Spectral and geometric problems in global analysis

全局分析中的谱和几何问题

• 批准号: 0605247
• 负责人: Carolyn Gordon
• 金额: $ 20.95万
• 依托单位: Dartmouth College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0605247&HistoricalAwards=false
• 关键词: Spectral; geometric; problems; global; analysis

Inverse spectral geometry is the study of the extent to which the geometry of a Riemannian manifold can be recovered from spectral data. The principal investigator, along with her research collaborators, will study Inverse spectral problems in both the compact and noncompact settings. For compact Riemannian manifolds, the natural spectral data are the eigenvalues of the Laplacian. To obtain information concerning Riemannian invariants that are not spectrally determined, constructions of isospectral manifolds will be investigated. Continuing joint work with E. Makover and D. Webb, the principal investigator will study the extent to which the spectrum of a Riemann surface determines its Jacobian. She and Webb, along with E. Dryden and S. Greenwald, will also consider inverse spectral results on orbifolds. For noncompact manifolds, natural spectral data include the scattering resonances and scattering phase. The principal investigator and Webb, along with P. Perry, will study of obstacles with the same scattering resonances and scattering phase. They will also study isopolar and isophasal potentials for the Schrodinger operator.In spectroscopy, one attempts to understand the chemical composition of an object such as a star from the characteristic frequencies of light emitted. Analogously, the question phrased by Mark Kac as "Can one hear the shape of a drum?" asks whether one can determine the shape of a vibrating membrane such as a drumhead from its characteristic frequencies of vibration. In earlier work, the principal investigator, along with D. Webb and S. Wolpert, constructed examples of polygonal shaped "membranes" (bounded domains in the plane) that have exactly the same characteristic frequencies, thus answering Kac's question in the negative. Kac's question generalizes to nonplanar surfaces and higher dimensional objects (Riemannian manifolds), with the analog of the characteristic frequencies being the Laplace spectrum. The principal investigator, along with her collaborators, will continue her investigation of the extent to which spectral data determines the geometry of a Riemannian manifold. Constructions of manifolds with the same spectrum will be studied to identify geometric properties that are not spectrally determined. Singular spaces (orbifolds) will also be considered.

逆光谱几何形状是对可以从光谱数据中恢复riemannian歧管的几何形状的程度的研究。 首席研究人员以及她的研究合作者将研究紧凑和非划界设置的逆频谱问题。 对于紧凑的riemannian歧管，自然光谱数据是拉普拉斯的特征值。 为了获得有关未确定的黎曼不变的信息，将研究同一歧管的构造。 与E. Makover和D. Webb继续进行联合合作，主要研究人员将研究Riemann表面频谱决定其Jacobian的程度。 她和韦伯以及E. Dryden和S. Greenwald也将考虑Orbifolds的逆光谱结果。 对于非绘制歧管，自然光谱数据包括散射谐振和散射阶段。 首席研究员和韦伯以及P. Perry将研究具有相同散射共振和散射阶段的障碍。 他们还将研究Schrodinger操作员的同极和同性含量。 类似地，马克·卡克（Mark Kac）的问题称为“一个人可以听到鼓的形状吗？”询问是否可以根据其特征性的振动频率来确定振动膜的形状，例如鼓头的形状。 在较早的工作中，主要研究者与D. Webb和S. Wolpert一起构建了具有完全相同的特征频率的多边形“膜”（平面中有界域）的例子，从而在负面回答了KAC的问题。 KAC的问题概括为非平面表面和更高的尺寸对象（riemannian歧管），而特征频率的类似物是拉普拉斯光谱。 首席调查员以及她的合作者将继续调查光谱数据确定riemannian歧管的几何形状的程度。 将研究具有相同光谱的歧管的构造，以识别未确定未确定的几何特性。 还将考虑奇异空间（Orbifolds）。