Function theory in CR geometry and partial differential equations

复分析中的函数理论在数学的其他分支，包括代数几何、应用数学，以及其他学科，如物理和工程中有着重要的应用。例如，在由电场引起的问题中，各种几何条件总是可以用复数变量的函数来表示。通过采用适当的保角映射，可以将许多不方便的几何构型转化为相当容易的格式，从而完全解决。一些其他类型的现实世界现象被建模为某些偏微分方程的解。在复杂分析中发展起来的方法，如傅立叶分析，在解决这些方程和给出解的解释方面发挥了基本作用。本课题将解决全纯函数理论中的相关问题。该项目的进展将有助于我们更精确地了解高维共形映射的性质，并探索可应用于其他领域的新方法。PI提议与她的合作者一起研究几个复杂变量和相应的偏微分方程中的各种问题。更具体地说，PI将研究一些特殊类型的CR流形（如（广义）海森堡超曲面）之间CR嵌入的刚性和分类问题。由于CR映射的横截性与CR嵌入性密切相关，她还计划沿着Baouendi-Huang猜想的高协维Levi非简并超曲面继续研究CR横截性。在偏微分方程方面，PI将通过积分表示理论的方法研究伪凸域上Cauchy-Riemann方程解的正则性。该方法将涉及全纯函数理论与区域几何之间的强相关性。PI还对某些Beltrami方程解的正则性问题，以及海森堡超曲面上某些一般类型的非线性椭圆型复偏微分方程的可解性和正则性问题感兴趣。

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.