A Short Course in Computational Geometry and Topology

Herbert Edelsbrunner has distinguished himself as one of the pioneers in computational topology. He has already written, with J. L. Harer, a longer textbook titled Computational Topology, published by the AMS, and he continues to be widely published in the area.

Continuing his attempt to promote computational aspects of both geometry and topology, Edelsbrunner’s current book under review is written “with the purpose to bring the subject of Computational Geometry and Topology to a wider audience.” It is based on several years of courses he has co-taught at the Institute of Science and Technology Austria. The book makes a concerted effort to build a strong bridge between computational aspects of geometry and topology, often showing the line is quite blurred. At only slightly over 100 pages long, it is divided into four main parts consisting of I) tessellations II) complexes III) homology and IV) persistence, with each part concluding with roughly half a dozen exercises. Each part is divided into 3 to 4 short sections which can be taught during two 75 minute per week lectures.

Readers who enjoyed Edelsbrunner’s informal style in Computational Topology will continue to enjoy his relaxed prose in this book. While statements of theorems are clearly distinguished from the rest of the surrounding text, the proofs are sometimes informally discussed or simply taken for granted. There are numerous figures and illustrations throughout the text. Roughly every other page boasts plane and 3-dimensional pictures to help the reader visualize the geometrical and topological constructions.

The material culminates well in the last part by applying the previous material to persistence, piecewise linear manifolds, and matrix reduction algorithms. While the book is intended for an advanced undergraduate or graduate course in computational geometry and topology, it can be used for informal self-study by a motivated reader who wants a thorough overview of many of the basic constructions and techniques in the area. It is an excellent and welcome addition to the growing field of computational geometry and topology.

Nick Scoville (nscoville@ursinus.edu)is an assistant professor of mathematics at Ursinus College. His areas of interest are homotopy theory, discrete topology, and the history of topology. He considers himself an amateur scholastic. His website can be found at http://webpages.ursinus.edu/nscoville/.