RUI: Geometric Estimates for Complex Analysis and Applications

Calvin University, Grand Rapids MI

Investigators

Abstract

This project involves the study of problems in complex analysis in one and several variables, and in their applications to problems in conformal mapping. In particular, the project concerns properties of holomorphic reproducing kernels (e.g., the Cauchy kernel in one dimension and its generalizations to the Leray and Bochner-Martinelli kernels in higher dimensions). The first component of the project will be to use analytic methods to find geometric estimates for the Cauchy transform. The second component of the project will be to identify and study the Moebius invariant curvatures for real hypersurfaces in complex Euclidean space. These curvatures will be used for estimating integral operators in higher dimensions. The third component of the project will be to extend the Kerzman-Stein method of conformal mapping from the setting of simply connected, smoothly-bounded domains to the context of domains with finite connectivity and domains with corners. For this part, undergraduate students will be involved with projects to which they can make significant contributions. Conformal mapping has played an important role in physics and engineering, and it continues to find new applications in the natural sciences. For instance, conformal maps can be used for understanding the flow of air past the wing of an aircraft or for computing the electrostatic potential of a thin metal plate. More recently, conformal maps have been used to provide accurate two-dimensional representations of the brain that can serve as guides for researchers in their quest to understand that complex organ. There are different ways for computing conformal maps that are appropriate for different applications. The proposed research in conformal mapping is fundamental, and its immediate applications are not apparent. It is expected that some aspects of the project will lead to new connections between different areas of mathematics, in particular, between differential geometry and several complex variables. Finally, there are important training aspects to the project, as undergraduate students will participate directly in the research. In this way, the project will also contribute to the greater mathematical and scientific infrastructure.