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.
 A: As an additional source of ideas, may I point out that vol. 157 of the Springer Applied Mathematical Sciences series is entitled Computational Homology (it is by Kaczinski, Mischaikow and Mrozek). This was related to the CHOMP project which is well worth checking out.
These relate more to applied mathematics than to data analysis.  Both threads probably deserve more attention by the mathematical community.
A: There is a very nice application of homology in data mining and computer science called "topological data analysis". 
It is mainly based on the computation of a homology theory called "persistent homology" which describes those topological features that are "persistent" while varying the parameter which is used in the clustering analysis (for example, the radius of balls around the points in the data cloud and giving classical topological complexes).
This homological theory gives a qualitative description of the topology of the underlying data cloud. In this sense it is a new approach to clustering, which is classically seen as an optimization problem subject to the  choice of a metric or metric like function for the data cloud under examination.
Topological data analysis is particularly interesting in presence of "Big Data", i.e. extremely high numbers of often uncorrelated data, such as those from medical sciences, social networks or meteorology. 
You can check Gunnar Carlsson's et al. works on the arxiv to get a better idea of it.
A great introduction to the topic is 
http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/S0273-0979-09-01249-X.pdf
A: There are definite real world applications. I would look at the website/work of Gunnar Carlsson (http://comptop.stanford.edu/) and Robert Ghrist (http://www.math.upenn.edu/~ghrist/). Both are excellent mathematicians. 
The following could be completely wrong: Carlsson is one of the main proponents of Persistent Homology which is about looking at what homology can tell you about large data sets, clouds, as well as applications of category theory to computer science. Ghrist works on stuff like sensor networks. I don't understand any of the math behind these things.
Also there are some preprints by Phillipe Gaucher you might want to check out. Peter Bubenik at cleveland state might also have some fun stuff on his website.
A: 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. 
