Exploratory data analysis is the search for patterns in sampled data. In this talk, I will concentrate on geometric data, where structure often takes the form of certain shapes representing constraints in the system generating the data. For example, a sphere for normalized data or a closed curve representing a periodic system. Recent topological techniques can help us find this type of structure.

A natural question then is: what is a reasonable null hypothesis for this type of structure? In this talk, I will describe recent results characterizing how much structure we expect to find if we sample noise, which we take here to be a Poisson point process, also known as a completely spatially random model. No previous knowledge of point processes or topology will be assumed. I will overview classical results from random geometric graphs and present our results for higher dimensional analogues.