic equations:
\smallskip

$\bullet$ Weak maximum principle for classical second order
elliptic equations;

$\bullet$ Hopf Lemma;

$\bullet$ Definition of weak solution;

$\bullet$ Lax-Milgram Theorem;

$\bullet$ Existence of weak solution;

$\bullet$ Regularity of weak solution.\\

3. Subelliptic analysis:
\smallskip

$\bullet$ Definition of pseudo-differential operator;

$\bullet$ Hormander's hypoelliptic Theorem.\\

\bigskip
\begin{thebibliography}{99}

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

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

\bibitem[Oh]{Oh} Takeo Ohsawa, Analysis of Several Complex
Variables.

\bibitem[Ev]{Ev} Evans, Lawrence C, Partial Differential
Equations.


\end{thebibliography}
\end{document}
