Topological data analysis (TDA) has emerged recently as a viable tool for analyzing complex data, and the area has grown substantially both in its methodologies and applicability. Providing a computational and algorithmic foundation for techniques in TDA, this comprehensive, self-contained text introduces students and researchers in mathematics and computer science to the current state of the field. The book features a description of mathematical objects and constructs behind recent advances, the algorithms involved, computational considerations, as well as examples of topological structures or ideas that can be used in applications. It provides a thorough treatment of persistent homology together with various extensions – like zigzag persistence and multiparameter persistence – and their applications to different types of data, like point clouds, triangulations, or graph data. Other important topics covered include discrete Morse theory, the Mapper structure, optimal generating cycles, as well as recent advances in embedding TDA within machine learning frameworks.

'A must-have up-to-date computational account of a vibrant area connecting pure mathematics with applications.'

Herbert Edelsbrunner - IST Austria

'This book provides a comprehensive treatment of the algorithmic aspects of topological persistence theory, both in the classical one-parameter setting and in the emerging multi-parameter setting. It is an excellent resource for practitioners within or outside the field, who want to learn about the current state-of-the-art algorithms in topological data analysis.'

Steve Oudot - Inria and Ecole polytechnique

‘There are many things to appreciate about this book, including the abundance of excellent and helpful figures, the extensive reference list, and the variety of instructive exercises for students to work through … Thanks to its inclusion of so much cutting-edge recent work and state-of-the-art algorithms, this is an ideal book for mathematicians or computer scientists looking to dive into this exciting and still very young area of research.’

Ellen Gasparovic Source: Mathematical Association of America Reviews

‘This is a much needed update and contribution to the vast and rapidly growing area of computational topology. It will be the new go-to text for years to come.’

Nicholas A. Scoville

‘As a complete package, the book is an ideal text for a first course on topological data analysis for beginning graduate students … Highly recommended.’

M. Clay Source: Choice