Mathematics 212: Several Complex Variables, Spring 2012

TuTh 11:00AM - 12:29PM, 891 Evans




Professor  Maciej Zworski
email: zworski@math.berkeley.edu
Office: 801 Evans Hall
Office hours: Tuesday 1-2 PM, or by appointment.

Prerequisites:
202AB or equivalent + some basic knowledge of differential forms and smooth manifolds.

Textbooks:
Lars Hörmander, An Introduction to Complex Analysis in Several Variables, 3rd edition, (Chapters I and II)
                        Lars Hörmander, The Analysis of Linear Partial Differential Operators II (Chapter 15.1)

Syllabus:

  1. Review of the theory of functions of one complex variables;
  2. Holomorphy, power series in several complex variables, Hartogs property;
  3. Domains of holomorphy, plurisubharmonic functions, pseudoconvex domains;
  4. Entire functions, weighted L2 spaces, Bergman projectors for L2 spaces with quadratic plurisubharmonic weights;
  5. Applications: Berezin-Toeplitz quantization, Catlin-D'Angelo-Quillen Theorem, analytic proof of the Nullstallensatz;
  6. Solving the equation in L2 spaces with global plurisubharmonic weights;
  7. Fefferman/Boutet de Monvel-Sjöstrand asymptotics for Bergman projectors for "nonlinear" weights;
  8. Quick introduction to complex manifolds, complex line bundles and powers of line bundles;
  9. Applications of Bergman kernel asymptotics: Kodaira's embedding theorem and the Catlin-Tian-Yau-Zelditch asymptotics.

Grading:

based on biweekly homework