MA 224 (i.e., the first course in Complex Analysis)

preferably, some exposure to complex dynamics in one variable (the latest iteration of the topics course MA 324, Topics in Complex Analysis in One Variable, for instance).

Students who have not seen any one-dimensional complex dynamics earlier but are highly interested in this course are encouraged to speak to the instructor.

This topics course is being run as an experiment in approaching the basic concepts in several complex variables with the eventual aim of studying some topics in multi-variable complex dynamics. By “complex dynamics”, we mean the the study of the dynamical system that arises in iterating a holomorphic map.

The course will begin with a complete and rigorous introduction to holomorphic functions in several variables and their basic properties. This will pave the way to motivating and studying a concept that is, perhaps, entirely indigenous to several complex variables: the notion of plurisubharmonicity.

Next, we shall look at some of the motivations behind the study of complex dynamics in several variables. Using the tools developed, we shall undertake a crash-course in currents, which are objects central to the study of some aspects of complex dynamics. We shall then cover as much of the following topics as time permits:

Properties of fixed points

The existence of proper subdomains of $C^n$, $n \geq 2$, that are holomorphically equivalent to $C^n$

The Fatou and the Julia set for a dominant holomorphic self-map of $CP^n$, $n \geq 2$

The Green current associated to a dominant holomorphic self-map of $CP^n$, and the dynamical information that it provides.

Suggested books and references:

L. Hormander, Complex Analysis in Several Variables, 3rd edition, North-Holland Publishing Co. Amsterdam, 1990.

J.E. Fornaess, Dynamics in Several Complex Variables, CBMS Series, No. 87, American Mathematical Society, Providence, Rhode Island, 1996.