Topological Data Analysis by Larry Wasserman.
From the post:
Topological data analysis (TDA) is a relatively new area of research that spans many disciplines including topology (in particular, homology), statistics, machine learning and computation geometry.
The basic idea of TDA is to describe the “shape of the data” by finding clusters, holes, tunnels, etc. Cluster analysis is special case of TDA. I’m not an expert on TDA but I do find it fascinating. I’ll try to give a flavor of what this subject is about.
Just in case you want to get in on the ground floor of a new area of research.
Larry has citations to the literature in case you need to pick up beach reading.