Arithmetic Statistics via Harmonic Analysis by Brandon Alberts
I will present arithmetic statistics of the first Galois cohomology group H^1(K,T), with elements ordered by a discriminant-like invariant. The Galois cohomology groups H^1(K,T) parametrize extensions of K with certain Galois groups and fixed resolvents, directly relating this question to Malle's conjecture. These results will be proven using harmonic analysis on adelic cohomology, modelled after the work in Tate's thesis. This method gives a canonical decomposition of the generating Dirichlet series, produces generalized results with ease (including restricted local behavior and alternate orderings), and makes partial progress towards harder problems in number field counting. This work is joint with Evan O'Dorney.