RUI: Spectral Theory and Geometric Analysis in Several Complex Variables

Complex numbers have long played an important role in mathematics, physics, and engineering such as in designing electrical circuits. The study of functions defined on complex numbers has led to several new fields including the area of several complex variables, which is a major branch of mathematics where algebra, analysis and geometry intertwine. The principal goal of this project is to systematically study problems of a geometric nature in several complex variables.

Many of the proposed problems have their roots in the physical sciences. This project aims to understand how geometric structures interplay with analytic properties in several complex variables. This project supports learning and research activities for undergraduate and graduate students, especially those from underrepresented groups.

These learning and research experiences will better prepare the students for graduate school or a professional career.

The main thrust of the proposed research is the several complex variables analogue of Mark Kac’s problem "Can one hear the shape of a drum?" That is, to what extend is the geometry of a complex or CR manifold determined by spectral behavior of the complex Laplacian? The complex Laplacian could either be the complex Neumann Laplacian on a complex manifold with boundary or the Kohn Laplacian on a CR manifold.

These complex Laplacians are the natural outgrowth in several complex variables of the classical Laplace operator with the Neumann boundary condition. Problems to be studied include positivity, discreteness, and stability of the spectrum. The spectral theory of the complex Laplacian is intimately related to problems in quantum mechanics.

The principal investigator plans to combine ideas and techniques from various fields of mathematics and physics to further study these problems in geometric function theory of several complex variables.

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.