Recent work on van der Waerden’s conjecture

Last summer, there was a lot of activity regarding an old conjecture of van der Waerden, culminating in its solution by Bhargava, and including joint work by Sam Chow and myself on which I want to report in this talk: We showed that the number of irreducible monic integer polynomials of degree n, with coefficients in absolute value bounded by H, which have Galois group different from S_n and A_n, is of order of magnitude O(H^{n-1.017}), providing that n is at least 3 and is different from 7,8,10. Apart from the alternating group and excluding degrees 7,8,10, this establishes the aforementioned conjecture to the effect that irreducible non-S_n polynomials are significantly less frequent than reducible polynomials.