Learning topology: bridging computational topology and machine learning

Topology is a classical branch of mathematics, born essentially from Euler's studies in the XVII century, which deals with the abstract notion of shape and geometry. Last decades were characterized by a renewed interest in topology and topology-based tools, due to the birth of computational topology and topological data analysis (TDA). A large and novel family of methods and algorithms computing topological features and descriptors (e.g., persistent homology) have proved to be effective tools for the analysis of graphs, 3D objects, 2D images, and even heterogeneous datasets. This survey is intended to be a concise but complete compendium that, offering the essential basic references, allows you to orient yourself among the recent advances in TDA and its applications, with an eye to those related to machine learning and deep learning.