The Parisi Variational Problem is a challenging non-local, strictly convex variational problem over the space of probability measures whose analysis is of great interest to the study of mean field spin glasses. In this talk, I present a conceptually simple approach to the study of this problem using techniques from PDEs, stochastic optimal control, and convex optimization. We begin with a new characterization of the minimizers of this problem whose origin lies in the first order optimality conditions for this functional. As a demonstration of the power of this approach, we study a prediction of de Almeida and Thouless regarding the validity of the 1 atomic anzatz. We generalize their conjecture to all mixed p-spin glasses and prove that their condition is correct in the entire temperature-external field plane except for a compact set whose phase is unknown at this level of generality. A key element of this analysis is a new class of estimates regarding gaussian integrals in the large noise limit called ``Dispersive Estimates of Gaussians’’ . This is joint work with Ian Tobasco (U. Michigan).