Mathematical Sciences: Geometry of the Fundamental Group

数学科学：基本群几何

• 批准号: 8801158
• 负责人: Robert Brooks
• 金额: $ 4.61万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-01 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8801158&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Fundamental; Group

Robert Brooks will investigate connections between the geometry of a surface and the eigenvalues of the Laplacian. Work on these problems was popularized some years ago by questions such as "Can you hear the shape of a drum?". If you have perfect pitch then you can in theory detect the harmonics of the drum, that is the eigenvalues of the Laplacian. So the question is how much geometry of a surface can be detected from knowledge of these eigenvalues. Brooks' work will focus on the study of the small eigenvalues of hyperbolic manifolds and isospectral problems on compact manifolds. There are already estimates for the low eigenvalues of compact manifolds. Brooks will work to extend these to the finite volume case. This is a non-trivial extension but certainly a natural one since a key ingredient of earlier estimates is the the notion of injectivity radius which is zero for non-compact finite volume manifolds. His work on isospectral problems is concerned with a study of the topology of sets of isospectral metrics in a given conformal class. The intention here is to apply the solution of the Yamabe problem to the geometry of three dimensional manifolds.

罗伯特布鲁克斯将调查之间的联系 曲面的几何和拉普拉斯算子的特征值。工作 关于这些问题的讨论在几年前被一些问题所普及， 比如“你能听到鼓的形状吗？".如果你有完美的 那么理论上你就可以检测到鼓的谐波， 也就是拉普拉斯算子的特征值所以问题是如何 表面的许多几何形状可以通过以下知识来检测： 这些特征值。 布鲁克斯的工作将集中在研究小 双曲流形的特征值与等谱问题 紧流形已经有人估计 紧流形的特征值布鲁克斯将努力扩大 这是有限体积的情况。这是一个非平凡的扩展 但肯定是天然的，因为早期的一个关键成分 估计是内射半径的概念， 非紧有限体积流形他在等光谱方面的工作 问题是关于一个研究的拓扑结构的集合， 在给定的共形类中的等谱度量。的意图 这里是将Yamabe问题的解决方案应用于 三维流形的几何