Representation Theory and Arithmetic Statistics of Generalized Configuration Spaces

In this thesis, we extend the work of Church-Ellenberg-Farb on FI-modules, representation stability of configuration spaces, and arithmetic statistics. We study two generalizations of the category FI: namely FI_G for G a group, first studied by Sam-Snowden, and FI^m, first studied by Gadish. We use these to study two types of generalized configuration spaces: the orbit configuration spaces Conf^G_n(M), associated to a G-cover M, and the space of ordered 0-cycles Z^{(d_1, ... d_m)}_n(X) introduced by Farb-Wolfson-Wood. After establishing basic properties of FI_G- and FI^m-modules, we obtain representation stability results for the cohomology of these generalized configuration spaces. We establish subexponential bounds on the growth of unstable cohomology, and the Grothendieck-Lefschetz trace formula then allows us to translate these topological stability phenomena to stabilization of arithmetic statistics for generalized configuration spaces over finite fields.