Geometry, Dynamics, and PDEs in Riemannian and Finsler Spaces

奖项：DMS 1510611，首席研究员：Dmitri burago这个研究项目从曲线长度的概念开始，并在有限假设下发展了物理或动力系统的熟悉和重要的思想。这些概念包括稳定性、不稳定性和熵的概念，它们处理较大几何性质的变化，因为度量受到小的扰动。该项目有一个与几何数据的离散形式和度量度量空间上的拉普拉斯算子有关的组成部分。首席研究员的两名研究生将对不寻常形状的振动体的频率进行实验测量。已知黎曼流形上的拉普拉斯-贝尔特拉米算子的特征值和特征函数是由epsilon-net上邻近图的拉普拉斯算子的特征值和特征向量来近似的。其中一个项目试图将该结果推广到微分形式上的拉普拉斯算子，并对度量度量空间上的拉普拉斯算子进行更好的估计，包括稳定性结果。理解哪些具有有界几何的空间可以通过顶点度和边长度具有统一边界的图近似为加性误差，这将是非常有趣的。到目前为止，我们只能在2平面上这样做，甚至不能在3空间上这样做，但是我们不知道在这个领域的主要猜想的一个反例。一个项目的目的是在合理的假设下证明米歇尔关于边界刚性的猜想。另一个项目将研究具有正熵的动力系统，其中熵是局部产生的，也就是说，正熵是在一个轨迹周围的任意小管中产生的。另一个目标是证明布斯曼表面积的椭圆性。

AbstractAward: DMS 1510611, Principal Investigator: Dmitri BuragoThis research project takes its origin at the concept of length of curves and goes on to develop familiar and important ideas for physical or dynamical systems under limited hypotheses. Among those ideas are notions of stability, instability, and entropy that address the variations of larger geometric properties as a metric is subjected to small perturbations. The project has a component related to discrete forms of geometric data and the Laplace operator on metric measure spaces. Two graduate students of the principal investigator will be working on experimental measurements of frequencies of vibrating bodies of unusual shapes.It is known that the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net. One of these projects seeks to extend that result to the Laplacian on differential forms and to get better estimates for the Laplacian on metric measure spaces, including stability results. It would be very interesting to understand which spaces with bounded geometry can be approximated to an additive error by graphs with uniform bounds on degrees of vertices and lengths of edges. So far, we can do that only for the 2-plane and not even for 3-space, but we do not know a single counterexample to the main conjecture in this area. One project aims to prove Michel's Conjecture on boundary rigidity under reasonable hypotheses. Another project will study dynamical systems with positive entropy in which entropy is generated locally, that is, positive entropy is generated in arbitrarily small tubes around one trajectory. Another goal is to prove ellipticity of the Busemann surface area.