A Short Course in Computational Geometry and Topology

by Herbert Edelsbrunner

This monograph presents a short course in computational geometry and topology. In the first part the book covers Voronoi diagrams and Delaunay triangulations, then it presents the theory of alpha complexes which play a crucial role in biology. The central part of the book is the homology theory and their computation, including the theory of persistence which is indispensable for applications, e.g. shape reconstruction. The target audience comprises researchers and practitioners in mathematics, biology, neuroscience and computer science, but the book may also be beneficial to graduate students of these fields.

Genres: Mathematics

119 pages, Paperback

First published June 4, 2014

Reviews

[Kevin K. Gillette · Rating 5 out of 5]

Recently I've begun to explore the nuts and bolts of a new type of data analysis called "topological data analysis" (TDA). The basis of TDA is that there natural shapes that high-dimensional data can be mapped to, which permit rather unusual and interesting connections - e.g., clusters - to emerge in a manner that is reminiscent of the notion of "dimension" (both the Hausdorff and the fractal sense).

To work with TDA computationally, a background in algebraic topology is necessary. More concretely, one must be acquainted with homology theory, as the idea of these emergent "shapes" rests on a framework known as persistent homology.

Prof. Edelsbrunner has written a marvelous monograph from course notes that he and a colleague assembled recently. The course is a graduate-level course at Duke University. Indeed, the book is organized by lecture, so if one is amenable to auto-didacticism, then one can learn this subject quite readily from the book. Of course, as I mention above, it definitely helps to have a solid background in algebraic topology, but even if one isn't already acquainted with the subject, an undergraduate-level background in modern algebra (groups and graphs) will suffice.

I give this book very high marks, as it was well-written and accompanied with numerous useful illustrations and examples. My only complaint, and it's a minor one, is that there were some terms introduced in the text before they were formally defined. The most salient example of this is the nerve of a collection of sets - this term arises several chapters before it is defined in its natural set-theoretic setting in section 10.4.

The audience for this monograph is data scientists eager to explore an interesting and visually appealing mechanism for finding relationships among data points (referred to as a point cloud in the literature).