This will be an introduction to schemes and the ways they generalize things you may or may not already know. I will quickly go over the concept of sheaves in order to define schemes. We will go through some specific examples, like varieties and curves, and try to show why schemes are natural things to study. Depending on time we will talk about rank 1 bundles over schemes and the Picard group, and how they relate to arithmetic. The goal is to show how schemes unify many ideas and actually make them easier to understand.
The talk will be widely accessible -- I will assume a technical mastery of algebra, geometry, analysis, number theory, and scheme theory.