Komplex analys i flera variabler

**Valid from:** Spring 2019**Decided by:** Professor Thomas Johansson**Date of establishment:** 2019-09-12

**Division:** Mathematics**Course type:** Course given jointly for second and third cycle**The course is also given at second-cycle level with course code:** MATP22**Teaching language:** English

The aim of the course is to give an overview of the field of complex analysis in several variables, in particular hightlighting the differences from and the similarities with complex analysis in one variable,

*Knowledge and Understanding*

For a passing grade the doctoral student must

- be able to give different conditions for a function of several complex variables to be holomorphic and to explain why they are equivalent.
- be able to account for the theory of domains of holomorphy. In particular be able to give some examples.
- be able to explain how analytic continuation in several variables is different from the situation in one variable, in particular account for some versions of Hartogsâ€™ extension theorem.
- be able to describe the local structure of zero sets of holomorphic functions and how they relate to analytic sets and complex submanifolds.
- be able to explain how pluripotential theory, in particular the study of plurisubharmonic functions, is needed to develop general methods for solving d-bar equations.
- be able to compare and contrast different notions of convexity, in particular geometric convexity and pseudoconvexity.
- be able to give some examples of integral representation formulas for holomorphic functions in several variables.
- be able to give some examples of the relevance of the solvability of d-bar equations.

Basic properties of holomorphic functions. Analytic continuation and power series in several variables. The inhomogeneous Cauchy-Riemann equation. The Weierstrass preparation theorem, zero sets and singularities. Hartog's theorem. Holomorphic mappings and complex manifolds. Convexity and holomorphic convexity. Domains of holomorphy. The first Cousin problem. The Levi problem. Pluripotential theory. Pseudoconvex domains. Integral representation formulas. Solutions of the d-bar equation for pseudoconvex domains.

Korevaar, J. & Wiegerinck, J.: Lecture notes in several complex variables. 2017.

Freely available from
https://staff.science.uva.nl/j.j.o.o.wiegerinck/edu/scv/scvboek.pdf

**Type of instruction:** Lectures

**Examination format:** Oral exam**Grading scale:** Failed, pass**Examiner:**

**Assumed prior knowledge:** Complex analysis in one variable, Fourier analysis and Functional Analysis.

**Course coordinators:** **Web page:** http://www.maths.lth.se/matematiklth/personal/frankw/#/scv19