Isospectrality: Length vs. Laplace Spectra and Isospectral Families

项目摘要-Ruth Gornet-DMS-0204648本提案涉及逆谱几何中的几个主题。在第一个项目中，首席调查者将证明经典的迹公式，它联系了一般流形的拉普拉斯谱和长度谱，提供的关于等谱流形的信息比以前推测的要少。在与P.Perry的合作中，将显式计算Heisenberg流形上的波迹，以了解长度对Laplace谱的行为.此外，还将研究长度谱的新概念，以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.另一个项目(与J.McGowan合作)研究透镜空间上的p-型谱。主要研究者已经构造了p-型谱对某些p相等但函数谱不等的F-ens空间的例子。这一行为将被进一步研究，以构造具有不等绝对长度谱的p-等谱透镜空间，即不同长度的闭测地线。在最后的项目中(与R.Brooks联合)，将使用对称群表示理论中的工具来构造可由Sunada方法构造的固定亏格的等谱Riemann曲面的个数的显式上界。当这个最后的项目完成时，使用不同的Zeta函数的不同构数域的个数将得到一个明确的上限。在1966年，Mark Kac推广了这个问题，“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，即研究等谱族和/或它们可能共享或可能不共享的几何性质，影响谱几何之外的区域；因此，由这一提议资助的研究支持了这些区域的纯数学基础。闭合等谱流形的第一个例子，Milnor的平坦环面，出现在物理学中的弦理论(与镜像对称有关)中。光谱学的经验科学研究了原子和分子的频率，以提供关于振动物体的信息。在医学成像、地球物理勘探和无损检测中也会出现相反的光谱问题。

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.