I received a PhD from the University of Washington, Seattle, in 1974, and I spent 16 years on the faculty at Princeton University before joining Michigan in 1991. I have supervised 6 Ph. D students at Princeton and have one now at the University of Michigan, Greg Buzzard, as well as being exterior Ph. D advisor for a student in Mexico City, Araceli Bonifant.
I have published about 80 research papers, plus one monograph joint with Berit Stensones: Counterexamples in Several Complex Variables (Princeton University Press) as well as two Proceedings volumes also in Princeton University Press. My work is mostly in higher dimensional complex analysis.My masters thesis from the University of Oslo was on function algebras. Much of my work has been devoted to function theoretic questions on domains in C^n. The approach has been to a large extent of a geometric nature.
In recent years I have worked also on problems in complex dynamics, mostly in higher dimension. This is a field which is undergoing a rapid growth for the moment, and is likely to do so in the foreseeable future. There are vast areas of open and very accessible problems. So I consider this a good area for graduate students. This is in contrast to complex dynamics in one dimension where the theory is highly developped, and results are hard to come by. Complex Dynamics in higher dimension uses tools from the theory of Several Complex Variables, it uses also general dynamical systems theory and algebraic geometry. The theory, as we develop it, is a 100 % rigorous mathematical theory although the computer is frequently used to search for the right conjectures and the computer is also occcasionally integrated with great care as a tool in some proofs, but only when this leaves the proofs mathematically rigorous.
If you want to get a better feeling for these areas, you might have a look at the Counterexample Book. For complex dynamics, you might get an idea from Devaneys book, An Introduction to Chaotic Dynamical Systems, at least as an introduction to the one variable theory. Upon request, I will be glad to supply a copy of my list of publications and/or my students thesis titles.