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).