The Brauer group is a fundamental invariant classifying the central division algebras over a field, with numerous applications in geometry and arithmetic. In the case of the function field of a variety, the period-index conjecture proposes a precise bound on the most basic invariant of a division algebra, its dimension, in terms of its order in the Brauer group. I will explain recent progress on this conjecture, including its proof for unramified division algebras over the function field of an abelian threefold, based on joint work with James Hotchkiss. This depends on a reinterpretation of the conjecture in terms of a version of the integral Hodge conjecture for "noncommutative" varieties, which for noncommutative Calabi-Yau threefolds can be approached using enumerative geometry.