Galois cohomology classes arise naturally in a wide range of contexts, as invariants of algebraic objects, as obstructions in algebraic and arithmetic geometry, and more generally as tools for understanding field arithmetic. While the resolution of the Bloch-Kato conjectures have given us important insights into Galois cohomology, fundamental questions remain as to how complicated the description of classes can be for particular fields of interest, in particular for function fields of varieties.

One of the most important cases is that of degree 2 cohomology, which corresponds to the Brauer group. Here these questions relate to the possible structure of division algebras over a field.

In this talk, I’ll give a survey of some of the progress and open questions on the complexity of descriptions of Galois cohomology class, including the period-index problem, the symbol length problem and approaches through local-global principles.