Multidimensional Persistence: Invariants and Parameterization

TL;DR

This paper provides a complete classification and parameterization of algebraic objects in multidimensional persistence, revealing the limitations of discrete invariants beyond one dimension.

Contribution

It extends the theory of multidimensional persistence by classifying algebraic invariants and demonstrating the impossibility of discrete invariants in higher dimensions.

Findings

Complete classification of algebraic objects for multifiltered complexes

Discrete invariants exist only in one-dimensional persistence

Higher dimensions lack discrete and complete invariants

Abstract

This article grew out of the theoretical part of my Master's thesis at the Faculty of Mathematics and Information Science at Ruprecht-Karls-Universit\"at Heidelberg under the supervision of PD Dr. Andreas Ott. Following the work of G. Carlsson and A. Zomorodian on the theory of multidimensional persistence in 2007 and 2009, the main goal of this article is to give a complete classification and parameterization for the algebraic objects corresponding to the homology of a multifiltered simplicial complex. As in the work of G. Carlsson and A. Zomorodian, this classification and parameterization result is then used to show that it is only possible to obtain a discrete and complete invariant for these algebraic objects in the case of one-dimensional persistence, and that it is impossible to obtain the same in dimensions greater than one.