Abstract: Suppose you have a vector space with the action of a Lie group. If you give it to a string theorist they might build a gauged linear sigma model, and if you give it to an algebraic geometer they might take the geometric invariant theory quotients. It turns out the string theory can then make interesting predictions for the algebraic geometer, particularly about how the derived categories of the different quotients are related. I'll explain some of this picture and what we understand about it mathematically.