Mathematical Sciences: Geometry of the Fundamental Group

首席研究员将研究流形的拉普拉斯谱与该流形的几何之间的关系。这将包括研究谱与几何性质之间的关系，如直径和内射半径，以及黎曼流形的迹公式。该项目将包括在获得等谱流形的有限性和紧性结果的背景下对特征值估计的研究。在这个项目中要解决的问题是Mark Kac著名的问题的延伸，“你能听到鼓的形状吗？”数学家会问，鼓辐射的声音频率频谱是否可以用来区分不同形状的鼓。在两个维度中，这些频率可以被用来做出这样的区分，但在更高的维度中，它们不能。首席研究员将研究是否可以通过分析物体的声谱来确定物体之间的其他数学区别，而不是它们的确切形状。

The principal investigator will study the relationship between the spectrum of the Laplacian of a manifold and the geometry of that manifold. This will include an investigation of the relationship between the spectrum and geometric properties such as diameter and injectivity radius, as well as a trace formula for Riemannian manifolds. The project will include a study of eigenvalue estimates in the context of obtaining finiteness and compactness results for isospectral manifolds. The problems to be addressed in this project are extensions of Mark Kac's famous question, "Can you hear the shape of a drum?" Mathematicians would ask if the spectrum of sound frequencies radiating from a drum can be used to distinguish between different shaped drums. In two dimensions these frequencies can be used to make such distinctions, but in higher dimensions they can not. The principal investigator will study whether or not other mathematical distinctions between objects other than their exact shapes, can be determined by an analysis of their sound spectrums.