In the theory of Brauer groups, the longstanding period-index problem asks for a bound on one measure of complexity of a central simple algebra (its index) in terms of another (its period). I will discuss some recent progress on this problem which relies on a mixture of ideas from Hodge theory, noncommutative/derived algebraic geometry, and enumerative geometry.

This is based on joint works with James Hotchkiss and with Johan de Jong.