Appears in collection : Balzan Lectures

This lecture will review how linear gravity can be obtained from integral formulae from twistor space and how the fully nonlinear self dual sector of 4d gravity can be built using complex analysis. It will focus on global problems in split signature where the self-dual sector can be generated from a Hamiltonian deformation of the real slice of twistor space. The space-time is reconstructed from holomorphic discs in twistor space with boundary on the deformed real slice.