Mathematical Sciences: Geometry of the Fundamental Group

数学科学：基本群几何

• 批准号: 9000631
• 负责人: Robert Brooks
• 金额: $ 5.89万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-06-01 至 1992-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9000631&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Fundamental; Group

The principal investigator will study the relationship between the spectrum of the Laplacian of a manifold and the geometry of that manifold. This will include an investigation of the relationship between the spectrum and geometric properties such as diameter and injectivity radius, as well as a trace formula for Riemannian manifolds. The project will include a study of eigenvalue estimates in the context of obtaining finiteness and compactness results for isospectral manifolds. The problems to be addressed in this project are extensions of Mark Kac's famous question, "Can you hear the shape of a drum?" Mathematicians would ask if the spectrum of sound frequencies radiating from a drum can be used to distinguish between different shaped drums. In two dimensions these frequencies can be used to make such distinctions, but in higher dimensions they can not. The principal investigator will study whether or not other mathematical distinctions between objects other than their exact shapes, can be determined by an analysis of their sound spectrums.

首席研究员将研究流形拉普拉斯函数的谱与流形几何的关系。这将包括光谱和几何性质（如直径和注入半径）之间关系的研究，以及黎曼流形的迹公式。该项目将包括在获得等谱流形的有限性和紧性结果的背景下研究特征值估计。这个项目要解决的问题是Mark Kac著名问题的延伸，“你能听到鼓的形状吗？”数学家们会问，从鼓中辐射出来的声频频谱是否可以用来区分不同形状的鼓。在二维中，这些频率可以用来做出这样的区分，但在更高的维度中，它们不能。首席研究员将研究，除了物体的确切形状外，物体之间的其他数学区别是否可以通过分析它们的声谱来确定。