Course description

This course is an introduction to the theory of functions of several complex variables, emphasizing the part of the theory that intersects with analysis and with partial differential equations.
Here are some of the topics to be discussed.

• multivariable power series
  • Reinhardt domains
  • domains of convergence
  • the Hartogs phenomenon
  • entire functions
• integral representations
  • the Cauchy integral
  • the Bochner–Martinelli integral
  • the Bergman kernel function
• notions of convexity
  • linear convexity
  • polynomial convexity
  • holomorphic convexity
  • pseudoconvexity
• the Levi problem
• the ∂ problem
• holomorphic mappings

Course objectives

By the end of the course, you should be able to

• describe the similarities and the differences between one-dimensional function theory and multidimensional function theory;
• explain the concept of domain of holomorphy;
• read the research literature on multidimensional function theory.

Prerequisites

You should have some acquaintance at the first-year graduate level with both real analysis and single-variable complex analysis. The official prerequisites for this course are Math 608 and Math 618 (or equivalents).

Textbook

There is no required textbook. A list of references is available.

Meeting time and place

The course meets 12:40–1:30 on Monday, Wednesday, and Friday afternoons in room 163 of the Blocker Building.

Grading

Grades will be based on class participation and homework. There will be no examinations (in particular, no final examination).

Course website

https://www.math.tamu.edu/~boas/courses/650-2019c/

Office hours

During the Fall 2019 semester, my office hour in Blocker 601L is 1:00–2:00 in the afternoon on Tuesday and Thursday; I am available also by appointment. The best way to contact me is via email to boas@tamu.edu. Telephone messages can be left at the main office of the Department of Mathematics, 979-845-7554.