Wall-Crossing implies Brill-Noether. Applications of stability   conditions on surfaces
  Over the last few years, wall-crossing for Bridgeland stability conditions has led to a large number of results in algebraic geometry, particular on birational geometry of moduli spaces.   We illustrate some of the methods behind these result by reproving Lazarsfeld's Brill-Noether theorem for curves on K3 surfaces via wall-crossing. We conclude with a survey of recent applications of stability conditions on surfaces.   The intended reader is an algebraic geometer with a limited working knowledge of derived categories. This article is based on the author's talk at the AMS Summer Institute on Algebraic Geometry in Utah, July 2015. 
Arend Bayer
math.AG