Geometric Interpretation of Persistence

Explore the geometric interpretation of persistence in this comprehensive lecture from the Applied Algebraic Topology Network. Delve into how persistent homology provides a multi-scale representation of metric spaces, examining the reconstruction of homotopy types at small scales and the information about hole sizes at increasing scales. Investigate the homotopy equivalence between Vietoris-Rips complexes and the nerve of appropriate space covers, leading to reconstruction results at small scales. Analyze the classification of one-dimensional persistent homology in geodesic spaces and its approximation through finite samples. Discover how geometric features generate higher-dimensional homological features, including the detection of contractible geodesics in geodesic spaces using persistent homology. Cover topics such as compact Romanian manifolds, critical triangles, finite subspaces, global vs. local contraction, deformation contraction, and homology restrictions in surfaces.

Introduction
Overview
Setting
Example
Reconstructing spaces
Results
Chapter description
Compact Romanian manifold
Critical triangles
Finite samples
Finite subspace
Technical details
Ideal case
Classical stability
Higher dimensional persistence
Global vs local contraction
Deformation contraction
Surfaces
homology restrictions
neighborhood thing