Function theory in several complex variables and partial differential equations

本数学研究项目涉及由CR流形和复流形引起的各种嵌入问题。Zhang将重点研究不同维流形之间的固有全纯映射的分类问题，以及有界对称域之间局部共形嵌入的刚性问题。张和她的合作者也一直在研究CR嵌入问题。通过提出一种解析方法，他们找到了CR流形嵌入模型流形的准则，并将映射的CR横截性与CR流形的几何形状联系起来。Zhang将继续探索有效的方法来研究CR的横向性和有限射流的确定。这个项目的另一个方面集中在几个复杂变量的柯西-黎曼方程。由伪微分算子理论可知，Cauchy-Riemann算子的正则性与区域边界的几何形状密切相关。张和她的合作者将研究边界上的各种几何类型条件。同时，他们将通过积分表示理论的方法研究伪凸域上柯西-黎曼方程的正则性问题。由于解是用由边界条件导出的奇异核来显式表示的，因此它提供了一种直接的方法来理解全纯函数理论与定义函数的区域几何之间的关系。本数学研究项目涉及现代数学的两个基本分支学科——若干复变量和偏微分方程。在这个项目中提出的新思想可能会导致计算数学的新应用，以及其他学科的新应用，如流体动力学，其中复杂变量的使用已经被称为将复杂的流动减少到非常简单的系统，以及电气工程，其中复杂变量已被广泛用于分析交流电电路。由于该项目中使用和开发的技术对研究生开放，因此与其他初级研究人员和研究生的合作将是该项目不可分割的一部分。在她的教学活动中，首席研究员将继续指导本科生，特别是女学生，帮助他们培养对数学科学的兴趣。

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.