PLEASE NOTE SPECIAL DAY (WEDNESDAY, MARCH 13).  We discuss recent works with Jean Bourgain in which we establish new global well-posedness results along Gibbs measure evolutions for the radial nonlinear wave and Schrodinger equations posed on the unit ball in $\mathbb{R}^N$.  In this framework, we consider initial data given as Gaussian random processes lying in the support of the Gibbs measures associated to the equations, and results are obtained almost surely with respect to these probability measures.  Our results include the treatment of nonlinearities for which the available Strichartz bounds do not allow the implementation of fixed-point iterations.  In particular, for the 3D nonlinear wave equation and the 2D nonlinear Schrodinger equation, our approach allows to cover the full range of power-type nonlinearities admissible to treatment by Gibbs measure, while for the nonlinear Schrodinger equation on the 3D ball, we provide the first instance of a probabilistic global well-posedness result treating supercritical data.   Our techniques are based on a delicate analysis of convergence properties of solutions to sequences of finite-dimensional projections of the equations.  Key tools include a class of probabilistic a priori bounds and a delicate analysis of fine frequency interactions for solutions of the projected equations.  When combined with the invariance of the Gibbs measure, these estimates allow us to obtain a.s. convergence of the sequence of approximate solutions.