Elementary Applied Topology

by Robert Ghrist

This text gives a brisk and engaging introduction to the mathematics behind the recently established field of Applied Topology. Over a century of development of principles and techniques in algebraic topology has of late crossed over to a variety of application domains. This text gives a completely novel introduction to these methods in the context of the applications. "Elementary Applied Topology" is short (250 pp. plus bibliography and index) and richly illustrated, with 268 figures. It is perfect for both self-study, and as the basis for a course in applied topology. This book is also well-suited for use as a supplementary text in a more traditional algebraic topology course, providing both context and motivation for the tools to be learned. The progression of mathematical techniques is a fresh approach. The book begins with a quick trip through manifolds and cell complexes. The segue to algebraic topology comes in the form of the Euler characteristic and the Euler calculus born from it. Passing from this to homology, exact sequences, and cohomology sets the stage for the innovative content to come. This is comprised of modern Morse theory (including discrete Morse theory, Conley index, and stratified Morse theory), sheaf theory (with an emphasis on cellular sheaves and cosheaves), and, finally, category theory and categorification. Every tool and topic is paired with an application. These range in scope across the biological, economic, engineering, material, physical, and statistical sciences. Of particular note are the applications to topological data analysis, including persistent homology and barcodes. "Elementary Applied Topology" is the first comprehensive text on applied algebraic topology for students of all mathematical sciences.

Genres: Mathematics

276 pages, Paperback

First published September 1, 2014

Reviews

[Rachel Childers · Rating 4 out of 5]

"Perhaps this text will best be used if simply read, for pleasure" (p.6). This is not so much a math textbook as an extended infomercial for the modern field of topology, with applications and intuition stressed over formal results. As the author's intention appears to be to give a broad overview of the field and its potential uses which will encourage readers to pursue deeper study, I believe that for me it succeeded admirably in this respect, largely due to its engaging style and abundance of examples, but admittedly in part due to inducing befuddlement over descriptions and occasionally definitions which were left somewhat imprecise.

[Chris · Rating 5 out of 5]

The one and only. If you want to get interested in topology, buy this book. Full stop.


Also check out Justin Curry's phd thesis on arXiv. (Ghrist's student)





If you're just a general non-mathematical person with $30 to spare on exposing yourself to a new world, give this a look, it will be much easier than most other books to get a sense of what mathematicians are on about.