The first lecture of the summer school was by David Ambrose. He gave a wide ranging talk on aspects of the initial value problem, for irrotational waves in 2D and 3D using a geometric formulation of the water wave problem. The first half of the lectures was on 2D waves. Although useful for water waves, the setting was the more general case of two fluids of differing densities. The talk started with a review of the paper of Hou, Lowengrub & Shelley (1994). The aim of HLS was a numerical scheme, but the lectures showed how it is also a starting point for a rigorous analysis of the IVP. The key is to evolve geometric properties of the surface (curvature, length) combined with careful analysis of the Birkhoff-Rott integral. The second part of the lectures showed how to generalize to 3D. Background work includes Ambrose & Masmoudi (2005) and Ambrose & Masmoudi (2007). The slides from the lectures are available for downloading here. Lecture notes will be forthcoming.