The focus of this dissertation is on the invertibility of certain topological summary statistics for metric objects. The first set of results concern persistence diagrams arising from distance functions on metric graphs, and is joint work with Steve Oudot. In this context, we prove both local and global injectivity results, culminating with a theorem that our statistic is generically injective in the appropriate topology. The second set of results, joint work with Steve Oudot and Clément Maria, uses persistence diagrams coming from eigenfunctions of the distance kernel operator on metric measure spaces. In this setting, we provide injectivity and inverse stability results, demonstrating that “nearby” topological statistics come from “nearby” metric spaces.
Solomon, Yitzchak Elchanan,
"Inverse Problems for Topological Transforms"
(2019).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/fjza-tr66
