Recent years have seen exciting new applications of topology in the handling of big data. TDA — Topological Data Analysis — has caused quite a buzz, with an impressive flow of new applications and new techniques. TDA rests upon the wealth of knowledge comprising topology, primarily tools from algebraic topology. It seeks to identify components of data that are more amenable to computational investigation in order to analyse and visualise data. The book under review is, as the author indicates, strongly based on his habilitation thesis reviewing recent research in algorithmic topological techniques in the visualisation of data.

I will start my review by commenting on the blurb, noting where I find it to be accurate and where less so. The first paragraph of the blurb mentions that

…combining theoretical and practical aspects of topology, this book delivers a comprehensive and self-contained introduction to topological methods for the analysis and visualisation of scientific data.

I find the book to be much more of a review of algorithms with very little theory. Moreover, I find the “self-contained” tag to be a bit misleading.

The second paragraph reads:

Theoretical concepts are presented in a thorough but intuitive manner, with many high-quality color illustrations. Key algorithms for the computation and simplification of topological data representations are described in details, and their application is carefully illustrated in a chapter dedicated to concrete use cases.

I find that to be quite precise, except for the claim about the thorough presentation of theoretical concepts.

Finally, the blurb concludes like this:

With its fine balance between theory and practice, *Topological Data Analysis for Scientific Visualisation* constitutes an appealing introduction to the increasingly important topic of topological data analysis, for lecturers, students and researchers.

Perhaps it is all in the eye of the beholder, but to me the appeal of the book lies in its review of the algorithms and not so much as an introduction to theoretical aspects.

What the book does quite well is provide a very accessible and interesting review of recent algorithms. Largely, each of the chapters 3–5 presents algorithms that accomplish a data visualisation task, accompanied with some good discussions and many illustrative pictures. Chapter 6 is a discussion of the emerging computational challenges related to this current body of research, and chapter 1 is a very hurried introduction, consisting of just two pages. So, the bulk of the book, chapters 3–6, forms a useful introduction to new and emerging computational aspects motivated by topological ideas in data visualisation.

The missed opportunity of creating an engaging and enticing introduction to the subject in chapter 1 is unfortunately dwarfed by the misfortunes plaguing chapter 2, the one intending to provide topological background. First, the presentation runs along numerous definitions quite rapidly, bunching together trivial definitions (such as that of a function) together with significantly intricate ones (such as Betti numbers) in one lump of text.

The real trouble, however, are the inaccuracies and mistakes in the definitions. For instance, a path, in Definition 2.27, is defined to be a homeomorphism \(p\) from an open interval \((a,b)\) to the space. This is claimed to be a path from \(p(a)\) to \(p(b)\), but \(a\) and \(b\) are explicitly removed from the domain of the path. Definition 2.31 attempts to describe simply connected spaces but neglects to require that the homotopy preserves the end points. Definition 2.32 attempts to define the boundary of a topological space rather than the boundary of a subspace of a given space, resulting in an empty definition.

The reader already familiar with these concepts will no doubt understand the erroneous thinking that led to these unfortunate formulations, and will know what is actually meant. But the readers encountering these notions for the first time will quickly drown in confusion when confronted with the ensuing bombardment of definitions leading to that of homology. The rest of the chapter does not fare much better.

To enjoy the bulk of the book the reader should come prepared with some background knowledge of algebraic topology, and maybe glance at the notions introduced in chapter 2 together with a reliable accompanying text for proper definitions.

Often, when I finish reading a book, I ponder whether the price of the book is realistic. In this case though, with thanks to whoever Springer-Verlag appointed as copy editor for this book, you don’t need to worry about the price. Simply download the author’s habilitation thesis. It seems the thesis and the published book are very (very!) similar, pretty pictures and careless mistakes alike. Perhaps Springer-Verlag’s idea of value for money is in the two page introduction you get when you pay for the book version.

Ittay Weiss is a Teaching Fellow at the University of Portsmouth, UK.