# Course Descriptions

## 16:640:523 - Functions of Several Complex Variables I

Xiaojun Huang

Subtitle: Several Complex Variables

Text: Lecture Notes by Xiaojun Huang to be distributed in the class

Prerequisites: Real and complex analysis, functional analysis

### Description:

A function with $$n$$ complex variables $$zin { f C}^n$$ is said to be holomorphic if it can be locally expanded as power series in $$z$$. An even dimensional smooth manifold is called a complex manifold if the transition functions can be chosen as holomorphic functions. Roughly speaking, a Cauchy-Riemann manifold (or simply, a CR manifold) is a manifold that can be realized as the boundary of a certain complex manifold. Several Complex Variables is the subject to study the properties and structures of holomorphic functions, complex manifolds and CR manifolds. Different from one complex variable, if $$n>1$$ one can never find a holomorphic function over the punctured ball that blows up at its center. This is the striking phenomenon that Hartogs discovered about 100 years ago, which opened up the firstpage of the subject. Then Poincar'e, E. Cartan, Oka, etc, further explored this field and laid down its foundation. Nowadays as the subject is intensively interacting with other fields, providing important examples, methods and problems, the basic materials in Several Complex Variables have become mandatory for many investigations in pure mathematics. This class tries to serve such a purpose, by presenting the following topics from Several Complex Variables.

1. Pseudoconvex domains and domain of holomorphy, Levi problem
2. L^2-estimates for dbar equations
3. Introduction to complex analytic varieties

The course materials will be largely taken from the following:

1. L. Hormander, {it An introduction to complex analysis in several variables}, Third edition, North-Holland, 1990.
2. Xiaojun Huang, Topics in Several Complex Variables, Lecture Notes to be distributed.
3. Xiaojun Huang, {it Subelliptic analysis in Cauchy-Riemann Geometry and Complex Geometry}, Lecture Notes on the national summer graduate school of China, 2007. (to appear)
• Prerequisites: One complex variable and the basic Hilbert space theory from real analysis.