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