This dissertation is composed of two parts. The first part applies techniques from Harmonic and nonlinear Fourier Analysis to the nonlinear Schrödinger equation, and therefore tools from the study of Dispersive Partial Differential Equations (PDEs) will also be employed. The dissertation will apply the ℓ2 decoupling conjecture, proved recently by Bourgain and Demeter, to prove polynomial bounds on the growth of Sobolev norms of solutions to polynomial nonlinear Schrödinger equations. The first bound which is obtained applies to the cubic nonlinear Schrödinger equation and yields an improved bound for irrational tori in dimensions 2 and 3. For the 4 dimensional case an argument of Deng that relies on the upside-down I-method and long time Strichartz estimates on generic irrational tori is applied to get bounds in for the energy-critical nonlinear Schrödinger equation, assuming small energy. The second part of the thesis proves unique continuation at the boundary for solutions to a class of degenerate elliptic PDEs. Specifically, there is a weight at each point on the domain which is bounded by some fractional power of the distance to the boundary. Techniques from Calculus of Variations and Geometric PDEs are used to show that solutions to this class of PDEs are uniquely defined simply by specifying their values and some notion of their normal derivative on an open set with positive measure on the boundary
