%% slides for the lecture bergman.tex
\documentclass{slides}

\usepackage{amsmath,amsthm,graphics}

\newtheorem*{theorem}{Theorem}
\newcommand{\dbar}{\overline\partial}

\begin{document}

\large

\begin{slide}

Cauchy integral for disk
\begin{equation*}
f(w)=\frac{1}{2\pi i}\int_{|z|=1} \frac{f(z)}{z-w}\,dz
\end{equation*}

Equivalent form
\begin{equation*}
f(w)=\frac1\pi \int_{|z|<1} \frac{f(z)}{(1-w\bar z)^2}\,dx\,dy,
\end{equation*}

Bergman kernel function for disk
\begin{equation*}
K(w,z)=\frac1{\pi(1-w\bar z)^2}
\end{equation*}

\end{slide}

\begin{slide}
Bergman kernel of a domain
\begin{equation*}
K(w,z)=\sum_j \phi_j(w)\overline\phi_j(z),
\end{equation*}
where $\{\phi_j\}$ is an orthonormal basis for the
square-integrable holomorphic functions.

In the class of holomorphic functions~$f$ such that
$\displaystyle f(z)\ge\int|f|^2$, the one with the maximal value at~$z$ is
$K(w,z)$. 

\end{slide}

\begin{slide}
The Riemann map~$f$ is related to the Bergman kernel function~$K$
via
\begin{equation*}
f'(z)=K(z,a)\sqrt{\frac{\pi}{K(a,a)}}.
\end{equation*}

The Bergman metric
\begin{equation*}
g_{jk}(z)=\frac{\partial^2}{\partial z_j\partial \bar z_k} \log K(z,z).
\end{equation*}

\end{slide}


\begin{slide}

\begin{center}
Bergman Prize winners
\end{center}

\setlength{\parskip}{5pt}

\begin{description}
\item [David Catlin] (1989)\\
(\emph{Princeton}, Chicago,
\emph{Princeton}, \\
Purdue)
\item [Steve Bell and Ewa Ligocka] (1991) \\
(MIT, \emph{Princeton},
Purdue; \\
Warsaw, Polish Acad.\ Sci.)
\item [Charles Fefferman] (1992) \\
(\emph{Princeton}, Chicago,
\emph{Princeton}) 
\item [Yum-Tong Siu] (1993) \\
(\emph{Princeton}, Purdue, Notre
Dame, Yale, Stanford, Harvard)
\item [John Erik Forn{\ae}ss] (1994)\\
(Washington, \emph{Princeton},
Michigan)

\end{description}

\end{slide}

\begin{slide}
\begin{center}
\large\bfseries  Diederich-Forn{\ae}ss
worm domain
\end{center}

\includegraphics[1in,3in][5in,7in]{worm.ps}

A smoothly bounded pseudoconvex domain lacking a Stein
neighborhood basis.
\end{slide}

\begin{slide}
\begin{theorem}[Fefferman]
Let $f:D\to G$ be a biholomorphic mapping between bounded
strongly pseudoconvex domains in~$\mathbf{C}^n$ with boundaries
of class~$C^\infty$. Then $f$~extends to a diffeomorphism of the
closures. 
\end{theorem}

\begin{theorem}[Bell and Ligocka]
Let $f:D\to G$ be a biholomorphic mapping between bounded
domains in~$\mathbf{C}^n$ with boundaries of class~$C^\infty$,
and suppose the Bergman projections preserve the space of
functions that are $C^\infty$ on the closure. Then $f$~extends to
a diffeomorphism of the closures.
\end{theorem}

\end{slide}

\begin{slide}
\begin{theorem}[Catlin]
A necessary and sufficient condition for the $\dbar$-Neumann
problem to be subelliptic on a (weakly) pseudoconvex domain is
that the boundary have bounded order of contact with complex
varieties. 
\end{theorem}

\begin{theorem}[Boas and Straube]
The Fefferman mapping theorem holds for all (weakly) convex domains.

More generally, the $\dbar$-Neumann problem is regular in both
$C^\infty$ and each Sobolev space~$W^k$ for domains that are
defined by plurisubharmonic functions.
\end{theorem}

\end{slide}

\begin{slide}
  \begin{theorem}[Michael Christ, 1995]
    The $\dbar$-Neumann operator and the Bergman projection are
    not $C^\infty$ globally regular in the Diederich-Forn{\ae}ss
    worm domain.
  \end{theorem}

\includegraphics[1in,3in][5in,7in]{worm.ps}
\end{slide}

\begin{slide}
We are dealing with a field of applied
mathematics where the interaction between mathematics and physics
is highly stimulating and fruitful for both sciences. 
-- Bergman and Schiffer, 1953

%The actual computation of a
%complete orthonormal system is a rather time-consuming procedure;
%but 
\ldots\ with the aid of modern computational technique, these
processes come well within the range of practical application.
-- Bergman, 1950

Punch-card
machine methods applied to the solution of the torsion problem,
\emph{Quart.\ Appl.\ Math.}\ \textbf{5} (1943), 69--81.
\end{slide}

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