The Topological Part of the Period-Index Conjecture

The period-index problem is a classical problem about finite-dimensional division algebras over a field. When the base field is the function field of a complex variety, there is a longstanding conjecture, which is wide open for function fields of threefolds and beyond. I will discuss recent perspectives on the conjecture from topology, Hodge theory, and moduli theory. Finally, I will explain a result showing that, roughly speaking, topological solutions to the conjecture exist for complex function fields of arbitrary dimension.