Complex analysis in several variables I

During the first period of the spring semester 2010, I will give a course on complex analysis in several variables, intended for Ph.D. students.

The course will consist of seven lectures, starting on January 20th. We will meet once a week, probably on Wednesdays between 1-3pm. The lectures will be given in Swedish or English depending on the audience. Note: If you plan to attend the lectures and do not speak Swedish, please let me know in advance to help with my preparations!

Prerequisites

Participants will be assumed to have taken some course in complex analysis in one variable. At the very least, the course Funktionsteori (FMAF01) or Komplex analys (FMA037) is required. It will be helpful if you have taken Analytiska funktioner (MATC11) or Analytiska funktioner (FMA130).

Content

Preliminary content

• Review of elementary theory for holomorphic functions of one complex variable.
• Complex vector spaces, complex structures and the d-dbar operator
• Holomorphic functions of several complex variables and their basic properties
• Integral formulas in one and several variables: Cauchy's integral forumla, the Bochner-Martinelli formula
• Hartog's extension theorem
• Approximation of holomorphic functions by polynomials, Runge's theorem and polynomial convexity
• Domains of existence for holomorphic functions. Introduction to pseudoconvexity
• Holomorphic and biholomorphic mappings. The Riemann mapping theorem and why it does not generalize to several variables
• Automorphism groups and invariant distances

Literature

• Lecture notes (in Swedish)
  • Overview of the lectures (tentative)
  • Lecture 1
  • Lecture 2
  • Lecture 3
  • Lecture 4
  • Lecture 5
  • Lecture 6
  • Lecture 7
• Steven G. Krantz, Function theory in several complex varaibles, 2nd ed, AMS-Chelsea
• Your personal choice of textbook in one complex variable as a reference

Exam

The exam consists of a set of problems and an oral exam. I expect you to solve at leat five or six of the problems.