FSF3610 Analysis in Several Complex Variables 9.0 credits

We will mainly follow Hörmander’s book Complex Analysis in Several Variables, the lecture notes by Bo Berndtsson entitled An Introduction to Things d-bar, and the lecture notes An Introduction to Weighted Pluripotential Theory by Norm Levenberg.

Basic topics:

• Domains of holomorphy

• Pseudoconvexity

• Reinhardt Domains

• Polynomial approximation and Runge Domains

• Hörmander’s d-bar estimates

• Stein Manifolds

Advanced topics:

• Elements of Complex Geometry

• Complex Monge-Ampere Equation

• Weighted pluripotential theory, Monge-Ampere Measures

• Bergman Kernel Asymptotics

After completion of the course, the students should:

• Have a good understanding of basic concepts in the theory of several complex variables

• Be familiar with the language of complex geometry

• Be familiar enough with more advanced concepts to be able to independently read and understand current research in the field

• Have an idea of the focus of current research in the field

The course is aimed towards students at a graduate level, in particular, PhD students in mathematical analysis. A good understanding of the basic theory of analysis in one complex variable is required, as is background in general mathematical analysis such as measure theory, differential geometry and functional analysis.

• Complex Analysis in Several Variables (3rd Edition), Lars Hörmander, North Holland (1990).

• An Introduction to things d-bar, Bo Berndtsson, online lecture notes.

• An Introduction to Weighted Pluripotential Theory, Norm Levenberg, online lecture notes.

If the course is discontinued, students may request to be examined during the following two academic years.

• HEM1 - Home assignments, 9.0 credits, grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

The students should display their proficiency in the subject during meetings with the examiner. Homework problems may be given, and should be graded and discussed by the students themselves.

The students should, via examination as discussed above, have demonstrated that they have reached the learning outcomes.

Håkan Hedenmalm

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.