This project aims to develop both theoretical results and practical techniques in the study of Partial Differential Equations. A fundamental tool in the study of these equations is harmonic analysis, in which local behaviour of a system is used to analyze global properties, using techniques such as the Fourier transform. The project attacks central problems in the area, revealing deep connections between analysis and geometry, and apply these to study the long-term behaviour of the solutions to some non-linear equations. Expected outcomes of the project include theoretical results and practical techniques for the solution of certain equations called non-linear dispersive equations, which arise for example in quantum and fluid mechanics.