Function Theory in Several Complex Variables

摘要：本项目主要研究几个复杂变量和Cauchy-Riemann几何领域的一些问题。这些问题不仅从复杂分析的角度来看，而且从微分几何、非线性分析和偏微分方程、经典动力学和力学的角度来看，都是有趣和重要的。更具体地说，黄希望进一步了解目前在几个复杂变量中的各种刚性现象。这包括不同复空间中球间全纯映射的分类。他想了解复空间的一类实子流形的局部全纯不变量，即所谓的Bishop流形。他打算继续他之前在实解析超曲面之间CR映射的正则性和不变度量的研究，包括它们在全纯映射动力学性质研究中的应用。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。事实上，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好理解。例如，从事平面空气动力学研究的工程师必须处理垂直速度和水平速度之间的相互作用。现在很好地理解了，如果一个人采取简单的步骤，把这两个速度组合成一个矢量，然后把这个矢量表示成一个复数，空气动力流动的许多方面变得更容易可视化，结果是许多问题变得更容易解决。同样，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。例如，结果表明，材料中裂纹的传播方向与与这些复数或函数相关的某些方程的性质有着神秘的关系。在这个项目中进行的研究结果可能会导致这些方程解的新性质的发现，这将转化为对材料相关力学性质的更深层次的理解。

Proposal: DMS-9970493Principal Investigator: Xiaojun HuangAbstract: The Principal Investigator proposes in this project to work on a number of problems in the area of several complex variables and Cauchy-Riemann geometry. These problems are interesting and important not just from the point of view of complex analysis, but also from the perspective of differential geometry, nonlinear analysis and PDEs, classical dynamics, and mechanics. More specifically, Huang wishes to further the present understanding of various rigidity phenomena in several complex variables. This includes the classification of holomorphic mappings between balls in different complex spaces. He would like to understand the local holomorphic invariants of a certain class of real submanifolds of complex spaces, the so-called the Bishop manifolds. He intends to continue his previous work on the regularity of CR mappings between real analytic hypersurfaces and his investigation of invariant metrics, including their applications to the study of dynamical properties of holomorphic mappings.Complex numbers and functions of complex variables have become, since the 19th century, indispensable tools in many areas of mathematics and in its application to other areas of science and engineering. Indeed, the solutions of many problems in the applied sciences could ultimately depend on an improvement in these complex analytic tools and a better understanding of their basic properties. For instance, engineers working in planar aerodynamics must contend with the interplay between vertical and horizontal speeds. It is now well understood that if one takes the simple step of combining these two speeds into a vector and then representing that vector as a complex number, many aspects of the aerodynamic flow become much easier to visualize, with the result that many problems become easier to solve. Similarly, 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, for instance, that the direction of the propagation of cracks in materials is mysteriously 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 could lead to the discovery of new properties of solutions of these very equations, which would then translate to a deeper understanding of the related mechanical properties of materials.