RUI: Spectral Theory and Geometric Analysis in Several Complex Variables

许多物理和社会现象可以用偏微分算子进行数学建模。拉普拉斯算子是一种微分算子，长期以来在力学、物理和数学中发挥着重要作用。复拉普拉斯算子是经典拉普拉斯算子在多变量复分析中的自然发展，是代数、分析和几何相互交织的数学分支。本研究项目探讨复拉普拉斯算子的解析和几何性质，特别是它的谱。光谱分析是科学研究的重要工具，众所周知，复拉普拉斯算子的光谱性质与物理学中的某些量子现象密切相关。这个项目的目标是了解底层复杂空间的代数、解析和几何结构是如何相互作用的。该项目结合了几个数学分支的思想和方法，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。这个项目让本科生参与到研究活动中来，扩大了数学领域中代表性不足群体的参与。复诺伊曼拉普拉斯算子是具有非强制边界条件的椭圆算子的一个原型。自1960年代Kohn和Hörmander的工作以来，人们对复诺伊曼-拉普拉斯算子的正则性理论进行了广泛的研究，在偏微分方程和一些复变量中都有了重要的发现。本文的主要目的是研究复诺伊曼拉普拉斯算子的谱理论，重点研究算子的谱行为与底层几何结构之间的相互作用。本课题研究的问题包括底层结构变形时谱的稳定性以及复拉普拉斯算子具有离散谱的复流形的表征。还研究了复流形上Cauchy-Riemann算子的正则性理论，再现核，不变度量及其在复杂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，促进新课程的开发，吸引学生进入数学，并促进跨学科研究。

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.