Professor Maciej Zworski

email: zworski@math.berkeley.edu

Office: 801 Evans Hall

Office hours: by appointment

email: zworski@math.berkeley.edu

Office: 801 Evans Hall

Office hours: by appointment

Prerequisites:

Textbooks:

- Lars Hörmander,
*An Introduction to Complex Analysis in Several Variables,*3rd edition, (Chapters I and II) (A helpful commentary by John Erik Fornaess can be found here)

- Lars Hörmander,
*The Analysis of Linear Partial Differential Operators I*(Section 9.4)

- Lars Hörmander,
*The Analysis of Linear Partial Differential Operators II*(Section 15.1)

- Dmitry Khavinson,
*Holomorphic Partial Differential Equations and Classical Potential Theory*(Chapters 1-8; these cover Section 9.4 of 2 but with more examples and background)

- Robert Berman, Bo Berndtsson, Johannes Sjöstrand,
*Asymptotics of Bergman kernels*### Syllabus:

- Review of the theory of functions of one complex variables
- Holomorphy, power series in several complex variables, the Hartogs principle
- Domains of holomorphy, plurisubharmonic functions, pseudoconvex domains
- Solving non-homogeneous Cauchy--Riemann equations in spaces defined using global strictly plurisubharmonic weights (Hörmander's L2 estimates)
- Solving partial differential equations with holomorphic coefficients: theorems of Cauchy--Kovalevskaya, Zerner and Bony--Shapira
- Local Bergman kernel asymptotics: a semiclassical version of the theorems of Fefferman and Boutet de Monvel--Sjöstrand
- Bergman kernel asymptotics for powers of positive line bundles over compact complex manifolds; the Catlin--Tian--Yau--Zelditch asymptotics, the Kodaira embedding theorem.

### Grading:

There will be biweekly homework assignments covering basic aspects of the course.