Virtually all polynomials are irreducible
It has been known for almost a hundred years that most polynomials with integral coefficients are irreducible and have a big Galois group.
For a few dozen years, people have been interested in whether the same holds when one considers sparse families of polynomials—notably, polynomials with plus-minus 1 coefficients.
In particular, “some guy on the street” conjectures that the probability for a random plus-minus 1 coefficient polynomial to be irreducible tends to 1 as the degree tends to infinity
(a much earlier conjecture of Odlyzko-Poonen is about the 0-1 coefficients model) . In this talk, I will discuss these types of problems, their connection with analytic number theory.