The aim of this talk is to present some recent work concerning global existence and scattering results for small data problems of Schrödinger type in one space dimension.This is joint work with Mihaela Ifrim.

Choosing favourable gauges is a crucial step in the study of nonlinear geometric dispersive equations. A very successful tool, that has emerged originally in work of Tao on wave maps, is the use of caloric gauges, defined via the corresponding geometric heat flows. The aim of this talk is to describe two such flows and their associated gauges, namely the harmonic heat flow and the Yang-Mills heat flow.