Mathematical Sciences: "Spectrum and geodesic flow"

本文研究紧致黎曼流形上的拉普拉斯谱理论与测地线流的动力学之间的关系。第一个项目涉及等谱问题的一个限制：假设拉普拉斯由一个酉傅里叶积分算子缠绕在一起。广义地解释，这一限制包括几乎所有已知的例子。第二个问题涉及拉普拉斯对关联函数。该函数在平均连续间隙的尺度上测量特征值之间的间隙分布。该函数由测地线流的定性动力学决定。最近，首席研究者确定了某些Zoll曲面上的第一个已知对函数。这项工作将继续下去。这个项目是在几何分析领域，处理谱理论和动力系统之间的接口。这项研究可能会对物理学和几何学有深远的应用。

This research concerns relations between the spectral theory of the Laplacian on a compact Riemannian manifold and the dynamics of the geodesic flow. The first project concerns a restriction of the isospectral problem: an assumption that the Laplacians are intertwined by a unitary Fourier integral operator. Broadly interpreted, this restriction includes almost all known examples. The second problem concerns the pair correlation function of the Laplacian. This function measures the distribution of gaps between eigenvalues on the scale of the mean consecutive gap. This function is determined by the qualitative dynamics of the geodesic flow. Recently the principal investigator determined the first known pair function on certain Zoll surfaces. This work will be continued. This project is in the area of geometric analysis and deals with the interface between spectral theory and dynamical systems. This research may have profound applications to physics and geometry.