Hermitian Analysis and CR Geometry

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The PI will continue his study of analysis and geometry as it relates to complex numbers. Complex numbers and functions of complex variables have become very useful tools in the study of other areas of pure mathematics and in the application of mathematics. For instance, the study of airflow over a wing uses complex numbers and complex functions to model this flow. The subject continues to be a vital and important topic in the mathematical sciences. Realizing this, the PI has recently begun teaching what is called Advanced Engineering Mathematics. Teaching this material shows in a precise sense how the proposed research develops higher dimensional analogues of fundamental mathematics currently used throughout physics and engineering. In addition to the mathematical research, the PI will also work with the Engineering College at the University of Illinois at Urbana-Champaign on fine-tuning the curriculum in Advanced Engineering Mathematics. Hermitian analysis has its roots in 19th century mathematics. The subject began, two centuries ago, with the work of Fourier on heat diffusion.It has developed using the theory of Hilbert spaces, by its applications to quantum mechanics, and via its role in signal processing. Modern Hermitian analysis both relies upon and informs complex analysis and CR geometry, the PI's primary research areas. The modern point of view emphasizes the tools of orthonormal expansion and orthogonal projection. The PI has introduced an algebraic technique called orthogonal homogenization which connects these ideas to the geometry of proper mappings between balls. The PI has reformulated one of his older results, on volumes of proper images of balls, as a variational problem, thereby extending the results to considerably more general situations. The PI's work on proper mappings between balls in different dimensional complex spaces has also uncovered unexpected connections to representation theory and algebraic combinatorics. The primary purposes of this research are to extend the ideas of orthogonal homogenization to the rational case, to establish additional variational inequalities, to study homotopy for proper holomorphic mappings of positive codimension, and to further develop the subject of CR complexity. In the process the PI will continue to mentor young mathematicians in these topics and to organize and attend conferences.