Elementary Applied Topology
Algebraic topology tools have been used more and more recently in
various applied area (numerical analysis, imaging, neuroscience,
evolutionary biology, computer vision, complexity theory, statistics,
machine learning, and what not). There is a wonderful book by Robert
Ghrist devoted to such a pedestrian course with applied examples on
basic topics such as manifolds, complexes, homotopy, Euler characteristics,
(co-)homology, Morse theory, sheaves, some homological algebra,
maybe a little bit of persistence theory. It is not an in-depth and/or
rigorous math course (and thus does not replace our topology and geometry
graduated classes), but rather a pedestrian introduction for students
with applied directions to main concepts with example and applications
that would entice users to a further study.
The book (besides being sold as a paperback at a very low price) is
available freely at
https://www.math.upenn.edu/~ghrist/notes.html
The prerequisites are general familiarity with abstract algebra
(groups, etc.) and basic notions of topology.