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拉普拉斯-贝尔特拉米算子。她将尝试确定拉普拉斯谱在多大程度上决定流形的几何形状。研究等光谱流形的一个目标是找出不由光谱决定的几何不变量。此外，她将尝试找出其他的自然算子，它们的谱可以区分这些度规。如果两个流形或曲面上的拉普拉斯算子对应的特征值相等，则称它们是等谱的。如果把这两个表面想象成鼓的表面，这就意味着两个鼓敲起来的声音是一样的。首席研究员将研究两个表面“听起来一样”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.