Abstract: In recent years, topological and geometric data analysis (TGDA) has emerged as a new and promising field for processing, analyzing and understanding complex data. Indeed, geometry and topology form natural platforms for data analysis, with geometry describing the ”shape” and ”structure” behind data; and topology characterizing / summarizing both the domain where data are sampled from, as well as functions and maps associated to them. In this talk, I will show how the topological objects from discrete Morse theory and persistent homology can be used to reconstruct hidden geometric graphs; how such an approach can be extended to handle high-dimensional point clouds data (with sparsification); and how they can be then combined with machine learning pipelines for further data analysis tasks. This talk is based on multiple projects with multiple collaborators and references will be given during the talk.