Isospectrality: Length vs. Laplace Spectra and Isospectral Families

项目摘要- Ruth Gornet - DMS-0204648本提案涉及逆光谱几何学的几个主题。在第一个项目中，首席研究员将表明，经典的迹公式，它涉及的拉普拉斯和长度谱的一般流形，提供更少的信息isospectral流形比previouslyspecified。在与P.佩里的合作工作中，为了理解长度与拉普拉斯谱的关系，我们将明确地计算Heisenberg流形上的波迹，另外，为了证明等谱流形上闭测地线的长度必须满足的一个必要条件，我们将研究一个新的长度谱的概念。另一个项目（与J. McGowan联合）研究透镜空间上的p型谱。主要研究者已经构造了透镜空间的例子，其p-形式谱对于某些p是相等的，但在函数上具有不等谱。这种行为将进一步研究，以构建具有不等绝对长度谱的p-等谱透镜空间;即，不同长度的闭合测地线。在最后的项目（与R。布鲁克斯）工具的对称群的表示理论将被用来构造一个显式的上界的isospectralRiemann曲面的数量的一个固定的亏格，可以构造的Sunada方法。当这个最后的项目完成时，一个明确的上界的非同构数域的数量与给定zeta函数的结果。在1966年，马克·卡茨普及的问题，“一个人能听到鼓的形状吗？“这个问题的数学表述是：”黎曼流形的谱中包含什么几何信息？"等谱性，即，对光谱族和/或它们可能共享或不共享的几何特性的研究影响了光谱几何学之外的领域;因此，由该提案资助的研究支持了这些领域的纯数学基础。封闭等谱流形的第一个例子，米尔诺尔的平坦环面，已经出现在物理学的弦论中（与镜像对称有关）。光谱学的经验科学研究了原子和分子的频率，以提供有关振动物体的信息。逆光谱问题也出现在医学成像、地球物理勘探和无损检测中。

Project Abstract - Ruth Gornet - DMS-0204648This proposal addresses several topics in inverse spectralgeometry. In the first project, the Principal Investigator willshow that the classical trace formula, which relates the Laplaceand length spectra for generic manifolds, provides lessinformation about isospectral manifolds than was previouslyspeculated. In joint work with P. Perry, the wave trace onHeisenberg manifolds will be explicitly calculated in order tounderstand the behavior of the length vs. Laplace spectra.Additionally, a new notion of length spectrum will be studied inorder to prove a necessary condition that lengths of closedgeodesics on isospectral manifolds must satisfy. A further project(joint with J. McGowan) studies the p-form spectrum on lensspaces. The Principal Investigator has constructed examples oflens spaces whose p-form spectra are equal for certain p but withunequal spectra on functions. This behavior will be furtherstudied toward constructing p-isospectral lens spaces with unequalabsolute length spectrum; i.e., different lengths of closedgeodesics. In the final project (joint with R. Brooks) tools fromrepresentation theory of the symmetric groups will be used toconstruct an explicit upper bound on the number of isospectralRiemann surfaces of a fixed genus that can be constructed from theSunada method. When this final project is completed, an explicitupper bound on the number of nonisomorphic number fields with agiven zeta function will result.In 1966, Mark Kac popularized the question, "Can one hear theshape of a drum?" The mathematical formulation of this questionis: ``What geometric information is contained in the spectrum of aRiemannian manifold?'' Isospectrality, i.e., the study ofisospectral families and/or the geometric properties they may ormay not share, impacts areas outside of spectral geometry; theresearch funded by this proposal thus supports thepure-mathematical foundations of these areas. The first examplesof closed isospectral manifolds, Milnor's flat tori, have appearedin string theory in physics (related to mirror symmetry). Theempirical science of spectroscopy has studied frequencies of atomsand molecules to provide information about vibrating objects.Inverse spectral problems also arise in medical imaging,geophysical prospection, and non-destructive testing.