This book gives a thorough introduction to topological data analysis (TDA),
the application of algebraic topology to data science. Algebraic topology is
traditionally a very specialized field of math, and most mathematicians have
never been exposed to it, let alone data scientists, computer scientists, and
analysts. I have three goals in writing this book. The first is to bring people
up to speed who are missing a lot of the necessary background. I will describe
the topics in point-set topology, abstract algebra, and homology theory needed
for a good understanding of TDA. The second is to explain TDA and some current
applications and techniques. Finally, I would like to answer some questions
about more advanced topics such as cohomology, homotopy, obstruction theory,
and Steenrod squares, and what they can tell us about data. It is hoped that
readers will acquire the tools to start to think about these topics and where
they might fit in.Comment: 322 pages, 69 figures, 5 table