Derived categories of cubic 4-folds and their geometric applications

A fundamental problem in algebraic geometry is to determine whether a given algebraic variety is birational to projective space.  This is most prominently open for cubic 4-folds, i.e. hypersurfaces defined by a cubic polynomial inside a 5-dimensional projective space.  A decade ago Kuznetsov suggested an approach to this problem using the derived category of coherent sheaves.  I will explain recent applications of this perspective to fundamental questions in hyper-Kähler geometry and Hodge theory, which in turn shed light on the original problem about cubic 4-folds.