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、物理和计算机科学的其他分支的内在联系，使复杂分析成为几十年来数学中的领先研究领域之一。更具体地说，建议的研究涉及复欧氏空间中实子流形的性质的研究，或者更一般地，复流形的性质。它们的重要性来自于这样一个事实，即这种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，如辛拓扑和接触几何，但也出现在其他科学中，如理论物理和电气工程。例如，光滑复值函数用简单的可计算函数，如多项式的逼近，可以通过研究其图形的各种凸性问题来理解--实子流形的自然例子。另一方面，实子流形往往刻画了环境流形的一些有价值的几何性质，因此在理解复杂空间的几何和拓扑方面起着重要的作用。该提案由几个具体问题组成，这些问题是这一主题的核心。作者发表了许多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 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.