Professor Maciej Zworski

email: zworski@math.berkeley.edu

Office: 801 Evans Hall

Office hours: Tuesday 11-12, Thursday 11-12

email: zworski@math.berkeley.edu

Office: 801 Evans Hall

Office hours: Tuesday 11-12, Thursday 11-12

Prerequisites:

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*(Section 15.1)

- Dmitry Khavinson,
*Holomorphic Partial Differential Equations and Classical Potential Theory*(Chapters 1-8)

- Johannes Sjöstrand,
*Analytic microlocal analysis using holomorphic functions with exponential weights*### Syllabus:

- Review of the theory of functions of one complex variables
- Holomorphy, power series in several complex variables, Hartogs property
- 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
- Pseudodifferential operators in complex domains (no background in standard theory is expected or needed)
- Fourier integral operators, Egorov's theorem and applications

### Grading:

You are asked to write a short paper on a research topic related to the course subject, for instance, based on a topic from the texts not covered in class.