Una introducción a los módulos de persistencia

Abstract

Throughout this paper we introduce the concept and properties of persistence modules following the ideas presented by Steve and Oudot in [8]. following the ideas presented by Steve and Oudot in [8]. These are the mathematical object that constitute the core of the current field of topological data analysis. the core of the current field of topological data analysis, which in recent years has become a topic of great interest. has become a topic of great interest in recent years. This field is a mixture of computer science, algebraic topology and statistics. and statistics, it is based on the assumption that scientific datasets carry information in their internal structure and that this structure is sometimes internal structure and that sometimes this internal structure is topological. Persistence modules are designed to carry topological designed to carry topological information about a dataset at many different scales simultaneously. on many different scales simultaneously. This information can be extracted in the form of an invariant (the persistence diagram or barcode [1]). persistence diagram or barcode [3]) that can be efficiently computed. The paper presents the mathematical properties of persistence modules, and proves in detail the persistence theorem [4]. the Krull-Remak-Schmidt-Azumaya theorem is demonstrated in detail. This theorem guarantees the unique de- This theorem guarantees the unique composition of a persistence module as a direct sum of intervals except isomorphism. Finally it will be shown how a quiver is associated to a persistence module.