RUI: Spectral Theory and Geometric Analysis in Several Complex Variables

复数长期以来在数学、物理和工程中扮演着重要的角色，例如在电路设计中。定义在复数上的函数的研究导致了几个新的领域，包括多个复变量的领域，这是数学的一个主要分支，代数、分析和几何相互交织在一起。这个项目的主要目标是系统地研究几个复变量中的几何性质的问题。许多被提出的问题都源于物理科学。这个项目的目的是了解几何结构如何与几个复变量中的解析性质相互作用。该项目支持本科生和研究生的学习和研究活动，特别是那些来自代表性不足群体的学生。这些学习和研究经验将更好地为学生进入研究生院或职业生涯做准备。这项拟议研究的主要目的是用几个复杂变量类比马克·卡克的问题：人们能听到鼓的形状吗？也就是说，复数或CR流形的几何在多大程度上由复数拉普拉斯的谱行为决定？复拉普拉斯可以是带边界的复流形上的复Neumann拉普拉斯，也可以是CR流形上的Kohn拉普拉斯。这些复拉普拉斯算子是经典拉普拉斯算子在Neumann边界条件下多个复变量的自然产物。需要研究的问题包括谱的正性、离散性和稳定性。复拉普拉斯的谱理论与量子力学中的问题密切相关。首席研究员计划结合数学和物理的不同领域的想法和技术，进一步研究几个复变量的几何函数理论中的这些问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Complex numbers have long played an important role in mathematics, physics, and engineering such as in designing electrical circuits. The study of functions defined on complex numbers has led to several new fields including the area of several complex variables, which is a major branch of mathematics where algebra, analysis and geometry intertwine. The principal goal of this project is to systematically study problems of a geometric nature in several complex variables. Many of the proposed problems have their roots in the physical sciences. This project aims to understand how geometric structures interplay with analytic properties in several complex variables. This project supports learning and research activities for undergraduate and graduate students, especially those from underrepresented groups. These learning and research experiences will better prepare the students for graduate school or a professional career.The main thrust of the proposed research is the several complex variables analogue of Mark Kac’s problem "Can one hear the shape of a drum?" That is, to what extend is the geometry of a complex or CR manifold determined by spectral behavior of the complex Laplacian? The complex Laplacian could either be the complex Neumann Laplacian on a complex manifold with boundary or the Kohn Laplacian on a CR manifold. These complex Laplacians are the natural outgrowth in several complex variables of the classical Laplace operator with the Neumann boundary condition. Problems to be studied include positivity, discreteness, and stability of the spectrum. The spectral theory of the complex Laplacian is intimately related to problems in quantum mechanics. The principal investigator plans to combine ideas and techniques from various fields of mathematics and physics to further study these problems in geometric function theory of several complex variables.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.