# Student Probability Seminar

#### The Parisi Variational Problem: A New Approach

**Speaker:**
Aukosh Jagannath

**Location:**
Warren Weaver Hall 1314

**Date:**
Thursday, September 24, 2015, 2 p.m.

**Synopsis:**

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 will present a new, conceptually simple approach to the study of this problem using techniques from PDEs, stochastic optimal control, and convex optimization. We will 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 new approach, we will study a prediction of de Almeida and Thouless regarding the validity of the 1 atomic anzatz. We will 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 Gaussiansâ€™â€™ . This is joint work with Ian Tobasco (NYU).