Geometric Function Theory in Several Complex Variables

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

There are two main research directions in this project. The first one is the continuation of the work on a general theory of boundary values, in the sense of analytic functionals or hyperfunctions. In its present state this theory has clarified several points in the study of boundary values of functions, in particular viz the intrinsic character of the definitions. Points which they have clarified in the global setting need further study in the local version. A theory in the non real analytic setting is still to be done. The second direction is at the confluent of pluri-potential theory and complex geometry. A still remote goal is the possible extension of Poletsky's theory of disks to the case of almost complex manifolds, whose role in Mathematics is rapidly growing. It also leads to the study of approximate solutions to the d-bar operator. The primary field of this proposal is Several Complex Variables. This field is rich due to its connections to many fields in Mathematics. There is in particular a remarkable interplay between Several Complex Variables and the theory of Partial Differential Equations, which itself is the most basic tool in the study of many processes in Physics and Engineering. The first direction discussed above fits in that setting. In applications quantitative studies have often to be replaced by qualitative studies, either because the quantitative study is too difficult or because the qualitative study is in fact more illuminating. This is where the field of Topology is essential. Several important connections exist between Several Complex Variables and Topology. The second direction discussed above fits in that setting. Deformation theory is a major theme, in which the study of almost complex structures has already played a crucial role.