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.

逆谱几何是研究黎曼流形的几何在多大程度上可以从谱数据中恢复出来。 对于紧致黎曼流形，自然谱数据是拉普拉斯算子的特征值。 主要研究员沿着与各种研究合作者将解决对称和局部对称空间和黎曼曲面上的线丛的逆谱问题。研究了紧致K“ahler流形的谱在多大程度上决定了相应的广义Jacobian簇和Albanese tori的结构;对于非紧致流形，相关的谱数据是散射共振和散射相位. Gordon，和Webb，沿着与P.佩里一起，将继续他们对具有相同散射共振和散射相位的障碍物的研究。光谱几何学，它植根于光谱学，研究物体（如振动鼓）的几何形状与光谱数据（如振动的特征频率）之间的关系。 谱几何学从数学的许多领域，如几何分析、表示论和数论中汲取了知识，是数学和物理之间相互作用的活跃领域。 这个项目将主要集中在几何方面，经常在李群表示发挥核心作用的设置。 对称空间是黎曼几何的模型空间。 研究的一个焦点将是对称空间是否与其他几何对象在谱上可区分的问题。 其他方面的研究将是光谱数据和各种几何属性之间的关系，如封闭测地线的长度。