Computational Homology /by Tomasz Kaczynski, Konstantin Mischaikow and Marian Mrozek . This is an interesting and unusual book written with the intention of serving several purposes. One of them is to demonstrate that methods of algebraic topology, in particular homology theory, that have proved remarkably successful in several areas of pure mathematics can provide powerful, and in some cases indispensable, tools in a number of areas of applied mathematics and science. The second is to provide the necessary theory and "technology" for such applications. This means on the one hand providing all the necessary mathematical foundations of the subject, including definitions and theorems, and on the other hand efficient computational techniques capable of dealing with real life situations. Thus, the book stresses algorithmic and computational approaches; and in fact includes computer code written in a programming language specially designed for this purpose. It is addressed to a varied audience of computer scientists, experimental scientists and engineers while at the same time trying to retain the interest of mathematicians. With this in mind the authors have attempted to produce a modular book, which allows a number of different reading approaches. The basic subdivision of the book is into three parts. The last part contains all the basic pre-requisites from algebra and topology: the most essential facts about Euclidean spaces, point set topology, abelian groups, vector spaces and matrix algebras. This part also contains a description of the programming language used to describe the algorithms found in the book
