Mathematical Sciences: Spectral Geometry & Representation Theory on Higher Step Riemannian Nilmanifolds

9409209对于(M，g)闭黎曼流形，用SPEC(M，g)表示作用在光滑函数上的相关拉普拉斯-贝尔特拉米算子的具有重数的特征值的集合。戈尔尼特的研究集中在“流形的谱中包含什么几何信息？”特别是，她开发了一种产生等谱黎曼尼尔流形对的新结构。这种结构产生了新的等谱流形对，表现出许多以前没有发现的性质。这样的例子是发现不是由光谱决定的性质的唯一方法。Gornet目前正在研究的几何性质包括长度谱(闭测地线的长度集合)和标记长度谱(它还包括关于闭测地线的自由同伦类的信息)。戈尔内特的研究中使用的技术来自黎曼几何、表示论和李群。戈尔内特建议利用研究计划补助金探索表示理论和封闭测地线研究的新技术，以补充她目前在谱几何和黎曼几何方面的背景。***

9409209 GORNET For (M,g) a closed Riemannian manifold, denote by spec(M,g) the collection of eigenvalues with multiplicities of the associated Laplace-Beltrami operator acting on smooth functions on M. Gornet's research focuses on the question "What geometric information is contained in the spectrum of a manifold?" In particular, she has developed a new construction for producing pairs of isospectral Riemannian nilmanifolds. This construction has produced new pairs of isospectral manifolds exhibiting many properties not previously found. Such examples are the only way to discover properties not determined by the spectrum. Geometric properties Gornet is currently studying include the length spectrum (the collection of lengths of closed geodesics) and the marked length spectrum (which also includes information on the free homotopy classes of the closed geodesics.) Techniques used in Gornet's research come from Riemannian geometry, representation theory, and Lie groups. Gornet proposes to use the Research Planning Grant to explore new techniques in representation theory and in the study of closed geodesics to supplement her current background in spectral and Riemannian geometry. ***