Spectral Geometry of Non-Compact Domains and Riemannian Manifolds

摘要DMS -0100829主要研究者将研究非紧域和黎曼流形的谱几何，以阐明散射极的几何内容。首先，PI将继续研究双曲流形的谱几何及其扰动。他将研究作为变形空间上的函数的共振，定义并分析拉普拉斯行列式。其次，PI将研究两步幂零李群上的波动方程及其离散子群的子群的散射理论，导出Heisenberg和Heisenberg型群上的波动方程的新参数，Gordon，Wilson，而其他人将被用来获得对和家庭的`这将有助于确定几何信息的限制，可以从散射极点的知识推导出来。 谱几何的基本问题是阐明黎曼流形上的拉普拉斯谱的几何内容，对于所谓的散射流形，拉普拉斯的本征值与散射共振一起构成了流形的谱数据。 阐明这种光谱数据的几何内容推进了我们对量子化的理解，产生了新的分析工具，用于研究几何对象，并提供了深入了解逆问题的一个更“应用”的性质，其中的特征值和scatteringpoles是可测量的数量。目前的工作的目的是开始几何自然的例子，李理论，automorphicfunctions，和谐波分析的技术可能会使用，并进展到hardproblems，如目标识别雷达在这种技术是不可用的，但基本的数学问题是非常相似的。

Abstract for DMS - 0100829The principal investigator will study the spectral geometry of non-compactdomains and Riemannian manifolds in order to elucidate the geometriccontent of scattering poles. First, the PI will continue his study of thespectral geometry of hyperbolic manifolds and their perturbations. He willstudy resonances as functions on the deformation space of the underlyingdiscrete group, and define and analyze a determinant of the Laplacian.Secondly, the PI will study scattering theory for the wave equation ontwo-step nilpotent Lie groups and their quotients by discrete subgroups.New parametrices or the wave equation on the Heisenberg and Heisenberg-typegroups will be derived, and trace formula for certain quotients obtained.Riemannian submersion techniques of Gordon, Wilson, and others will be usedto obtain pairs and families of `isoscattering' manifolds which will helpdetermine the limits of geometric information which may be deduced from aknowledge of the scattering poles. Thirdly, the PI will study theisoscattering problem for exterior domains in Euclidean space.The fundamental problem of spectral geometry is to elucidate thegeometric content of the Laplace spectrum on a Riemannian manifold.For so-called scattering manifolds, the eigenvalues of the Laplaciantogether with scattering resonances constitute the spectral data for themanifold. Elucidating the geometric content of such spectral dataadvances our understanding of quantization, produces new analytic toolsfor the study of geometric objects, and provides insight into inverseproblems of a more `applied' nature where the eigenvalues and scatteringpoles are measurable quantities. The present work aims to begin withgeometrically natural examples where techniques of Lie theory, automorphicfunctions, and harmonic analysis may be used, and progress to harderproblems such as target identification by radar where such techniques are not available but the underlying mathematical problems are very similar.