Persistent Homology - a Survey

Persistent Homology - a Survey provides a mathematical foundation for analyzing shape using topology and geometry. This technical text outlines computational methods for studying the persistence of topological features across multiple scales. The book presents key concepts in algebraic topology and explains how they apply to data analysis and shape description. The focus remains on homology groups and their computation through filtrations and persistence diagrams. The work includes detailed proofs and algorithms, moving from basic definitions through advanced applications. Examples demonstrate the practical usage of persistent homology in fields like protein structure analysis and image processing. This synthesis of computational topology bridges pure mathematics and real-world applications while establishing persistence as a tool for quantifying and comparing shapes. The text serves as both a theoretical framework and a practical guide for researchers implementing these methods.

There are not enough internet reviews to create a summary of this book. Instead, here is a summary of reviews of Herbert Edelsbrunner's overall work: Readers find Edelsbrunner's textbooks and academic works technically rigorous but challenging to approach. His book "Computational Geometry: Algorithms and Applications" receives attention from computer science students and researchers. Liked: - Clear mathematical explanations and proofs - Comprehensive coverage of geometric algorithms - High-quality diagrams and illustrations - Real-world applications included alongside theory Disliked: - Dense writing style requires significant mathematical background - Limited introductory material for beginners - Some readers note outdated programming examples - High price point for textbooks Ratings: - Computational Geometry (3rd Ed): 4.1/5 on Goodreads (52 ratings) - Amazon shows limited reviews due to specialized academic audience One graduate student reviewer noted: "The content is excellent but requires serious mathematical maturity. Not for casual reading." Another mentioned: "The concepts are presented rigorously, but newcomers to computational geometry may struggle without additional resources."

Applied Topology by Robert Ghrist This text bridges persistent homology with concrete applications in signal processing and data analysis.

Computational Topology: An Introduction by Herbert Edelsbrunner This book provides the mathematical foundations for topological data analysis with detailed algorithms and implementation discussions.

Elementary Applied Topology by Robert Ghrist The text connects topology concepts to real-world applications through visualization and computation methods.

Topology for Computing by Afra J. Zomorodian This work demonstrates the practical implementation of topological algorithms in computer science applications.

Algebraic Topology by Allen Hatcher This text establishes the mathematical groundwork that underpins the theory of persistent homology.

🔹 Persistent homology, introduced in the early 2000s, helps mathematicians study the "shape" of data by tracking how topological features appear and disappear across different scales, similar to looking at a landscape from different heights. 🔹 Herbert Edelsbrunner won the 2018 Wittgenstein Award, Austria's highest academic honor, for his groundbreaking work in computational geometry and topology. 🔹 The mathematical concepts in persistent homology have practical applications in areas like protein folding, signal processing, and even analyzing social networks. 🔹 Edelsbrunner helped found Geomagic, a company that developed 3D modeling software used in industries from manufacturing to medicine, demonstrating the real-world impact of computational topology. 🔹 The "barcode" visualizations used in persistent homology allow complex topological data to be represented in an intuitive way, making abstract mathematical concepts accessible to researchers across different fields.