A Primer on Persistent Homology of Finite Metric Spaces

Topological data analysis (TDA) is a relatively new area of research related to importing classical ideas from topology into the realm of data analysis. Under the umbrella term TDA, there falls, in particular, the notion of persistent homology PH, which can be described in a nutshell, as the study of scale-dependent homological invariants of datasets. In these notes, we provide a terse self-contained description of the main ideas behind the construction of persistent homology as an invariant feature of datasets, and its stability to perturbations.