Introduction to Persistent Intervals

Abstract

We give an accessible introductory exposition on the concept of persistence intervals from topological data analysis. By carefully crafting numerous examples as well as adopting pedagogical tricks that minimize the direct use of topological tools, we are able to give an efficient, highly readable presentation of the standard theory of persistence intervals that only requires linear algebra as a prerequisite. This is principally an expository thesis, but we also make some minor extensions to the existing literature by giving some basic insights on the nature of local and global barcodes (going a bit beyond what is stated in the conclusion of [LM15]), as well as by showing that the clearing and compression optimizations in [BKR14] can be used to simplify certain aspects of the parallelization algorithm in [LM15] in the case of a simplicial cover of size two.