There is a canonical pairing on the Brauer group of a surface over a finite field, which is the analogue of the Cassels-Tate pairing on the Tate-Shafarevich group of a Jacobian variety. An old conjecture of Tate predicts that this pairing is alternating. In this talk I will present a proof of Tate’s conjecture. The key new ingredient is a circle of ideas originating in algebraic topology, centered around the Steenrod operations, that is imported to algebraic geometry on the ships of eětale homotopy theory. The talk will advertise these new tools (while assuming minimal background in algebraic topology)