In this talk I will introduce several families of signatures for compact Riemannian manifolds or finite metric approximations of these. Our signatures are defined as persistence diagrams of carefully chosen functions over the space, and they provide a multi-scale description of the structure of the space. Some of them are global and can be used in shape classification applications, while others have a more local nature and can be used in (partial) shape matching applications.

All our signatures are stable under small perturbations of the space in the Gromov-Hausdorff distance. I will also illustrate their practicality through a series of experiments.