Title: The quadratic Bateman-Horn conjecture over function fields

Abstract: Are there infinitely many natural numbers n with n^{2}+1 a prime?
In a joint work in progress with Will Sawin we show that for some finite fields F, there are infinitely many monic polynomials f in F[u] for which f^{2}+u is prime (i.e. monic irreducible).
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 certain sheaves on the complement of a hyperplane arrangement in affine space.
Video of talk