PLEASE NOTE SPECIAL DAY AND LOCATION.  I will give an overview of some recent work in the subjects of "arithmetic invariant theory" and "arithmetic statistics." The first has to do with using representations of groups to study moduli spaces of arithmetic or geometric objects. These types of results have been used to obtain statistics for arithmetic objects, e.g., bounds for average ranks of elliptic curves. Time permitting, I will also talk about some examples that have applications to dynamics on K3 surfaces.