Research in Several Complex Variables

多个复杂变量的研究

• 批准号: 1600371
• 负责人: Debraj Chakrabarti
• 金额: $ 10.23万
• 依托单位: Central Michigan University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600371&HistoricalAwards=false
• 关键词: Research; Several; Complex; Variables

This project combines two very powerful ideas of mathematics -- that of complex numbers and that of the calculus. Complex numbers include quantities such as the square root of negative one. The calculus studies how physical or geometric quantities vary in space or time. The combination of these ideas (called complex analysis) leads to far-reaching and beautiful results about smoothly varying complex quantities (called holomorphic functions). One may use these ideas to model various natural phenomena such as electrical attraction or the motion of liquids. These considerations also have surprising consequences in other parts of mathematics, such as the properties of prime numbers, the geometry of higher dimensional spaces, and the study of equations (called partial differential equations) used to describe many physical processes such as heat conduction and the propagation of waves. This research project studies the behavior of holomorphic functions as one approaches the boundary of the region in higher dimensional space where the function is defined. The investigator will also continue his successful mentoring of undergraduate student research in related topics.The behavior of holomorphic functions of several variables is well-understood on strongly pseudoconvex domains. In this project, the goal is to study the extension of such results to new and more general types of domains. Among the questions under study are estimates for the solutions of the solutions of the inhomogeneous Cauchy-Riemann equations, understanding what happens when there are no such estimates, and the boundary behavior of holomorphic functions. The domains to be studied include piecewise smooth domains (in particular product domains), Levi-flat Stein domains in complex manifolds, and non-pseudoconvex domains, both with and without "holes." The research will involve both the investigation of general questions and the careful study of particular examples, which can exhibit unexpected phenomena associated to these domains. Tools employed in this project to study these problems include a priori estimates, integral formulas, and algebraic methods based on sheaf theory, as well as insights from other parts of mathematics such as partial differential equations, functional analysis, and differential geometry.

该项目结合了两个非常强大的数学思想——复数和微积分。复数包括诸如负一的平方根之类的量。微积分研究物理量或几何量如何随空间或时间变化。这些思想的结合（称为复分析）导致了关于平滑变化的复量（称为全纯函数）的深远而美丽的结果。人们可以利用这些想法来模拟各种自然现象，例如电吸引力或液体运动。这些考虑因素在数学的其他部分也产生了令人惊讶的后果，例如素数的性质、高维空间的几何形状以及用于描述许多物理过程（例如热传导和波的传播）的方程（称为偏微分方程）的研究。该研究项目研究当全纯函数接近定义函数的高维空间区域的边界时的行为。研究人员还将继续成功指导本科生在相关主题的研究。多个变量的全纯函数的行为在强赝凸域上已得到很好的理解。在这个项目中，目标是研究将这些结果扩展到新的和更通用的领域类型。正在研究的问题包括非齐次柯西-黎曼方程解的估计、了解没有此类估计时会发生什么，以及全纯函数的边界行为。要研究的域包括分段平滑域（特别是乘积域）、复杂流形中的列维平坦斯坦因域以及带或不带“孔”的非伪凸域。该研究将涉及对一般问题的调查和对特定示例的仔细研究，这些示例可能会展示与这些领域相关的意想不到的现象。该项目用于研究这些问题的工具包括先验估计、积分公式和基于层理论的代数方法，以及来自数学其他部分（例如偏微分方程、泛函分析和微分几何）的见解。