Spectral theory of Complex Laplacians and Applications

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Complex analysis of several variables is a branch of mathematics where analysis, algebra, and geometry intertwine. The complex Laplacians studied in this project are the second-order differential operators associated with the Cauchy-Riemann or tangential Cauchy-Riemann complexes. The main goal of the project is to study the spectral theory of these Laplacians, in particular, the interplay between the spectral behavior of the complex Laplacians and the geometric structure of the underlying spaces. Problems to be addressed in this project include explicit computations, positivity, pure discreteness, and stability of the spectrum. The principal investigator will also study boundary behavior of the Bergman and Szego kernels, as well as the relationship between these kernels.Complex analysis is an essential tool in physics and engineering. The classical differential operators known under their technical names as the Dirichlet- and Neumann-Laplacians have been used to study heat diffusion and fluid dynamics by physicists and engineers for more than two centuries. The so-called complex Laplacians are the analogues of these classical operators in the setting of complex analysis in several variables. Spectral theory of differential operators plays an important role in many areas of biological, medical, and physical sciences. For example, the inverse problem--the problem of determining the shape from spectral properties--has applications in fields such as medical imaging. Pure discreteness of the spectra of the complex Laplacians is intimately related to diamagnetism and paramagnetism, topics widely studied in quantum physics and chemistry. Ideas and techniques developed in this project can potentially have repercussions in other branches of mathematics and sciences. The project also has significant impact on the development of human resources: the principal investigator will develop topics courses and engage graduate and undergraduate students in research activities. Research in this project will be incorporated into the teaching and learning process. In addition, this project will also facilitate interdisciplinary research activities among mathematicians, biologists, and computer scientists.