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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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