The quadratic Bateman-Horn conjecture over function fields

Abstract: Are there infinitely many natural numbers n for which n2+1 is a prime number? In a joint work with Will Sawin we show that over certain finite fields F, there exist infinitely many monic polynomials f in F[u] for which f2+u is a prime polynomial (i.e. a monic irreducible polynomial). After surveying some earlier works, I'll explain how to reduce the problem to a question of cancellation in an incomplete exponential sum. Via the Grothendieck-Lefschetz trace formula, this will lead us to bounding the cohomology of lisse sheaves on the complement of a hyperplane arrangement in affine space.