Analysis and Dynamics in Several Complex Variables

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Complex numbers, an extension of real numbers, are indispensable in science, engineering, and economics. For instance, a circuit driven by an alternating current, such as that in a generator that delivers power to a household, can be analyzed using complex numbers. This project is in the setting of complex numbers and complex-valued functions. The fundamental theorem of calculus, which is in the setting of real numbers, uses an integral formula to establish a connection between functions and their derivatives. The principal investigator will study the smoothness of complex-valued functions using an integral formula representation in higher-dimensional spaces, or in more general spaces, and consider some applications of this theory. The project will also support the training of graduate students through their thesis work. This project has four research topics. The principal investigator will study the regularity of Cauchy-Riemann equations on strictly pseudoconvex domains, or domains that have a sufficient number of positive or negative Levi eigenvalues in a complex manifold. The research will focus on minimum regularity requirements for the domains while seeking solutions that have higher-order regularity. More generally, the principal investigator will study the integral representation of complex differential forms and use it to study the stability of deformation of complex structures on strictly pseudoconvex or Z(1) domains. The principal investigator will study the classification of neighborhoods of a compact complex manifold using methods from complex analysis and dynamical systems, and will continue to study the CR singularity of real submanifolds in complex Euclidean spaces.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.