Spectral theory of Complex Laplacians and Applications

多变量复分析是分析、代数和几何相互交织的数学分支。本课题研究的复拉普拉斯算子是与柯西-黎曼或切向柯西-黎曼复形相关的二阶微分算子。该项目的主要目标是研究这些拉普拉斯算子的谱理论，特别是复杂拉普拉斯算子的谱行为与底层空间几何结构之间的相互作用。本课题要解决的问题包括谱的显式计算、正性、纯离散性和稳定性。首席研究员还将研究Bergman和Szego核的边界行为，以及这些核之间的关系。复杂分析是物理学和工程学中必不可少的工具。两个多世纪以来，物理学家和工程师们一直用经典的微分算子来研究热扩散和流体动力学，它们的专业名称是狄利克雷算子和诺伊曼-拉普拉斯算子。所谓复拉普拉斯算子是这些经典算子在多变量复分析中的类似物。微分算子的谱理论在生物、医学和物理科学的许多领域起着重要的作用。例如，反问题——从光谱特性确定形状的问题——在医学成像等领域有应用。复拉普拉斯谱的纯粹离散性与抗磁性和顺磁性密切相关，这是量子物理和化学中广泛研究的主题。在这个项目中发展的思想和技术可能对数学和科学的其他分支产生潜在的影响。该项目对人力资源的发展也有重大影响：首席研究员将开发主题课程，并吸引研究生和本科生参与研究活动。该项目的研究将被纳入教学过程。此外，该项目还将促进数学家、生物学家和计算机科学家之间的跨学科研究活动。

Complex analysis of several variables is a branch of mathematics where analysis, algebra, and geometry intertwine. The complex Laplacians studied in this project are the second-order differential operators associated with the Cauchy-Riemann or tangential Cauchy-Riemann complexes. The main goal of the project is to study the spectral theory of these Laplacians, in particular, the interplay between the spectral behavior of the complex Laplacians and the geometric structure of the underlying spaces. Problems to be addressed in this project include explicit computations, positivity, pure discreteness, and stability of the spectrum. The principal investigator will also study boundary behavior of the Bergman and Szego kernels, as well as the relationship between these kernels.Complex analysis is an essential tool in physics and engineering. The classical differential operators known under their technical names as the Dirichlet- and Neumann-Laplacians have been used to study heat diffusion and fluid dynamics by physicists and engineers for more than two centuries. The so-called complex Laplacians are the analogues of these classical operators in the setting of complex analysis in several variables. Spectral theory of differential operators plays an important role in many areas of biological, medical, and physical sciences. For example, the inverse problem--the problem of determining the shape from spectral properties--has applications in fields such as medical imaging. Pure discreteness of the spectra of the complex Laplacians is intimately related to diamagnetism and paramagnetism, topics widely studied in quantum physics and chemistry. Ideas and techniques developed in this project can potentially have repercussions in other branches of mathematics and sciences. The project also has significant impact on the development of human resources: the principal investigator will develop topics courses and engage graduate and undergraduate students in research activities. Research in this project will be incorporated into the teaching and learning process. In addition, this project will also facilitate interdisciplinary research activities among mathematicians, biologists, and computer scientists.