Multiparameter persistence modules in the large scale

TL;DR

This paper studies the large scale behavior of multiparameter persistence modules, classifies indecomposables in low dimensions, and shows the complexity of full classification in higher dimensions.

Contribution

It provides a classification of indecomposable diagrams for 2D modules and demonstrates the wild complexity of classifying higher-dimensional modules.

Findings

Classifies indecomposables in 2D persistence modules.

Shows higher-dimensional classification problem is wild.

Introduces a notion of equivalence based on large scale behavior.

Abstract

A persistence module with $m$ discrete parameters is a diagram of vector spaces indexed by the poset $\mathbb{N}^m$. If we are only interested in the large scale behavior of such a diagram, then we can consider two diagrams equivalent if they agree outside of a ``negligeable'' region. In the $2$-dimensional case, we classify the indecomposable diagrams up to finitely supported diagrams. In higher dimension, we partially classify the indecomposable diagrams up to suitably finite diagrams, and show that the full classification problem is wild.