Records from the S-Matrix Marathon: Tasty Bits of Several Complex Variables

TL;DR

This paper provides an overview of several complex variables, focusing on holomorphic functions, pseudoconvexity, CR functions, and the $ar ext{d}$-problem, connecting domain geometry with function theory.

Contribution

It synthesizes key concepts in several complex variables and links geometric properties of domains to holomorphic function theory, based on a series of lectures.

Findings

Characterization of domains of holomorphy

Connection between boundary geometry and holomorphic functions

Extension of holomorphic functions from subspaces

Abstract

We will cover the basics of several complex variables in 4 lectures: Basic properties of holomorphic functions in several variables, the notion of pseudoconvexity, CR functions and CR geometry, and the $\bar\partial$-problem. The main underlying idea is to connect various characterizations of domains of holomorphy, that is, the natural domains of definition for holomorphic functions. In the process we will connect the function theory on the domain to geometric properties of the boundary, and discuss the relationship between the boundary values and the functions themselves and extension of holomorphic functions from subspaces. These notes are based on a series of lectures held during the S-Matrix Marathon workshop at the Institute for Advanced Study on 11-22 March 2024.