Moduli of non-pure sheaves, contractions and flops

I will explain how to contract subvarieties of a given smooth projective variety using non-standard moduli of sheaves. We will also see how to perform "wall-crossing" to obtain a surgery diagram. This is used to prove instances of Kawamata's DK hypothesis for the behavior of derived categories under birational transformations. This talk is based on joint work with Svetlana Makarova.