Autumn Quarter will begin with basic ideas about holomorphic functions of several complex variables and holomorphic maps. Having established some fundamentals, we will study analytic continuation, which in several variables is a much richer subject than in one variable. In particular, we will discuss envelopes of holomorphy. This discussion will lead us to consider holomorphic convexity and pseudoconvexity with its attendant theory of plurisubharmonic functions. Towards the end of the quarter, we will consider integral formulas in the multidimensional case together with some applications. Autumn quarter will conclude with a discussion of analytic functionals (which are generalizations of distributions tailored to the analytic context).

The first part of Winter Quarter will be devoted to local analysis: The Weierstrass Preparation Theorem and its consequences. This includes the analysis of the local structure of the zero locus of a holomorphic function. This part of the subject is rather algebraic and parallels certain parts of algebraic geometry. After these local investigations, we will turn to the study of complex manifolds. We will consider the Cousin problems in this context. (These are the higher dimensional analogues of the classical problems of finding meromorphic functions with prescribed poles and holomorphic functions with prescribed zeros.) The quarter will finish with a discussion of the theory of Stein manifolds, which are complex manifolds holomorphically equivalent to complex submanifolds of complex Euclidean space.

Prerequisites: A graduate-level in complex analysis.