Three Inverse Problems for Persistent Homology

How much information is lost when we apply the persistence map PH? We investigate this question by studying the level sets of PH in three different settings. In the first, we observe that path components of space of Morse functions on a given manifold with the same PH (from the sublevel set filtration) are orbits of Morse functions under the identity component of the diffeomorphism group. When barcode endpoints are distinct and the manifold is an orientable surface, this observation allows us to compute the homotopy type of path components of level sets of PH, by leveraging the work of Maksymenko. The second setting we study the persistence map in is that of continuous functions on the geometric realization of a tree. By establishing a homotopy equivalence between level sets of PH and a certain configuration space, we are able to compute the number of path components in level sets of PH in this setting, for barcodes with distinct endpoints. Lastly, we study the subspace of point clouds with the same barcodes (from the Čech or Vietoris-Rips filtration). Here we establish upper and lower bounds on the dimension of this space, and, in the Vietoris-Rips case, show that computing the dimension of this space is challenging by connecting the problem to rigidity theory. In particular, we show that a point cloud being locally identifiable under Vietoris-Rips persistence is equivalent to a certain graph being rigid on the same point cloud.