Applications for Homology The Question: Are there any ways that "applied" mathematicians can use Homology theory?  Have you seen any good applications of it to the "real world" either directly or indirectly?
Why do I care?  Topology has appealed to me since beginning it in undergrad where my university was more into pure math.  I'm currently in a program where the mathematics program is geared towards more applied mathematics and I am constantly asked, "Yeah, that's cool, but what can you use it for in the real world?"  I'd like to have some kind of a stock answer for this.
Full Disclosure. I am a first year graduate student and have worked through most of Hatcher, though I am not by any means an expert at any topic in the book.  This is also my first post on here, so if I've done something wrong just tell me and I'll try to fix it.

You may want to check out "Topological and Statistical Behavior Classifiers for Tracking Applications" abstract preprint
This has the first unified theory for target tracking using Multiple Hypothesis Tracking, Topological Data Analysis, and machine learning. Our string of innovations are 1) robust topological features are used to encode behavioral information (from persistent homology), 2) statistical models are fitted to distributions over these topological features, and 3) the target type classification methods of Wigren and Bar Shalom et al. are employed to exploit the resulting likelihoods for topological features inside of the tracking procedure. 
