On Multidimensional Persistence

Abstract

The main goal of this project is the study of Multidimensional Persistence. We begin by reviewing Persistent Homology, a well-studied tool in Topological Data Analysis that completely classifies filtered spaces through a discrete invariant, the barcode. We then discuss some recent advancements made in this field using Commutative Algebra, namely persistent ideals and a new (more powerful) barcode, introduced in [14]. Next, we carefully thread through Multidimensional Persistence theory, a generalization of Persistent Homology, that identifies multifiltrations with multigraded multivariate polynomial ring modules, stating a fundamental result (the Zomorodian-Carlsson correspondence, Theorem 3.5) and showing that, unlike the single-dimensional case, no complete discrete invariant exists for the multidimensional case. We end the discussion with a brief commentary on the classification of the objects studied. Finally, we give a result of our own, a generalization of the advancements made in [14] for the multidimensional case.