The use of persistent homology in data science for topological data analysis

Topological data analysis (TDA) is a young field that has been rapidly growing over the last years and which blends algebraic topology, statistics and computer science. It arises motivated by the fact that the topology of a space gives useful information about it. In particular, when working with scientific data, information about their internal structure may provide important properties about the phenomena that it represents or its behaviour. The goal of topological data analysis is to provide well-founded mathematical, statistical and algorithmic methods to infer the topological structure underlying the data. The aims of this thesis are two. On the one hand, to introduce this field and its mathematical tools, focusing on persistent homology and its representation by persistence diagrams. On the other hand, the study of the latter using techniques from optimal transport, proposing a vectorization based on transport maps.