Inverse Problems in Topological Persistence

TL;DR

This survey reviews inverse problems in topological persistence, focusing on surjectivity and injectivity, highlighting key tools, open problems, and applications in the field.

Contribution

It provides a comprehensive overview of the current state of inverse problems in topological persistence, emphasizing theoretical tools and open challenges.

Findings

Analysis of surjectivity in persistence

Investigation of injectivity conditions

Identification of open problems in the field

Abstract

In this survey, we review the literature on inverse problems in topological persistence theory. The first half of the survey is concerned with the question of surjectivity, i.e. the existence of right inverses, and the second half focuses on injectivity, i.e. left inverses. Throughout, we highlight the tools and theorems that underlie these advances, and direct the reader's attention to open problems, both theoretical and applied.