In this talk, we will summarize several recent results concerning the nonlinear Schrodinger equation (NLS). The first part of the talk is dedicated to the study of the low-to-high frequency cascade which occurs as a result of the NLS evolution. In particular, one wants to look at how the frequency support of a solution evolves from the low to the high frequencies. This phenomenon can be quantitatively described as the growth in time of the high Sobolev norms of the solution. We present a method to bound this growth using the idea of an almost conservation law, which was previously used in the low regularity context in the work of Bourgain and Colliander, Keel, Staffilani, Takaoka, and Tao.

In the second part of the talk, we will study the Gross-Pitaevskii hierarchy. This is an infinite system of linear partial differential equations which occurs in the derivation of the nonlinear Schrodinger equation from the dynamics of N-body Bose systems. We will study this hierarchy on the three-dimensional torus. We will show a conditional uniqueness result for the hierarchy in a class of density matrices of regularity strictly greater than 1. Our result builds on the previous study of this problem on R^3 by Erdos, Schlein, and Yau, as well as by Klainerman and Machedon and on the study of this problem on T^2 by Kirkpatrick, Schlein, and Staffilani. Finally, we will apply randomization techniques in order to study randomized forms of the Gross-Pitaevskii hierarchy at low regularities, as was done in the setting of nonlinear dispersive equations starting with the work of Bourgain. The second part of the talk is based on joint work with Philip Gressman and Gigliola Staffilani.