Date:

Mon, 23/05/202214:30-16:00

**Title**: Probabilistic Galois Theory, the square discriminant case.

**Abstract**: Consider the easiest model of random polynomials with integers coefficients,

where we fix the degree n=deg f and we choose the coefficients uniformly at random from a large box and let its size L go to infinity.

Probabilistic Galois theory, in its naviest form, asks for the distribution of the Galois group of f.

In 1936, Van der Waerden proved that the Galois group of the polynomial is the full symmetric group S_n asymptotically almost surely.

He conjectured that the second most probable group is S_{n-1}. This conjecture has seen a lot of progress along the years.

Recently, there was a big breakthrough by Bhargava who showed that the second most probable group is either S_{n-1} or A_n.

The breakthrough main achievement was the new upper bound C/L on the probability of the Galois group being A_n.

One may believe that, in fact, the probability for A_n must be much smaller; the goal of the talk is to discuss what should be

the probability for A_n, to give lower bounds, and new heuristics.