Multidimensional Interleavings and Applications to Topological Inference

This thesis concerns the theoretical foundations of persistence-based topological data analysis. The primary focus of the work is on the development of theory of topological inference in the multidimensional persistence setting, where the set of available theoretical and algorithmic tools has remained comparatively underdeveloped, relative to the 1-D persistence setting. The thesis establishes a number of theoretical results centered around this theme, some of which are new and interesting even for 1-D persistent homology. In addition, this work presents theory of topological inference formulated directly on the (topological) level of filtrations rather than on the (algebraic) level of persistent homology modules. The main mathematical objects of study in this work are interleavings. These are tools for quantifying the similarity between multidimensional filtrations and persistence modules. They were introduced for 1-D filtrations and persistence modules by Chazal et al. [8], where they were used to prove a strong and very useful generalization of the stability of persistence result of [14]; we generalize the definition of interleavings appearing in [8] in several directions and use these generalizations to define pseudometrics on multidimensional filtrations and multidimensional persistence modules called interleaving distances. The first part of this thesis, adapted from the preprint [32], studies in detail the theory of interleavings and interleaving distances on multidimensional persistence modules. We present six main results about the interleaving distance. First, we show that in the case of 1-D persistence, the interleaving distance is equal to the bottleneck distance on tame persistence modules. Second, we prove a theorem which implies that the restriction of the interleaving