Geometric Methods in Complex Analysis

复杂分析中的几何方法

基本信息

批准号：
RGPIN-2020-04432
负责人：
Shafikov, Rasul
金额：
$ 1.53万
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：
2022
资助国家：
加拿大
起止时间：
2022-01-01 至 2023-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758577
关键词：
Geometric Methods Complex Analysis

项目摘要

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 specifically, the proposed research is concerned with the study of the properties of real submanifolds in complex Euclidean spaces, or, more generally, complex manifolds. Their significance 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 and electrical engineering. For example, approximation of smooth complex-valued functions by simple computable functions, such as polynomials, can be understood through the study of various convexity problems of their graphs - natural example of real submanifolds. On the other hand, real submanifolds often capture valuable geometric properties of the ambient manifolds, and thus they can play an important role in understanding the geometry and topology of complex spaces. The proposal consists of several concrete problems that lie at the heart of this topic. 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——复变量函数的研究。这一领域的发展是现代数学的伟大贡献之一。事实上，现代数学的许多不同研究方向都源于复分析问题。相反，要研究复分析，需要许多其他领域的工具，例如微分方程、拓扑、泛函分析等。此外，复分析在其他科学中还有许多应用。正是这种与数学、物理学和计算机科学其他分支固有的相互联系，使得复杂分析几十年来一直成为数学的领先研究领域之一。更具体地说，所提出的研究涉及复杂欧几里得空间中实子流形的性质，或者更一般地说，复杂流形的性质。它们的重要性来自于这样一个事实：此类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（例如辛拓扑和接触几何）的应用中，而且也出现在其他科学（例如理论物理和电气工程）中。例如，通过简单的可计算函数（例如多项式）来逼近平滑复值函数，可以通过研究其图的各种凸性问题来理解 - 真实子流形的自然示例。另一方面，真实的子流形通常捕获环境流形的有价值的几何特性，因此它们可以在理解复杂空间的几何和拓扑方面发挥重要作用。该提案包含本主题核心的几个具体问题。作者发表了多篇专门讨论该主题的论文。提案中概述的进一步研究的新方向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，该提案的成功实现将会带来新的见解，并对学科未来的发展起到一定的作用。