Computational topology lies at the intersections of geometry, topology, computer science, and data analysis. Since its early years, ideas from pure mathematics have inspired the definition of computable invariants and the development of techniques for extracting topological information from data. Yet, the interaction between pure and applied topology is much deeper. On the one hand, topological and geometric theories offer tools and frameworks for understanding computable invariants. On the other hand, computational topology and data science raise new questions in pure topology and geometry, and even motivate the introduction of entirely new concepts.

In this talk, I will illustrate these mutual influences through examples related to shape comparison and the Gromov–Hausdorff distance, a dissimilarity measure between metric spaces. More specifically, I will focus on three directions:

1. Using dimension theory to describe situations in which stable invariants inevitably produce false positives;
2. Identifying conditions under which the Hausdorff distance between a metric graph and a subset thereof coincides with their Gromov–Hausdorff distance (joint work with Henry Adams, Sushovan Majhi, Fedor Manin, and Žiga Virk);
3. Introducing a generalisation of the Gromov–Hausdorff distance that enables the comparison of chromatic point clouds and labelled shapes, which is validated by proving the stability of the six-pack (joint work with Ondřej Draganov and Sophie Rosenmeier).

After motivating each problem, I will present the main results and conclude with open questions.