Comparing counting functions for determinantal point processes

I will describe a general method for comparing the counting functions of determinantal point processes in terms of trace class norm distances between their kernels (and review what all of those words mean). Then I will outline joint work with Elizabeth Meckes using this method to prove a version of a self-similarity property of eigenvalues of Haar-distributed unitary matrices conjectured by Coram and Diaconis.  Finally, I will discuss ongoing work by my PhD student Kyle Taljan, bounding the rate of convergence for counting functions of GUE eigenvalues to the Sine or Airy process counting functions.