Riemannian Geometry and Spectral Analysis

DMS-0072154John W.洛特黎曼几何是研究弯曲空间的。这种空间的例子是三维平面空间中的曲线和曲面。黎曼表明如何使精确的概念曲率的空间任意尺寸。 频谱分析可以粗略地描述为研究空间如何振动。 更确切地说，一个弯曲的空间与一个特定的偏微分算子，拉普拉斯算子相关联。 谱分析是研究拉普拉斯算子的特征值如何依赖于空间的基础几何的研究。在以前的工作中，主要研究者获得了微分形式拉普拉斯算子的谱和基础空间的几何之间的关系，后者受到其直径上界和曲率上界和下界的约束。特别是，他的特点是，当有统一的上界的第j个特征值的p-形式拉普拉斯算子，当有小的积极特征值的p-形式拉普拉斯算子。 他建议在几个方向上扩展这项工作。 一个方向是假设空间的曲率有一个下界。 在这种情况下，新的问题出现了，因为在假设的几何约束下，空间可以“折叠”到一个低维的高度奇异空间。 主要研究者以前的工作，在上下曲率界的情况下，也处理了在崩溃极限中出现的奇异空间。 然而，只有一个较低的曲率界，出现的奇异空间是一个不同的性质。他还建议延长以前的工作方向的频谱分析的狄拉克运营商，根据几何假设的上限直径的空间和上限和下限的曲率。

DMS-0072154John W. LottRiemannian geometry is the study of curved spaces. Examples of such spaces are curves and surfaces in three-dimensional flat space. Riemann showed how tomake precise the notion of curvature for a space of arbitrary dimension. Spectral analysis can be roughly characterized as the study of how a space vibrates. More precisely, to a curved space is associated a certain partial differential operator, the Laplacian. Spectral analysis is the study of how the eigenvalues of the Laplacian depend on the underlying geometry of the space.In previous work, the principal investigator obtained relationships between the spectrum of the differential form Laplacian and the geometry of the underlying space, the latter being constrained by upper bounds on its diameterand upper and lower bounds on its curvature. In particular, he characterized when there are uniform upper bounds on the j-th eigenvalue of the p-form Laplacian, and when there are small positive eigenvalues of the p-form Laplacian. He proposes to extend this work in several directions. One direction is to just assume that there is a lower bound on the curvature of the space. New issues arise in this case, as under the assumed geometric constraints, the space can ``collapse'' to a highly singular space of lower dimension. The principal investigator's previous work, in the case of upper and lower curvature bounds, also dealt with the singular spaces that arise in a collapsing limit. However, with just a lower curvature bound, the singularspaces that arise are of a different nature. He also proposes to extend the previous work in the direction of analyzing the spectrum of the Dirac operator, under the geometric assumptions of an upper bound on the diameter of the spaceand upper and lower bounds on its curvature.