Geometric Methods in Complex Analysis

复杂分析中的几何方法

• 批准号: RGPIN-2015-04765
• 负责人: Shafikov, Rasul
• 金额: $ 1.24万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675829
• 关键词: Geometric; Methods; Complex; Analysis

The proposal deals with geometric methods in complex analysis -- the study of functions of complex variables. Development of this field is one of the great contributions of modern mathematics. In fact, many different directions of research in modern mathematics stem from problems in complex analysis. Conversely, to study complex analysis one needs many tools from other areas - such as differential equations, topology, functional analysis, etc. In addition, complex analysis has numerous applications to other sciences. It is this inherent interconnection with other branches of mathematics, physics, and computer science that has made complex analysis one of the leading research areas in mathematics for many decades.******More precisely, the proposed research is concerned with the study of the so-called CR manifolds. It is built around three concrete problems in several complex variables which lie at the heart of the subject. These are: 1) analytic continuation of local equivalences between CR manifolds; 2) understanding boundary values of holomorphic functions on domains with nonsmooth boundary; and 3) determining complex analytic properties of singular submanifolds in complex euclidean spaces. The significance of these problems comes from the fact that such manifolds are models for many objects and processes that appear in applications in other areas of mathematics, such as symplectic topology and contact geometry, but also in other sciences such as theoretical physics, electrical engineering, and computer science. ******The author has published a number of papers dedicated to the subject. The new directions for further research outlined in the proposal are based on the original ideas and innovative techniques that the author has developed in these papers. In general terms, successful realization of the proposal will lead to new insights and will play a role in the future development of the subject.

该提案涉及复分析中的几何方法-复变函数的研究。这一领域的发展是现代数学的重大贡献之一。 事实上，现代数学中许多不同的研究方向都源于复分析中的问题。相反，研究复分析需要其他领域的许多工具-例如微分方程，拓扑学，泛函分析等，此外，复分析在其他科学中有许多应用。正是这种与数学、物理学和计算机科学的其他分支的内在联系，使复分析成为几十年来数学的主要研究领域之一。更确切地说，所提出的研究涉及所谓的CR流形的研究。它围绕着几个复杂变量中的三个具体问题构建，这些变量是该学科的核心。 这些是：1）CR流形之间局部等价的解析延拓; 2）了解具有非光滑边界的区域上全纯函数的边值; 3）确定复欧氏空间中奇异子流形的复解析性质。这些问题的重要性来自于这样一个事实，即这些流形是许多对象和过程的模型，这些对象和过程出现在其他数学领域的应用中，例如辛拓扑和接触几何，但也出现在其他科学中，例如理论物理，电气工程和计算机科学。 * *建议中概述的进一步研究的新方向是基于作者在这些论文中开发的原创思想和创新技术。总的来说，成功地实现该建议将导致新的见解，并将在该主题的未来发展中发挥作用。