RUI: Spectral Theory and Geometric Analysis in Several Complex Variables

RUI：多个复杂变量的谱理论和几何分析

• 批准号: 1500952
• 负责人: Siqi Fu
• 金额: $ 17.89万
• 依托单位: Rutgers University Camden
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2020-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500952&HistoricalAwards=false
• 关键词: RUI; Spectral; Theory; Geometric; Analysis

Many physical and social phenomena can be modeled mathematically using partial differential operators. The Laplace operator is a differential operator that has long played an important role in mechanics, physics, and mathematics. The complex Laplace operator is a natural outgrowth of the classical Laplace operator in complex analysis of several variables, a branch of mathematics where algebra, analysis, and geometry intertwine. This research project investigates analytic and geometric properties of the complex Laplace operator, in particular, its spectrum. Spectral analysis is a major tool in scientific research, and spectral properties of the complex Laplace operator are known to be closely related to certain quantum phenomena in physics. The goal of this project is to understand how algebraic, analytic, and geometric structures of the underlying complex space interact with each other. The project combines ideas and methods from several branches of mathematics, and the techniques under development could potentially have applications in other areas of mathematics and physical sciences. This project involves undergraduate students in research activities and broadens participation of underrepresented groups in mathematics. The complex Neumann Laplace operator is a prototype of an elliptic operator with non-coercive boundary conditions. Since the work of Kohn and Hörmander in the 1960's, there have been extensive studies on regularity theory of the complex Neumann Laplace operator that led to important discoveries in both partial differential equations and several complex variables. The main thrust of this proposal is to study spectral theory of the complex Neumann Laplace operator, with emphasis on the interplay between the spectral behavior of the operator and the underlying geometric structures. Among the problems studied in this project are stability of the spectrum as the underlying structures deform and characterization of complex manifolds whose complex Laplace operator has discrete spectrum. Also investigated are regularity theory of the Cauchy-Riemann operator on complex manifolds, reproducing kernels, invariant metrics, and their applications to problems in complex algebraic geometry. This project supports research activities of undergraduate and graduate students, facilitates the development of new courses that attract students into mathematics, and fosters interdisciplinary research.

许多物理和社会现象可以用偏微分算子来数学建模。 拉普拉斯算子是一种微分算子，长期以来在力学、物理学和数学中发挥着重要作用。复拉普拉斯算子是多元复分析中经典拉普拉斯算子的自然产物，多元复分析是代数、分析和几何相互交叉的数学分支。本研究计画探讨复拉普拉斯算符的解析与几何性质，特别是其频谱。 谱分析是科学研究中的一个主要工具，而复拉普拉斯算符的谱特性与物理学中的某些量子现象密切相关。 该项目的目标是了解复杂空间的代数，分析和几何结构如何相互作用。该项目结合了数学几个分支的思想和方法，正在开发的技术可能在数学和物理科学的其他领域有潜在的应用。该项目涉及研究活动的本科生，并扩大了数学代表性不足的群体的参与。复Neumann拉普拉斯算子是具有非强制边界条件的椭圆算子的原型。 自20世纪60年代Kohn和Hörmander的工作以来，人们对复Neumann拉普拉斯算子的正则性理论进行了广泛的研究，导致了偏微分方程和几个复变量的重要发现。 这个建议的主要目的是研究复诺依曼拉普拉斯算子的谱理论，重点是算子的谱行为和基本几何结构之间的相互作用。在这个项目中研究的问题是稳定性的频谱作为底层结构变形和表征复杂的流形，其复杂的拉普拉斯运营商具有离散频谱。 还研究了复流形上的Cauchy-Riemann算子的正则性理论，再生核，不变度量及其在复代数几何问题中的应用。 该项目支持本科生和研究生的研究活动，促进吸引学生进入数学的新课程的开发，并促进跨学科研究。