This topics course is being run as an experiment in approaching the properties of holomorphic maps in several complex variables (SCV) in a self-contained manner (i.e., without requiring any prior exposure to SCV).
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 some objects that are, perhaps, entirely indigenous to SCV: e.g., plurisubharmonic functions and invariant metrics. This will allow us to discuss the inequivalence of the (Euclidean) ball and the polydisc in higher dimensions, and to discuss appropriate analogues of the one-variable Riemann Mapping Theorem in higher dimensions.
Next, we shall study the properties of the Kobayashi metric (which is one of the invariant metrics mentioned above) and the Kobayashi distance. This will be used to study the behaviour of automorphisms of bounded domains and refinements of some of the results hinted at above – to the extent that time permits.