On the ubiquity of twisted sheaves

Abstract

We describe some recent work on the uses of twisted sheaves in algebra, arithmetic, and geometry. In particular, we touch on the role of twisted sheaves in: The geometry of the period-index problem for the Brauer group.The connection between finiteness of the u-invariant and Colliot-Thelene's conjecture on 0-cycles. The link between the Tate conjecture for K3 surfaces and finiteness of the set of isomorphism classes of K3 surfaces over a finite field. The geometry of rational curves on the moduli spaces of supersingular K3 surfaces.