Syzygies in algebraic geometry and geometric group theory

Determining the structure of the equations of an algebraic curve in its canonical embedding (given by its holomorphic forms) has been a central question in algebraic geometry from the beginning of the subject. In 1984 Mark Green put forward a very elegant conjecture linking the complexity of the curve in its moduli space to the structure of its equations (syzygies). I will discuss how novel ideas coming from geometric group theory led to a surprising solution of this conjecture for generic curves of arbitrary characteristic and what implications these methods have to important questions in topology or group theory.