Math 823: Advanced Topics in Complex Analysis

Xianghong Gong, Fall 2018:

The course covers a few classical topics in several complex variables. The main topics are selected to demonstrate some useful methods in the study of domains and their boundaries in C^n.

1. Basics of holomorphic functions. Hartogs extension theorem. d-bar homotopy formula for strictly pseudoconvex domains. Newlander-Nirenberg theorem and the KAM rapid iteration.

2. Kobayashi metrics. Biholomorphisms of strictly pseudoconvex domains

3. Segre varieties and Schwarz reflection principle

4. Chern-Moser theory for real hypersurfaces

5. CR singularities, small divisors, and the Moser-Webster theory