I will present some of my contributions to Topological Data Analysis through the problem of building rigorous and usable TDA pipelines. Starting from a function, mesh, or point cloud, one builds a filtration and extracts a topological summary, such as a persistence diagram, merge tree, or Reeb graph. I will discuss how these summaries encode coordinate-free and invariant information, and why turning this idea into a robust mathematical pipeline requires carefully designed metrics, stability results, and estimation procedures, using tools from topology, geometry, statistics, and optimal transport.

This perspective also exposes a second difficulty: even once stable summaries have been constructed, many of them naturally live in non-linear metric spaces, making standard statistical and machine-learning methods difficult to apply directly. I will therefore conclude with recent work on Persistence Spheres, an explicit Hilbert-space embedding of persistence diagrams with provable bi-continuity, aimed at making topological summaries compatible with modern machine-learning pipelines.