Birational Geometry of moduli spaces of sheaves on K3 surfaces via Bridgeland stability

We use wall-crossing for Bridgeland stability conditions to systematically study the birational
geometry of a moduli space M of Gieseker-stable sheaves on a K3 surface. In particular, we show:
- Any K-equivalent birational model of M appears as a moduli of Bridgeland stable objects, such that
the birational transformation is induced by wall-crossing.
- We complete Markman's proof the Kawamata-Morrison cone conjecture on the moveable cone of M.
- We establish the Hassett-Tschinkel/Huybrechts/Sawon conjecture on the existence of birational
Lagrangian fibrations whenever there exists a divisor of square zero with respect to the Beauville-Bogomolov
form. 
This is based on joint work with Emanuele Macrì. The two main ingredients are the construction 
of a canonical determinant line bundle on moduli spaces of Bridgeland-stable objects, and a detailed
analysis of walls for which every object becomes strictly semistable.