Mathematical Sciences: Geometric Function Theory in Several Complex Variables

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The general area of mathematical research encompassed by this project relates to the theory of several complex variables and partial differential equations. Some of the work will involve real variable microlocal techniques, other parts will focus on mappings between domains in spaces of several complex variables of different dimensions. One question of particular concern is that of unique continuation in Cauchy-Riemann manifolds. Effectively these studies involve questions of whether or not a holomorphic function of several variables can be constant along various submanifolds of a domain, such as along a curve, and not be constant everywhere. A second thrust of this work will focus on properties of sets preserved under holomorphic mappings between domains in spaces of possibly different dimensions. Much of what is intuitive, and not surprising in one dimension, turns out to be false in higher. Mappings whose Jacobians have determinant one everywhere may map unbounded domains onto bounded ones. Some sequences are equivalent under automorphisms of complex space, others are not. Difficulty in visualizing the true state of affairs makes progress slow in this area. Fundamental to the analysis of first-order differential operators defined on domains is the determination of when the inhomogeneous d-bar equation can be solved for closed one-forms. There is considerable evidence to suggest that the answer lies in the geometric concept of polynomial convexity of the complementary set. If this can be shown to be the case, it will provide a major tool in establishing deep connections between the differential operators and the shape of domains in complex space.