New Frontiers in Several Complex Variables

The theory of calculus, which makes possible a precise understanding of continuously varying magnitudes, forms the mathematical underpinning of much of modern-day science and technology. The complex numbers are a system of generalized numbers providing greater flexibility with algebraic operations, such as allowing for the extraction of square roots of negative quantities. In this project, the investigator will apply the tools of calculus to smoothly varying complex outputs which depend on one or more complex inputs. Such objects are known in the mathematical literature as holomorphic functions. The study of holomorphic functions has applications to phenomena as diverse as the flow of liquids, heat conduction, the distribution of prime numbers, and the speed of computer programs. The project will employ two methods to construct holomorphic functions. The first is via a system of equations called the inhomogeneous Cauchy-Riemann equations. The second, the method of projections, interprets the collection of holomorphic functions geometrically as a sub-collection inside a larger collection of functions, and uses this geometric insight to construct holomorphic functions with specified properties. The deeper understanding of holomorphic functions to be gained in this project will be useful for applications both within and outside mathematics. The project will also involve undergraduate participants, who will work on both the computational and theoretical aspects of the problems. Such participation will expose students to challenging mathematical problems and will contribute to the training of future mathematicians and scientists.

Holomorphic functions of one or several complex variables constitute a natural and important class of functions in analysis and have important applications throughout science and engineering. This project investigates two classical techniques for constructing such functions: the inhomogeneous Cauchy-Riemann equations and projection operators. For the Cauchy-Riemann equations, the main thrust is to study the solution of these equations in annuli, where an annulus in complex Euclidean space is a domain obtained from a larger domain (the envelope) by removing a compact subset (the hole). Since annuli are not domains of holomorphy, new phenomena appear in this situation which are not found in the well-understood and classical setting of pseudoconvex domains. The analogy between complex analysis on annuli and Alexander duality (a circle of ideas in topology) is expected to forge novel methods of attack on these questions. Coupled with classical approaches based on a priori estimates for partial differential operators, these new methods will be employed to yield important new information on function theory of new classes of non-pseudoconvex domains. With regards to the geometric construction via projection operators, a primary focus of investigation will be a class of questions concerning the regularity of the Bergman projection, and other associated projection operators, in Lebesgue spaces on non-smooth domains. Such regularity depends on the geometry of the domain in ways that are not completely understood. Among the domains to be studied are quotients of balls by finite groups, and Reinhardt domains whose boundary contains the origin.

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.