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Computational Homologyby Tomasz Kaczynski
Synopses & Reviews
In recent years, there has been a growing interest in applying homology to problems involving geometric data sets, whether obtained from physical measurements or generated through numerical simulations. This book presents a novel approach to homology that emphasizes the development of efficient algorithms for computation. As well as providing a highly accessible introduction to the mathematical theory, the authors describe a variety of potential applications of homology in fields such as digital image processing and nonlinear dynamics. The material is aimed at a broad audience of engineers, computer scientists, nonlinear scientists, and applied mathematicians. Mathematical prerequisites have been kept to a minimum and there are numerous examples and exercises throughout the text. The book is complemented by a website containing software programs and projects that help to further illustrate the material described within.
Homology is a powerful tool used by mathematicians to study the properties of spaces and maps that are insensitive to small perturbations. This book uses a computer to develop a combinatorial computational approach to the subject. The core of the book deals with homology theory and its computation. Following this is a section containing extensions to further developments in algebraic topology, applications to computational dynamics, and applications to image processing. Included are exercises and software that can be used to compute homology groups and maps. The book will appeal to researchers and graduate students in mathematics, computer science, engineering, and nonlinear dynamics.
Table of Contents
Preview.- Cubical Homology.- Computing Homology Groups.- Chain Maps and Reduction Algorithms.- Preview of Maps.- Homology of Maps.- Computing Homology of Maps.- Prospects in Digital Image Processing.- Homological Algebra.- Nonlinear Dynamics.- Homology of Topological Polyhedra.- Topology.- Algebra.- Syntax of Algorithms.- References.- Symbol Index.- Subject Index.
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Science and Mathematics » Mathematics » Applied