Geometry, Dynamics, and PDEs in Riemannian and Finsler Spaces

摘要奖：DMS 1510611，首席研究员：Dmitri Burago这项研究项目起源于曲线长度的概念，并在有限的假设下为物理或动力学系统发展出熟悉而重要的概念。在这些想法中，有稳定性、不稳定性和熵的概念，这些概念解决了当度量受到小扰动时较大几何属性的变化。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件。我们知道黎曼流形上的Laplace-Beltrami算子的特征值和特征函数是由epsilon-net上邻近图的(适当加权的)图Laplace算子的特征值和特征向量来逼近的。其中一个项目试图将这一结果推广到微分形式上的拉普拉斯，并得到度量度量空间上的拉普拉斯的更好的估计，包括稳定性结果。了解哪些具有有界几何的空间可以通过具有一致的顶点度和边长的界的图来逼近为可加误差，这将是非常有趣的。到目前为止，我们只能对2-平面这样做，甚至不能对3-空间这样做，但我们还不知道这一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，在该系统中，熵是在局部产生的，也就是说，在围绕一个轨迹的任意小管中产生正熵。另一个目的是证明Busemann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.