Geometric Methods in Complex Analysis

该提案涉及复变函数研究中的几何方法。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杂分析中的问题。相反，要研究复杂分析，人们需要许多其他领域的工具--如微分方程、拓扑学、泛函分析等。此外，复杂分析在其他科学中有许多应用。正是这种与数学、物理和计算机科学其他分支的内在联系使复数分析成为几十年来数学中的主要研究领域之一。更准确地说，拟议的研究涉及到所谓的CR流形的研究。它是围绕三个具体的问题而建立的，这些问题涉及几个复杂的变量，而这些变量处于主题的核心。它们是：1)CR流形之间局部等价的解析延拓；2)理解具有非光滑边界的区域上全纯函数的边值问题；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，即这样的流形是许多对象和过程的模型，这些模型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。*作者发表了多篇关于这一主题的论文。提案中勾勒出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始思想和创新技术。总体而言，该建议的成功实现将带来新的见解，并将对该学科未来的发展起到作用。

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.