Function Theory in Several Complex Variables

本课程将对若干复变量和柯西-黎曼几何中的若干问题进行研究。更具体地说，将寻求更好地理解复杂分析和复杂几何中的各种刚性现象。研究二维复空间中实曲面的解析结构，并探讨其与经典力学中许多著名问题的内在联系。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方案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，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，例如，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发现这些方程解的新性质。

Research will be conducted on a number of problems in Several Complex Variables and Cauchy-RiemannGeometry. More specifically, the better understanding of various rigidity phenomena in complex analysis and complex geometry will be sought.The study the analytic structure of real surfacesin the complex space of dimension two, and the investigation of its intrinsic connections with many well-known problems in classical mechanicswill be pursued. Complex numbers and functions of complex variableshave become, since the 19th century, indispensabletools in many areas of mathematics and in itsapplication to other areas of science and engineering.Indeed, the solutions of many problems in the appliedsciences could ultimately depend on improvements in these complex analytic tools and a betterunderstanding of their basic properties. For instance,in material science the standard method for treatingmultidirectional stresses in a uniform way is to represent them as complex numbers or, in morecomplicated situations, as complex functions. It then turns out, for instance, that the direction of the propagation of cracks in materials is related to the properties of certain equations associatedwith these complex numbers or functions. Results of the research to be carried out in this project couldlead to the discovery of new propertiesof solutions of these equations.