Global solutions for cubic quasilinear Schroedinger flows in two and higher dimensions

In recent work the authors proposed a broad global well-posedness conjecture for cubic defocusing dispersive equations in one space dimension, and then proved this conjecture in two cases, namely for one dimensional semilinear and quasilinear Schr\"odinger flows. Inspired by the circle of ideas developed in the proof of the above conjecture, in this paper we expand the reach of these methods to higher dimensional quasilinear cubic Schr\"odinger flows. The study of this class of problems, in all dimensions, was initiated in pioneering work of Kenig-Ponce-Vega for localized initial data, and then continued by Marzuola-Metcalfe-Tataru (MMT) for initial data in Sobolev spaces. The outcomes of this work are (i) a new, potentially sharp local well-posedness result in low regularity Sobolev spaces, one derivative below MMT and just one half derivative above scaling, (ii) a small data global well-posedness and scattering result at the same regularity level, the first result of its kind at least in two space dimensions, and (iii) a new way to think about this class of problems, which, we believe, will become the standard approach in the future.