Isospectrality: Length vs. Laplace Spectra and Isospectral Families

同谱性：长度与拉普拉斯谱和同谱族

• 批准号: 0204648
• 负责人: Ruth Gornet
• 金额: $ 7.67万
• 依托单位: Texas Tech University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2003-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204648&HistoricalAwards=false
• 关键词: Isospectrality; Length; vs.; Laplace; Spectra

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.

项目摘要- Ruth Gornet - dms -0204648本提案涉及逆光谱几何中的几个主题。在第一个项目中，首席研究员将证明，与一般流形的拉普拉斯光谱和长度光谱相关的经典迹公式，提供的关于等光谱流形的信息比以前推测的要少。在与P. Perry的联合工作中，将明确计算海森堡流形上的波迹，以便了解长度与拉普拉斯谱的行为。此外，为了证明等谱流形上闭合测地线长度必须满足的一个必要条件，本文还研究了长度谱的新概念。另一个项目（与J. McGowan合作）研究了透镜空间上的p型谱。主要研究者构造了p型谱对某p相等但在函数上谱不相等的例子。在构造非等绝对长度的p-等光谱透镜空间时，将进一步研究这一特性；即不同长度的封闭测地线。在最后的项目中（与R. Brooks合作），将使用对称群表示理论的工具来构造固定属的等谱黎曼曲面数量的显式上界，这些曲面可以用sunada方法构造。当这个最终项目完成后，将得到具有给定zeta函数的非同构数域数目的显式上界。1966年，马克·卡茨（Mark Kac）普及了这个问题：“人能听到鼓的形状吗？”这个问题的数学公式是：“阿里曼流形的谱中包含什么几何信息？”“等谱性，即对等谱族和/或它们可能共享或不共享的几何特性的研究，影响着光谱几何之外的领域；因此，由该提案资助的研究支持了这些领域的纯数学基础。第一个闭合等谱流形的例子，米尔诺的平环面，已经出现在物理学的弦理论中（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。