Function theory in several complex variables and partial differential equations

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This mathematics research project concerns a variety of embedding problems arising from CR manifolds and complex manifolds. Zhang will focus on the classification problem of proper holomorphic maps between manifolds of different dimensions, and the rigidity problem of local conformal embeddings between bounded symmetric domains. Zhang and her collaborators have also been studying the CR embeddability problem. By suggesting an analytic approach, they find criterions for the embeddability of CR manifolds into the model manifolds and associate the CR transversality of the map with the geometry of the CR manifolds. Zhang will continue to explore effective methods to study the CR transversality and the finite jet determination in their full generality. Another aspect of this project focuses on Cauchy-Riemann equations in several complex variables. It is well known via the pseudo-differential operator theory that the regularity of the Cauchy-Riemann operator is closely related to the geometry of the boundary of the domain. Zhang and her collaborators will study a variety of geometric type conditions on the boundary. In the meanwhile, they will investigate the regularity problem of Cauchy-Riemann equations over pseudoconvex domains through the method of integral representation theory. Since the solutions are explicitly expressed in terms of singular kernels derived from the boundary conditions, it provides a direct way to understand the relationship between holomorphic function theory and the geometry of the domain where functions are defined.This mathematics research project is in the areas of several complex variables and partial differential equations, which are two fundamental sub-disciplines of modern mathematics. The new ideas presented in this project may lead to new applications in computational mathematics, as well as new applications to other disciplines such as fluid dynamics, where the use of complex variables has been known to reduce complex flows to very simple systems, and electrical engineering, where complex variables have been intensively used to analyze circuits with alternating current. Since the techniques to be used and developed in this project are accessible to graduate students, collaborations with other junior researchers and graduate students will be an integral part of this project. Throughout her teaching activities, the principal investigator will continue to mentor undergraduate students, especially female students, and help them to cultivate their interests in the mathematical sciences.