POWRE: Spectral Geometry of Nilmanifolds and Kleinian Groups

这项工作是通过妇女在研究和教育中的职业机会方案作为访问教授活动资助的。 1966年，马克·卡茨（Mark Kac）提出了一个问题：“人们能听到鼓的形状吗？”“将鼓面视为平面域，当鼓振动时产生的频率对应于拉普拉斯算子的本征值。因此，数学公式的问题所提出的卡茨是：“频谱的平面域确定其几何形状？“逆谱几何研究这个问题的推广到黎曼流形。 本文主要讨论逆谱几何中的三个问题：（1）经典迹公式与长度谱，（2）谱刚性与1-形式上的拉普拉斯谱，（3）双曲三维流形的谱分析。前两个主题涉及黎曼流形，它在证明谱不确定的性质方面发挥了至关重要的作用。 在第一个项目中，波的不变量，使用迹公式构造，激发了一个新的概念的长度谱，这将是研究证明一个必要条件，封闭测地线等谱流形必须满足。在第二个项目中，一维不变子空间的拉普拉斯算子将被检查，以表明1-形式的频谱检测的Sunada方法。 在第三个主题中，引入双曲几何和拓扑学的工具来阐明谱与几何之间的关系，特别是在几何无限双曲三维流形上，它们被实现为几何有限流形的极限。

This work is funded through the Professional Opportunities for Women in Research and Education (POWRE) program as a Visiting Professorship Activity. In 1966, Mark Kac popularized the question, "Can one hear the shape of a drum?" Viewing a drumhead as a plane domain, the frequencies produced when the drum vibrates correspond to the eigenvalues of the Laplace operator. Thus the mathematical formulation of the question posed by Kac is: "Does the spectrum of a plane domain determine its geometry?" Inverse spectral geometry studies the generalization of this question to Riemannian manifolds. This proposal addresses three topics in inverse spectral geometry: (1) the classical trace formula and the length spectrum, (2) spectral rigidity and the Laplace spectrum on 1-forms,and (3) spectral analysis of hyperbolic 3-manifolds. The first two topics concern Riemannian nilmanifolds, which have played a vital role in demonstrating properties not determined by the spectrum. In the first project, wave invariants, constructed using the trace formula, motivate a new notion of length spectrum, which will be studied toward proving a necessary condition that closed geodesics on isospectral manifolds must satisfy. In the second project, one-dimensional invariant subspaces of the Laplacian will be examined to show that the 1-form spectrum detects the Sunada method. In the third topic, the tools of hyperbolic geometry and topology are brought in to elucidate the relationship between spectrum and geometry, particularly on geometrically infinite hyperbolic 3-manifolds that are realized as limits of geometrically finite ones.