Function theory in CR geometry and partial differential equations

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Function theory in complex analysis has important applications to other branches of mathematics including algebraic geometry, applied mathematics, as well as to other disciplines, such as physics and engineering. For instance, in problems arising from electric fields, various geometric conditions can always be expressed in terms of functions of complex variables. By employing appropriate conformal maps, many inconvenient geometric configurations can be transformed to rather easy formats and hence be completely solved. Some other types of real-world phenomena are modeled as solutions to certain partial differential equations. Methods developed in complex analysis, such as Fourier analysis, have played fundamental roles in solving those equations and giving interpretations for solutions. The current project will address related problems in holomorphic function theory. Progress of the project will help us understand more precisely properties of conformal maps in higher dimensions and explore new methods that can be applied into other fields. The PI proposes to study a variety of problems in several complex variables and the corresponding partial differential equations, together with her collaborators. More specifically, the PI will investigate rigidity and classification problems of CR embeddings between some special types of CR manifolds, such as (generalized) Heisenberg hypersurfaces. Since transversality property of CR mappings is closely related to CR embeddability, she also plans to continue the work on CR transversality along the line of the conjecture of Baouendi-Huang for Levi non-degenerate hypersurfaces of higher codimension. On the aspect of partial differential equations, the PI will work on regularity of solutions to Cauchy-Riemann equations over pseudoconvex domains through the method of integral representation theory. The method will involve strong correlation between holomorphic function theory and the geometry of the domains. The PI is also interested in the regularity problem for solutions to some Beltrami equations, as well as solvability and regularity to some general types of nonlinear elliptic complex partial differential equations on Heisenberg hypersurfaces.