An old problem, dating back to Van der Waerden, asks about counting irreducible polynomials degree $n$ polynomials with coefficients in the box [-H,H] and prescribed Galois Group. Van der Waerden was the first to show that H^n+O(H^{n-\delta}) have Galois group S_n and he conjectured that the error term can be improved to o(H^{n-1}).
Recently, Bhargava almost proved van der Waerden conjecture showing that there are O(H^{n-1+\varepsilon}) non S_n extensions, while Chow and Dietmann showed that there are O(H^{n-1.017}) non S_n, non A_n extensions for n>=3 and n\neq 7,8,10.
In joint work with Lior Bary-Soroker, and Or Ben-Porath we prove a lower bound in the case of G=A_n, C_2 wreath S_{n/2} . Our lower bound for A_n is comparable to the work of Landesman, Lemke-Oliver and Thorne where they show a lower bound H^{n/4}, although the arguments are different. The proof can be viewed, on the geometric side, as constructing a morphism \varphi from A^{n/2} into the variety z^2=\Delta(f) where each varphi_i is a quadratic form. This in turn induces a linear map A^{n/2} to A^n/ A_n. At the end, time permitting, I will discuss other directions/ questions related to Van der Waerden's problem which boil down to understanding integral points on varieties