In this thesis, we study geometric aspects of exponential metrics associated with log-correlated Gaussian fields, as well as analytic properties of dispersive wave equations. The first part focuses on the structure of exponential metrics in arbitrary dimension, where we establish that Liouville quantum gravity (LQG) metrics can approximate any Riemannian metric equipped with a locally finite measure. In the second part, we investigate geometric features of LQG and related random planar maps, introducing a notion of Gaussian curvature for LQG surfaces and analyzing its discrete counterpart for mated-CRT maps. We obtain asymptotics and scaling limits for curvature observables. The final part focuses on channels of energy estimates for linearized wave equations. This type of estimates is essential for establishing soliton resolution via the channels of energy method of Kenig-Merle. We prove such estimates for the wave maps equation linearized around a ground state, and more generally for a class of equations with analytic nonlinearities in even dimensions greater than or equal to $8.$
