Sums in progressions over F_q[T], the symmetric group, and geometry

I will discuss some recent progress in analytic number theory for polynomials over finite fields, giving strong new estimates for the number of primes in arithmetic progressions, as well as for sums of some arithmetic functions in arithmetic progressions. The strategy of proof is fundamentally geometric, and I will explain some of the geometric ideas in the proof, including how we can use the representation of the symmetric group to handle many different arithmetic functions in a uniform way.