Problems in geometric analysis

逆光谱几何是研究在多大程度上的几何阿里曼流形可以从光谱数据恢复。对于紧黎曼流形，其自然谱数据是拉普拉斯流形的特征值。首席研究员将与各种研究合作者一起研究对称和局部对称空间以及黎曼曲面上的线束上的逆光谱问题。他们还将研究紧化K\ ahler流形的谱在多大程度上决定相关的广义雅可比变分和Albanese环面的结构。对于非紧凑流形，相关的光谱数据是散射共振和散射相位。Gordon、Webb和P. Perry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。光谱几何根植于光谱学，研究振动鼓等物体的几何形状与振动特征频率等光谱数据之间的关系。光谱几何借鉴了许多数学领域，如几何分析、表示理论和数论，是数学和物理相互作用的活跃领域。该项目将主要关注几何方面，经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。研究的一个重点将是对称空间是否在光谱上与其他几何物体区分。其他方面的研究将是光谱数据与各种几何性质之间的关系，如封闭测地线的长度。

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.