Abstract: In the last dozen years, topological methods have been shown to produce a new pathway to study arithmetic statistics over function fields, most notably in Ellenberg-Venkatesh-Westerland's work on the Cohen-Lenstra conjecture. More recently, Ellenberg, Tran and Westerland proved the upper bound in Malle's conjecture on the enumeration of function fields by studying the homology of braid groups with certain exponential coefficients. In this talk, we will give an overview of their framework and extend their techniques to study other questions in arithmetic statistics. As an example, we will demonstrate how this extension can be used to study character sums of the resultant of monic squarefree polynomials over finite fields, answering and generalizing a question of Ellenberg and Shusterman, and Malle's conjecture for function fields with prescribed ramification.