The twin primes conjecture predicts that there are infinitely many consecutive pairs of odd numbers that are both prime. It is closely related to Chowla's conjecture, which, among other things, counts the number of consecutive pairs of numbers with an odd number of prime factors. We can consider analogues of these questions  where the natural numbers are replaced with polynomials over a finite field. In joint work with Mark Shusterman, we prove strong forms of both analogous conjectures under mild conditions on the finite field. The proof relies on powerful geometric tools as well as, at a crucial point, a simple algebraic trick involving derivatives of polynomials.