Inverse Spectral Problems in Riemannian Geometry

摘要奖：DMS-0072534主要研究者：Carolyn S.戈登逆谱几何是研究曲面几何或更一般的黎曼流形几何可以从谱数据中提取的程度。 紧黎曼流形的基本谱数据是Laplace-Beltrami算子的特征值。 作者提出应用最近发展起来的方法来研究本征值谱在何种程度上决定了紧致黎曼流形的局部几何。 他们也会问在何种程度上额外的光谱数据，如光谱的拉普拉斯作用于不同程度的微分形式确定几何形状的流形. 逆谱问题将被认为是黎曼orbifolds以及流形; orbifolds是最易处理的奇异空间。 对于薛定谔算符“拉普拉斯+势”，本文研究了在环面上的线丛情况下从谱数据恢复势的问题。 类似于平面区域的情形，紧黎曼流形上的拉普拉斯算子的最低特征值可以被看作是基调。 本文利用Riemann曲面的谱与图的谱之间的联系，研究随机Riemann曲面是否具有大的第一特征值的问题。 对于非紧黎曼流形，主要的光谱数据是散射极;研究者们希望得到连续的等极度规族。在光谱学中，人们试图从发出的光或声的特征频率中恢复物体的化学成分或形状。 在振动膜的情况下，如鼓面，在数学上被看作是一个有界的区域在theplane，频谱的特征频率对应于数学概念的拉普拉斯频谱。 拉普拉斯谱也被定义为数学和物理中出现的称为流形的其他几何对象。 研究者们，沿着斯科特·沃尔珀特，在早些时候构建了第一个具有相同光谱的不同形状鼓面（平面区域）的例子。 平面区域可以在其全局形状上不同，但局部是相同的;即，如果你看一小块从其中一个地区切下来的，你不能告诉它来自哪个地区。 最近，主要的制造商开发的方法来构建几何对象具有相同的拉普拉斯频谱，但不同的地方，以及asglobal形状。 这些方法将用于研究流形的哪些局部几何性质不是谱决定的。 另外的光谱问题也将被考虑，如任意大的体积，但有界的基调表面的建设。

AbstractAward: DMS-0072534Principal Investigator: Carolyn S. GordonInverse spectral geometry is the study of the extent to which thegeometry of a surface or, more generally, of a Riemannianmanifold can be extracted from spectral data. The primaryspectral data associated to a compact Riemannian manifold are theeigenvalues of the Laplace-Beltrami operator. The investigatorspropose to apply recently developed methods to study the extentto which the eigenvalue spectrum determines the local geometry ofa compact Riemannian manifold. They will also ask the extent towhich additional spectral data such as the spectrum of theLaplacian acting on differential forms of various degreesdetermines the geometry of the manifold. Inverse spectralproblems will be considered on Riemannian orbifolds as well as onmanifolds; orbifolds are the most tractable singular spaces. Forthe Schrodinger operator "Laplacian plus potential", the problemof recovering the potential from spectral data will be studied inthe case of line bundles over tori. In analogy to the case ofplanar domains, the lowest eigenvalue of the Laplacian on acompact Riemannian manifold may be viewed as the fundamentaltone. The question of whether random Riemann surfaces have largefirst eigenvalue will be studied using connections betweenspectra of Riemann surfaces and spectra of graphs. Fornoncompact Riemannian manifolds, the primary spectral data arethe scattering poles; the investigators expect to exhibitcontinuous families of isopolar metrics.In spectroscopy, one attempts to recover the chemical compositionor the shape of an object from the characteristic frequencies oflight or sound emitted. In the case of a vibrating membrane suchas a drumhead, viewed mathematically as a bounded region in theplane, the spectrum of characteristic frequencies corresponds tothe mathematical notion of the Laplace spectrum. The Laplacespectrum is also defined for other geometric objects calledmanifolds which arise in mathematics and physics. Theinvestigators, along with Scott Wolpert, earlier constructed thefirst examples of differently shaped drumheads (planar regions)with the same spectrum. Planar regions can differ in theirglobal shape but locally are identical; i.e., if you look at asmall piece cut out from one of the regions, you can not tellwhich region it came from. Recently, the principal investigatordeveloped methods for constructing geometric objects with thesame Laplace spectrum but which differ in their local as well asglobal shape. These methods will be used to investigate whichlocal geometric properties of manifolds are not spectrallydetermined. Additional spectral problems will also be consideredsuch as the construction of surfaces of arbitrarily large volumebut having bounded fundamental tone.