Biparameter persistence for filtering functions and the matching distance

Multiparameter persistence is an area of topological data analysis that synthesises the geometric information of a topological space via filtered homology. Given a topological space and a filtering function on it, one can in fact consider a filtration given by the sublevel sets of the space induced by the function, and then take the homology of such filtration. In the case when the filtering function assumes values in the real plane, the homological features of the filtered object can be recovered through a "curved" grid on the plane called the extended Pareto grid of the function. In this talk, we exploit such a grid to understand the geometry of a metric between filtering functions and the homological invariants associated with them, called the matching distance. This talk is based on joint work with Marc Ethier, Patrizio Frosini and Nicola Quercioli.