The last decade saw an enormous boost in the field of computational topology: methods and
concepts from algebraic and differential topology, formerly confined to the realm of pure …

In topological data analysis (TDA), one often studies the shape of data by constructing a
filtered topological space, whose structure is then examined using persistent homology …

" 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 …

We define a new topological summary for data that we call the persistence landscape. Since
this summary lies in a vector space, it is easy to combine with tools from statistics and …

Although deep learning approaches have had tremendous success in image, video and
audio processing, computer vision, and speech recognition, their applications to three …

Combining concepts from topology and algorithms, this book delivers what its title promises:
an introduction to the field of computational topology. Starting with motivating problems in …

Persistence diagrams, the most common descriptors of Topological Data Analysis, encode
topological properties of data and have already proved pivotal in many different applications …

Our intention, at the beginning, was to write a short paper resolving some technical issues in
the theory of topological persistence. Specifically, we wished to present a clean easy-to-use …

This work introduces a number of algebraic topology approaches, including multi-
component persistent homology, multi-level persistent homology, and electrostatic …

Persistent homology is an algebraic tool for measuring topological features of shapes and
functions. It casts the multi-scale organization we frequently observe in nature into a …