Towards the “shape” of cosmological observables and the string theory landscape with topological data analysis

Persistent homology is a technique from Topological Data Analysis that computes the multiscale “shape” of a data set. We review the basic formalism of persistent homology as well as several applications to cosmology and string theory. We describe how persistent homology provides efficient and robust morphological summaries of cosmological observables including the Cosmic Microwave Background and Large-Scale Structure, and how statistical pipelines involving persistent homology can potentially constrain cosmological parameters. We also review the string theory landscape as an interesting data set displaying complex features in high-dimensional spaces, and thus an ideal setting for persistent homology. We describe the characterization of distributions of flux vacua using persistent homology.