The classical theorems of Noether and Petri on the ideals of canonically embedded curves are central in the theory of curves. In the 80s, Mark Green realized that these results should extend to a far broader statement about the entire resolution of the ideal. No major progress was made until Voisin resolved this conjecture for generic curves in 02 and 05. Voisin's proof was extremely sophisticated and uses in a deep way the geometry of the situation. We will give a very simple and short proof of her result, using nothing more than the basic yoga developed by Green, Ein and Lazarsfeld in the 80s. In the case of even genus, we will show how the proof our resolves a deeper (and previously open) conjecture, due to Schreyer, describing in depth the structure of the extremal syzygy space. Speaker(s): Michael Kemeny (University of Wisconsin)