Spatial statistics is a personal favorite subject. It has incredible power for modeling, but also requires thoughtfulness and creativity in approaching the construction of research questions, and subsequently the approach to modeling and analysis. The subject lends itself well to modeling of environmental phenomena where there is plenty of free access to GIS data. I was fortunate to be able to work on a project analyzing wildfire patterns in Northern Arizona as a result.
I'm working on a simplified write-up of the project, but in the meantime you can find a more technical write-up in the technical blog post.