Geometry, Dynamics, and PDEs in Riemannian and Finsler Spaces

AbstractAward：DMS 1510611，首席研究员：Dmitri Burago这个研究项目起源于曲线长度的概念，并在有限的假设下发展了物理或动力系统的熟悉和重要的思想。 在这些思想中，有稳定性、不稳定性和熵的概念，它们解决了度量受到小扰动时较大几何性质的变化。 该项目的一个组成部分涉及离散形式的几何数据和度量测度空间上的拉普拉斯算子。主要研究者的两名研究生将从事不寻常形状的振动体的频率的实验测量。众所周知，黎曼流形上的拉普拉斯-贝尔特拉米算子的本征值和本征函数可以用ε-网上的邻近图的（适当加权的）图拉普拉斯算子的本征值和本征向量来近似。 其中一个项目试图将该结果扩展到微分形式上的拉普拉斯算子，并获得度量测度空间上的拉普拉斯算子的更好估计，包括稳定性结果。 这将是非常有趣的了解，空间与有界几何可以近似为一个加性误差的图与一致的界限度的顶点和边长。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，但我们不知道在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，其中熵是局部产生的，即正熵是在围绕一个轨迹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.