There are applications of homology theory both to Topological Data Analysis and other parts of Applied Topology (work of [Gunnar Carlson and his coworkers](http://comptop.stanford.edu/) in Stanford), via the [CHOMP](http://chomp.rutgers.edu/) project to the structure of materials, dynamics of Evolution of Pattern Complexity during Phase Separation, etc. (The Stanford webpage mentions many more applications and they really merit a mention but I leave the 'reader' the joy of browsing around the links there.)  There is also work by [Robert Ghrist](http://www.math.upenn.edu/~ghrist/research.html) again on Applied Topology.