Networks, as efficient representations of complex systems, have appealed to scientists for a
long time and now permeate many areas of science, including neuroimaging (Bullmore and …

A weighted $ d-$ complex is a simplicial complex of dimension $ d $ in which each face is
assigned a real-valued weight. We derive three key results here concerning persistence …

The subject of persistent homology has vitalized applications of algebraic topology to point
cloud data and to application fields far outside the realm of pure mathematics. The area has …

Recently, multi-scale notions of local homology (a variant of persistent homology) have been
used to study the local structure of spaces around a given point from a point cloud sample …

Persistent homology is a leading tool in topological data analysis (TDA). Many problems in
TDA can be solved via homological--and indeed, linear--algebra. However, matrices in this …

arXiv:2301.11756v2 [math.AC] 5 Feb 2023 Page 1 arXiv:2301.11756v2 [math.AC] 5 Feb 2023 A
comment on the structure of graded modules over graded principal ideal domains in the context …

In this section, we will give an informal introduction to persistent homology, which we will
make more rigorous in the following sections. In topological data analysis, we are interested …

We explore two distinct topics in the field of topological data analysis: invariants and metrics
for multiparameter persistence modules, and the homology of random geometric simplicial …

We re-examine the theory and orthodox methods that underlie the study of persistent
homology, particularly in its calculation of homological cycle representatives that are …

[en] Topological Data Analysis (TDA) is a recently developed tool designed to study the
geometry of finite data sets. In these notes, we describe the theory of persistent homology …