Partial Differential Equations in Several Complex Variables

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Complex analysis is one of the most important areas of mathematics. Complex analysis in one variable is a mature and beautiful subject. It has broad applications to various branches of mathematics and physics, including algebra (Fundamental Theorem of Algebra), topology (Riemann Mapping Theorem), number theory (Prime Number Theorem), fluid mechanics, optics and electrostatics (Conformal Mappings). Holomorphic functions in several complex variables were studied more recently, in the past one hundred years. Function theory in several complex variables is fundamentally different from the theory of one complex variable. Its development has influenced modern analysis in partial differential equations, harmonic analysis and complex geometry. The results of the project will have wide appeal to a large section of the mathematics community. It has already found applications in modern technologies that involve non-smooth boundaries such as electrospray and scattering. The principal investigator is engaged in the development of a textbook on one complex variable theory for partial differential equations to make these interesting topics more accessible to a wide range of students not just in mathematics, but also in physical sciences and engineering. The project provides training opportunities for graduate students and the principal investigator is actively involved in teaching and training students at all levels. The goal of the project is to explore several diverse and interconnected areas in several complex variables. The focus is on two of the most important equations in several complex variables: the Cauchy-Riemann equations and the tangential Cauchy-Riemann equations. Classical results on these equations require that the domains have smooth boundary, or the ambient spaces are Euclidean or Stein. A large part of the project is on domains with non-smooth boundary and on complex manifolds. The principal investigator has initiated the study of the Cauchy-Riemann equations on Lipschitz domains using harmonic analysis and geometric measure theory. The problems include the Hausdorff property of Dolbeault cohomology groups, Levi-flat hypersurfaces and complex foliation, Runge approximation and function theory on non-smooth domains and complex manifolds. These problems are central to several complex variables, complex geometry and partial differential equations. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.