Mathematical Sciences: Inverse Spectral Problems in Riemannian Geometry

首席研究员将继续研究 紧黎曼流形上的Laplace-Beltrami算子她将 试图确定拉普拉斯算子的频谱 决定了流形的几何形状。研究的一个目标是 是研究等谱流形的例子， 几何不变量不是由光谱决定的。在 此外，她将尝试识别其他自然操作员， 光谱可以区分度量。 两个流形或曲面称为等谱的，如果 流形上拉普拉斯算子的相应特征值是 平等如果把这两个表面看作鼓的表面， 这意味着两个鼓在敲击时会发出相同的声音。的 首席研究员将研究两人 表面"听起来一样"，但形状不同。

The principal investigator will continue her study of the Laplace-Beltrami operator on compact Riemannian manifolds. She will try to determine the extent to which the spectrum of the Laplacian determines the geometry of the manifold. One goal of the research is to study examples of isospectral manifolds and identify geometric invariants which are not determined by the spectra. In addition she will try to identify other natural operators whose spectra might distinguish the metrics. Two manifolds or surfaces are said to be isospectral if the corresponding eigenvalues of the Laplacians on the manifolds are equal. If one thought of the two surfaces as surfaces of drums this would mean that the two drums would sound the same when struck. The principal investigator will study situations in which the two surfaces "sound the same" but have different shapes.