Problems in geometric analysis

逆谱几何是研究从谱数据中可以恢复黎曼流形几何的程度的学科。对于紧黎曼流形，自然谱数据是拉普拉斯流形的特征值。主要研究人员将与不同的研究合作者一起解决对称和局部对称空间上的逆谱问题，以及黎曼曲面上的线丛。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亚托夫的结构。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以及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环境中。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在光谱上可与其他几何对象区分的问题。研究的其他方面将是光谱数据与各种几何性质之间的关系，例如闭合测地线的长度。

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.