Title: Irreducibility of Random Polynomials with Rademacher Coefficients 
Abstract: 

Consider a random polynomial whose coefficients are independent Rademacher random variables (taking the values ±1 with equal probabilities). A central conjecture in probabilistic Galois theory predicts that such polynomials are irreducible asymptotically almost surely as their degree approaches infinity. Here irreducibility is considered over the field of rational numbers. In the first part of the talk I will discuss the recent progress that has shown that this conjecture follows from the Generalized Riemann Hypothesis and that the limiting infimum of the irreducibility probability is positive.

In the second part of the talk, we will explore ideas from the proof of the following result: the limiting supremum of the irreducibility probability is 1, unconditionally. Specifically, we demonstrate that along special sequences of degrees, the polynomial is irreducible asymptotically almost surely. This result is based on joint work with Hokken, Kozma, and Poonen.