The fiber of multiparameter persistent homology for simplicial complexes

Persistent homology is a powerful descriptor of filtered spaces, but it is generally not injective: many filtrations can have the same persistent homology. In this talk, I will introduce multiparameter persistent homology (MPH) and the associated inverse problem for sublevel-set filtrations on a fixed finite simplicial complex. I will then describe recent work in which we study this map by decomposing both its domain, the space of filters, and its codomain, a moduli space of essentially finite persistence modules, into simpler pieces, allowing us to view MPH as a stratified map. Using this structure, we show that the fibers of the MPH map are polyhedral complexes and bound their dimension in terms of multigraded Betti numbers, recovering the known one-parameter bound as a special case.