" In this chapter, we introduce some of the very basics that are used throughout the book.
First, we give the definition of a topological space and related notions of open and closed …

Zigzag persistence is a powerful extension of the standard persistence which allows
deletions of simplices besides insertions. However, computing zigzag persistence usually …

Comments• page viii, bottom of page: the following names should be added to the
acknowledgements:-Peter Landweber had an invaluable contribution to these notes. First …

In this article, we extend the notions of dominated vertex and strong collapse of a simplicial
complex as introduced by J. Barmak and E. Miniam. We say that a simplex (of any …

We present a geometric perspective on sparse filtrations used in topological data analysis.
This new perspective leads to much simpler proofs, while also being more general, applying …

We introduce a fast and memory efficient approach to compute the persistent homology (PH)
of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the …

A tower is a sequence of simplicial complexes connected by simplicial maps. We show how
to compute a filtration, a sequence of nested simplicial complexes, with the same persistent …

For a $ P $-indexed persistence module ${\sf M} $, the (generalized) rank of ${\sf M} $ is
defined as the rank of the limit-to-colimit map for the diagram of vector spaces of ${\sf M} …

Topological Data Analysis is a growing area of data science, which aims at computing and
characterizing the geometry and topology of data sets, in order to produce useful descriptors …

Zigzag filtrations of simplicial complexes generalize the usual filtrations by allowing simplex
deletions in addition to simplex insertions. The barcodes computed from zigzag filtrations …