Equations and syzygies of algebraic curves

Algebraic curves (Riemann surfaces) are among the most studied objects in mathematics due to the fact that they can be approached from the point of view of algebraic geometry, complex analysis or Galois theory. In 1984, Mark Green put forward a deceptively simple conjecture concerning the structure of the equations of an algebraic curve in its canonical embedding, amounting to the statement that the complexity of each curve of genus g can be recovered in a precise way from the equations among its canonical forms. I will present an introduction to this circle of ideas, then explain how ideas coming from topology and geometric group theory led to a recent solution of Green's Conjecture for generic curves in arbitrary characteristics.