Syzygies, gonality and symmetric products of curves

In the mid 1980s, Mark Green and I conjectured that one could read off the gonality of an algebraic curve $C$ from the syzygies among the equations defining any one sufficiently positive embedding of $C$. Ein and I recently noticed that a small variant of the ideas used by Voisin in her work on canonical curves leads to a quick proof of this gonality conjecture. The proof involves the geometry of certain vector bundles on the symmetric product of $C$. I will describe this circle of ideas. If time permits, I will also discuss a partial generalization, with Ein and Yang, to smooth varieties of all dimensions.