Spectral and geometric problems in global analysis

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。 主要研究员，沿着与她的研究合作者，将研究逆谱问题在紧凑和非紧凑设置。 对于紧致黎曼流形，自然谱数据是拉普拉斯算子的特征值。 为了获得关于黎曼不变量的信息，不是谱确定的，将研究等谱流形的构造。 继续与E. Makover和D.首席研究员韦伯将研究Riemann曲面的谱在多大程度上决定其Jacobian。 她和韦伯，沿着还有E。德莱登和S. Greenwald，也将考虑orbifolds的逆谱结果。 对于非紧流形，自然谱数据包括散射共振和散射相位。 首席研究员和韦伯，沿着与P.佩里，将研究具有相同散射共振和散射相位的障碍物。 他们还将研究薛定谔算符的等极和等相势。在光谱学中，人们试图从发射光的特征频率来理解物体（如星星）的化学成分。 类似地，马克·卡茨（Mark Kac）提出的问题是：“人们能听到鼓的形状吗？“的问题，是否可以确定振动膜的形状，如鼓面从其振动的特征频率。 在早期的工作中，主要研究者，沿着D. Webb和S. Wolpert，构造了多边形“膜”（平面中的有界域）的例子，它们具有完全相同的特征频率，从而否定了Kac的问题。 卡茨的问题推广到非平面表面和高维物体（黎曼流形），与模拟的特征频率是拉普拉斯频谱。 主要研究者，沿着与她的合作者，将继续她的调查在何种程度上光谱数据确定几何形状的黎曼流形。 将研究具有相同谱的流形的构造，以确定不是谱确定的几何性质。 奇异空间（orbifolds）也将被考虑。

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.