The persistent homology of a sampled map: on failed reconstructions

Abstract: This talk introduces a filtration analysis of sampled maps based on persistent homology. The aim is to reconstruct the underlying maps. The key idea is to extend the definition of homology-induced maps of correspondences using quiver representations. Our definition of homology-induced maps is given by the most persistent direct summands of representations. The direct summands uniquely determine a persistent homology for a sampled map. Compared to existing methods using eigenspace functors, our filtration analysis represents an important advantage that no prior information related to the eigenvalues of the underlying map is required. However, our reconstruction method does not always work. In this talk, we focus on examples of failed reconstructions.

This talk is based on the paper DOI:10.1007/s41468-021-00065-3 (in press). The preprint is available at arXiv:1810.11774.

computational geometryalgebraic topologycombinatoricsgeometric topologyprobability

Audience: advanced learners

Asia Pacific Seminar on Applied Topology and Geometry