Differential Operators in Several Complex Variables

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

ABSTRACT This project studies the complex and Kohn Laplacians in several complex variables, the d-bar-problem on complex manifolds, as well as their applications to and interplays with problems in algebraic geometry and quantum physics. One of the main thrusts of the proposed research is systematical study of the d-bar-Neumann and Kohn Laplacians along the lines of the classical spectral theory for the Dirichlet and Neumann Laplacians. The investigator will continue his work on the interplays between spectral behavior of these Laplacians and the underline geometric structures. A problem of particular interest is to characterize geometric structures on which these Laplacians have purely discrete spectrum. The investigator will also study the spectral theory of related Schrodinger operators with magnetic fields. Complex analysis and differential equations arise naturally in physics, engineering, chemistry, economics, and other sciences. Spectral analysis of differential operators plays an important role in quantum mechanics. The proposed research will further explore the intimate connections between the seemingly unrelated problems in several complex variables and quantum mechanics. It could lead to new understanding and discoveries in both fields. Ideas, tools, and techniques developed in this project will also have repercussions in other fields. This project will have direct impact on the development of human resources; it will support development of new courses and texts that attract students into mathematics and sciences. It will facilitate interdisciplinary research activities with biologists and computer scientists.