Differential Operators in Several Complex Variables

多个复变量中的微分算子

• 批准号: 0500909
• 负责人: Siqi Fu
• 金额: --
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500909&HistoricalAwards=false
• 关键词: Differential; Operators; Several; Complex; Variables

ABSTRACTThis project studies the complex and Kohn Laplacians in severalcomplex variables, the d-bar-problem on complex manifolds, as wellas their applications to and interplays with problems in algebraicgeometry and quantum physics. One of the main thrusts of theproposed research is systematical study of the d-bar-Neumann andKohn Laplacians along the lines of the classical spectral theoryfor the Dirichlet and Neumann Laplacians. The investigator willcontinue his work on the interplays between spectral behavior ofthese Laplacians and the underline geometric structures. A problemof particular interest is to characterize geometric structures onwhich these Laplacians have purely discrete spectrum. Theinvestigator will also study the spectral theory of relatedSchrodinger operators with magnetic fields.Complex analysis and differential equations arise naturally inphysics, engineering, chemistry, economics, and other sciences.Spectral analysis of differential operators plays an importantrole in quantum mechanics. The proposed research will furtherexplore the intimate connections between the seemingly unrelatedproblems in several complex variables and quantum mechanics. Itcould lead to new understanding and discoveries in both fields.Ideas, tools, and techniques developed in this project will alsohave repercussions in other fields. This project will have directimpact on the development of human resources; it will supportdevelopment of new courses and texts that attract students intomathematics and sciences. It will facilitate interdisciplinaryresearch activities with biologists and computer scientists.

本项目主要研究复变函数中的复拉普拉斯算子和Kohn Laplacian算子，复流形上的d-杆问题，以及它们在代数几何和量子物理中的应用和相互作用。本文的主要工作之一是沿着Dirichlet和Neumann Laplacian的经典谱理论的思路，沿着系统地研究d-bar-Neumann和Kohn-Laplacian。 研究人员将继续他的工作之间的相互作用的光谱行为ofthese拉普拉斯和下划线的几何结构。一个特别感兴趣的问题是描述几何结构，这些拉普拉斯有纯粹的离散谱。复变函数和微分方程是物理学、工程学、化学、经济学和其他科学中的自然现象，微分算子的谱分析在量子力学中起着重要的作用。 这项研究将进一步探索看似无关的多复变问题与量子力学之间的密切联系。 它可能会在这两个领域带来新的理解和发现。在这个项目中开发的思想、工具和技术也将在其他领域产生影响。 该项目将对人力资源的开发产生直接影响;它将支持开发新的课程和教材，吸引数学和科学学生。 它将促进生物学家和计算机科学家的跨学科研究活动。