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\title
Math 606, Spring 2002\vskip 10pt
Xavier
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\centerline {\bf SYLLABUS \rm}
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The Poisson kernel, The harmonic conjugate kernel, the Dirichlet problem
in the unit disc. Boundary behavior of harmonic and holomorphic
functions, non-tangential convergence. Fatou's theorem and other
boundary representation of harmonic functions satisfying various
integrability conditions. Intoduction to the Hardy spaces $H^p$. The F.
and M. Riesz theorem on measures orthogonal to $e^{in\theta}, \; n\geq
1$. The Privalov localization construction, the local fatou theorem, the
Privalov uniqueness theorem, The Marcinkiewicz- Zygmund-Spencer theorem on
existence of non-tangential limits in terms of the area integral.
A short introduction to the theory of minimal surfaces in differential
geometry and its connections to complex analysis. Applications to
minimal surfaces of Runge's theorem and of the theory of boundary
behavior of holomorphic functions; the role of curvature.
The Mittag-Leffler theorem, the Picard theorem, Beurling's theorem on the
invariant subspaces of the shift operator. Subharmonic functions, solution
of the Dirichlet problem by Perron's method. Definition of Riemann
surfaces, examples.
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Textbook: Conway's Functions of one complex variable, Rudin's Real and
complex Analysis and Koosis's Introduction to $H^p$ spaces.
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