" 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 …

In 2009, Chazal et al. introduced ϵ ϵ-interleavings of persistence modules. ϵ ϵ-
interleavings induce a pseudometric d_ I d I on (isomorphism classes of) persistence …

In topological data analysis and visualization, topological descriptors such as persistence
diagrams, merge trees, contour trees, Reeb graphs, and Morse–Smale complexes play an …

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 …

We generalize the persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer to the
setting of constructible persistence modules valued in a symmetric monoidal category. We …

When a category CC satisfies certain conditions, we define the notion of rank invariant for
arbitrary poset-indexed functors F: P → CF: P→ C from a category theory perspective. This …

We propose a metric for Reeb graphs, called the functional distortion distance. Under this
distance, the Reeb graph is stable against small changes of input functions. At the same …

This paper presents a unified computational framework for the estimation of distances,
geodesics and barycenters of merge trees. We extend recent work on the edit distance 104 …

This article presents a computational framework for the Principal Geodesic Analysis of
merge trees (MT-PGA), a novel adaptation of the celebrated Principal Component Analysis …