Dispersive equations are ubiquitous in nature.
(1) Some of the areas that give rise to these equations are water waves, optics, lasers, ferromagnetism, particle physics, general relativity, nonlinear elasticity and many others. These equations also have connections to geometric flows arising in Kahler and Minkowski geometries
(2) The nonlinear Schroedinger equations also arise as the equations governing Bose-Einstein condensates which is a fascinating phenomena predicted by quantum statistical mechanics.
The 2001 Nobel Prize in Physics was awarded for the experimental verification of BEC. There has been much recent progress in different directions, in particular advances towards the soliton resolution conjecture, the study of asymptotic stability/instability of physical system. In this project, we will study Fourier analysis, harmonic analysis and then use them to dynamics of dispersive equations.