Two lectures about mumford-tate groups

We define Hodge structures as representations, and introduce the Mumford-Tate group MT(H) of a rational Hodge structure. We give a characterization of MT(H) based on properties of reductive groups, and study it for elliptic curves and their powers. We define what it means for a Hodge structure to arise from an abelian variety (as is the case for K3 surfaces), and then focus on cases in which this is not true. The presentation includes a selection of exercises.