Function Theory of Several Complex Variables

在这个项目中，首席研究员将继续研究与微分几何、非线性分析和经典动力学研究密切相关的多元复杂分析中的几个基本问题。更具体地说，首席研究员希望继续他在几个复变量的等价问题上的研究，并通过使用附加全纯盘进一步研究复空间中实子流形的全纯壳的复结构。他将继续研究几种复杂变量的各种刚性问题，以及它们在复奇点理论和复几何中与超刚性问题的应用和相互作用。他打算研究柯西-黎曼（CR）族可嵌入、紧致、强伪凸、三维CR流形的同时嵌入和填充问题。最后，他将继续他的工作，以了解是否一个代数，强伪凸超曲面可以局部全纯嵌入到高维复空间的海森堡超曲面中。自19世纪以来，复数和复变量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，以统一的方式处理多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函数。结果表明，除其他因素外，材料中裂纹扩展的方向与与这些复数或函数相关的某些方程的性质有关。在这个项目中进行的研究结果可能导致这些方程解的新性质的发现。这个项目有重要的教育和培训方面：一些研究生和初级数学家将积极参与这个项目。最后，首席研究员将继续组织一些复杂变量和复杂几何的国际会议，将数学家聚集在一起讨论他们的研究和教学。

In this project, the principal investigator will continue his work on several basic problems in multivariate complex analysis that are closely related to research in differential geometry, nonlinear analysis, and classical dynamics. More specifically, the principal investigator would like to continue his research on the equivalence problem in several complex variables and carry further his work on the complex structure of the holomorphic hull of a real submanifold in a complex space through the use of the attached holomorphic disks. He will continue his investigation of 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 intends to investigate the simultaneous embedding and filling problem for a Cauchy-Riemann (CR) family of embeddable, compact, strongly pseudo-convex, three-dimensional CR-manifolds. Finally, he will continue his work to understand whether an algebraic, strongly pseudo-convex hypersurface can be locally holomorphically embedded into the Heisenberg hypersurface in a complex space of higher dimension. 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. This project has significant educational and training aspects: several graduate students and junior mathematicians will be actively involved in this project. Finally, the principal investigator will continue to organize international conferences on several complex variables and complex geometry, bringing together mathematicians to discuss their research and teaching.