Function Theory of Several Complex Variables

在这个项目中，首席研究员将致力于多元复数分析中的几个问题，这些问题与非线性分析、偏微分方程组和经典动力学的研究有关。更具体地说，主要研究人员希望继续研究多个复变量中的各种刚性问题，以及它们在复奇点和复杂几何理论中的应用及其与超刚性问题的相互作用。他将研究复空间中实子流形的等价问题，并进一步研究目前对复流形中实子流形的全纯局部壳的理解。他还打算研究一族可嵌入的紧致、强伪凸的三维CR流形的同时嵌入和填充问题，并通过利用d-bar方程的方法来探索Grauert-Siu-Ling直接映象理论。此外，还将探讨CR映射的各种几何和解析性质。自十九世纪以来，复变量的复数和复变函数已成为许多数学领域及其在其他科学和工程领域的应用中不可或缺的工具。应用科学中许多问题的解决最终可能取决于这些复杂分析工具的改进和对其基本性质的更深入理解。例如，在材料科学中，统一处理多方向应力的标准方法是将它们表示为复数，或者在更复杂的情况下表示为复杂函数。结果表明，材料中裂纹扩展的方向与某些方程的性质有关，这些方程与这些复数或函数有关。本课题研究的结果可能会导致这些方程解的新性质的发现。该项目具有重要的教育和培训方面：几名研究生将积极参与该项目。最后，首席研究员计划共同组织关于几个复变量和偏微分方程式的会议，将数学家(包括年轻人和代表性不足群体的成员)聚集在一起，讨论他们的研究和教学。

In this project the principal investigator will work on several problems in multivariate complex analysis that are related to research in nonlinear analysis, partial differential equations, and classical dynamics. More specifically, the principal investigator wishes to continue his research into various rigidity problems in several complex variables, as well as their applications and interactions with super-rigidity problems in the theory of complex singularities and complex geometry. He will study the equivalence problem for real submanifolds in complex spaces and further the present understanding of the local hull of holomorphy for a real submanifold in a complex manifold. He also intends to investigate the simultaneous embedding and filling problem for a CR family of embeddable compact, strongly pseudoconvex, three-dimensional CR manifolds and to explore the Grauert-Siu-Ling direct image theory through an approach that makes use of the d-bar equation. Various geometric and analytic properties for CR mappings will be pursued, as well. Complex numbers and functions of complex variables have become, since the nineteenth century, indispensable tools in many areas of mathematics and its application to other areas of science and engineering. The solutions of many problems in the applied sciences could ultimately depend on improvements in these complex analytic tools and a deeper understanding of their basic properties. For example, in materials science, the standard method for treating multidirectional stresses in a uniform way is to represent them as complex numbers or, in more complicated situations, as complex functions. It then turns out that, among other things, the direction of the propagation of cracks in materials is related to the properties of certain equations associated with these complex numbers or functions. Results of the research to be carried out in this project may lead to the discovery of new properties of solutions of these equations. The project has significant educational and training aspects: several graduate students will be actively involved in this project. Finally, the principal investigator is planning to coorganize conferences on several complex variables and partial differential equations, bringing together mathematicians (including young people and members of underrepresented groups) to discuss their research and teaching.