Problems in geometric analysis

几何分析中的问题

• 批准号: 0906168
• 负责人: Carolyn Gordon
• 金额: $ 22.53万
• 依托单位: Dartmouth College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906168&HistoricalAwards=false
• 关键词: Problems; geometric; analysis

Inverse spectral geometry is the study of the extent to which the geometry of aRiemannian manifold can be recovered from spectral data. For compactRiemannian manifolds, the natural spectral data are the eigenvalues of the Laplacian. The principal investigator along with various research collaborators will address inverse spectral problems on symmetric and locally symmetric spaces and on line bundles over Riemann surfaces. They will also study the extent to which the spectrum of a compact K\"ahler manifold determines the structure of the associated generalized Jacobian varieties and Albanese tori.For non-compact manifolds, the relevant spectral data are the scattering resonances and scattering phase. Gordon, and Webb, along with P. Perry, will continue their study of obstacles with the same scattering resonances and scattering phase.Spectral geometry, which is rooted in spectroscopy, studies the relationship between the geometry of an object such as a vibrating drum and spectral data such as the characteristic frequencies of vibration. Spectral geometry draws from many areas of mathematics such as geometric analysis, representation theory, and number theory and is an active area of interplay between mathematics and physics. This project will focus primarily on geometric aspects, frequently in settings in which Lie group representations play a central role. Symmetric spaces are the model spaces of Riemannian geometry. One focus of the research will be the question of whether symmetric spaces are spectrally distinguishable from other geometry objects. Other aspects of the research will be the relationship between the spectral data and various geometric properties such as the lengths of closed geodesics.

逆谱几何是研究从谱数据恢复黎曼流形几何形状的程度。 对于紧致黎曼流形，自然光谱数据是拉普拉斯算子的特征值。 首席研究员与各种研究合作者将解决对称和局部对称空间以及黎曼曲面上的线束上的反谱问题。他们还将研究紧致 K'ahler 流形的谱在多大程度上决定了相关的广义雅可比簇和阿尔巴尼环面的结构。对于非紧流形，相关的谱数据是散射共振和散射相位。 Gordon ，韦伯和 P.佩里将继续研究具有相同散射共振和散射相位的障碍物。谱几何植根于光谱学，研究物体的几何形状（例如振动鼓）与光谱数据，例如振动的特征频率。光谱几何源自数学的许多领域，例如几何分析、表示论和数论，并且是数学和物理学之间相互作用的活跃领域。通常，在李群表示发挥核心作用的环境中，对称空间是黎曼几何的模型空间。 研究的重点之一是对称空间是否在光谱上与其他几何对象有区别。 研究的其他方面将是光谱数据与各种几何特性（例如闭合测地线的长度）之间的关系。