In analytic number theory one needs to bound sums of oscillating functions, such as the Mobius function. Over function fields these are trace functions of sheaves, so their sums are controlled, in view of the Grothendieck--Lefschetz trace formula, by the cohomology groups of the sheaves. In joint work in progress with Will Sawin, building on arguments from works of Bombieri, Adolphson--Sperber, Deligne--Illusie, Katz, and a study of singularities, we obtain bounds on the cohomology of tame sheaves which leads to applications to the distribution of irreducible polynomials over finite fields in certain thin sets.