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\heading {\bf MATH 655,   Topics in Complex Analysis  }\endheading



\centerline{\bf Fall 2001, Mei-Chi Shaw}

\bigskip
In this course we will discuss several main topics in    several
complex variables and complex differential geometry using   partial
differential equations.
It  is intended as an  introductory course for students who are interested
in partial differential
equations arising from problems in    several complex variables and complex
geometry.  We will start  from
the beginning    with no assumption of previous knowledge other than
familiarity with  the
basic real analysis and one complex variable.
\medskip
Our first goal is to study   the Cauchy-Riemann equations and tangential
Cauchy Riemann
equations in the simplest setting, i.e., $\Bbb C^n$.      Several different
methods will be introduced to
solve these equations, including   the
$L^2$ technique,  the  $\bar\partial$-Neumann problem and the integral
representation method.
Applications will  also be given,  including function  theory in
pseudoconvex domains,
regularity of the Bergman projection up to the boundary.  Recent progress
on Lipschitz domains and  the
irregularity of the Bergman projection on the so-called worm domains will
also be discussed.
\medskip

The second half of the semester will be concentrated on  complex geometry.
Here the curvature of the manifolds will
come into play.   The topics include


1. Harmonic functions on Riemannian manifolds.


2. Stein and K\"ahler manifolds,
curvature of K\"ahler manifolds, Chern classes.

3.  Bergman metrics,   K\"ahler-Einstein
metrics and   the Calabi conjecture.

4.  Cauchy-Riemann equations in complex manifolds.






\medskip
\noindent
{\bf References}:
\medskip
\noindent
1. \lq\lq Partial Differential Equations in Several Complex Variables"
\newline
\noindent
  So-Chin Chen and Mei-Chi Shaw,   AMS/IP Studies in Advanced Mathematics,
volume 19,
American
Mathematical Society, Providence, RI; International Press, Boston, MA, 2001.
\medskip
\noindent
2.  \lq\lq  Canonical Metrics in K\"ahler Geometry"
\newline
\noindent
  Gang Tian, Lectures in Mathematics, Birkh\"auser Verlag, Basel, Boston,
Berlin. 2000.
\medskip
\medskip
\noindent
3.  \lq\lq  Complex Differential Geometry"
\newline
\noindent
  Fangyang  Zheng,   AMS/IP Studies in Advanced Mathematics, volume 18.
American
Mathematical Society, Providence, RI; International Press, Boston, MA, 2000.
\medskip
\medskip
\noindent {\bf Prerequisite}: Basic real   analysis and one complex variable.




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