Mathematical Sciences: Inverse Spectral Problems in Riemannian Geometry

首席研究员将继续研究紧致黎曼流形上的 Laplace-Beltrami 算子。她将尝试确定拉普拉斯算子的谱在多大程度上决定流形的几何形状。该研究的目标之一是研究等谱流形的示例并识别不是由谱确定的几何不变量。此外，她将尝试识别其他自然算子，其光谱可能会区分指标。 如果流形上拉普拉斯算子的相应特征值相等，则称两个流形或曲面是等谱的。如果将这两个表面视为鼓的表面，则意味着这两个鼓在敲击时发出的声音是相同的。首席研究员将研究两个表面“听起来相同”但形状不同的情况。

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.