" 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 …

This thesis develops the theory of sheaves and cosheaves with an eye towards applications
in science and engineering. To provide a theory that is computable, we focus on a …

We consider the question of defining interleaving metrics on generalized persistence
modules over arbitrary preordered sets. Our constructions are functorial, which implies a …

The Reeb graph is a construction which originated in Morse theory to study a real-valued
function defined on a topological space. More recently, it has been used in various …

Algorithms for persistent homology are well-studied where homomorphisms are induced by
inclusion maps. In this paper, we propose a practical algorithm for computing persistence …

We construct “barcodes” for the chain complexes over Novikov rings that arise in Novikov's
Morse theory for closed one-forms and in Floer theory on not-necessarily-monotone …

In topological data analysis, a point cloud data P extracted from a metric space is often
analyzed by computing the persistence diagram or barcodes of a sequence of Rips …

We study two methods for computing network features with topological underpinnings: the
Rips and Dowker persistent homology diagrams. Our formulations work for general …

Summarizing topological information from datasets and maps defined on them is a central
theme in topological data analysis. Mapper, a tool for such summarization, takes as input …

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 …