# Mathematics Colloquium

#### Mean-field disordered systems and Hamilton-Jacobi equations

**Speaker:**
Jean-Christophe Mourrat, Courant Institute of Mathematical Sciences

**Location:**
Warren Weaver Hall 1302

**Date:**
Monday, March 2, 2020, 3:45 p.m.

**Synopsis:**

The goal of statistical mechanics is to describe the large-scale behavior of collections of simple elements, often called spins, that

interact through locally simple rules and are influenced by some amount of noise. A celebrated model in this class is the Ising model, where

spins can take the values +1 and -1, and the local interaction favors the alignment of the spins.

In this talk, I will mostly focus on the situation where the interactions are themselves disordered, with some pairs having a

preference for alignment, and some for anti-alignment. These models, often called "spin glasses", are already surprisingly difficult to

analyze when all spins directly interact with each other. I will describe a fundamental result of the theory called the Parisi formula. I

will then explain how this result can be recast using suitable Hamilton-Jacobi equations, and what benefits this new point of view may

bring to the topic.

interact through locally simple rules and are influenced by some amount of noise. A celebrated model in this class is the Ising model, where

spins can take the values +1 and -1, and the local interaction favors the alignment of the spins.

In this talk, I will mostly focus on the situation where the interactions are themselves disordered, with some pairs having a

preference for alignment, and some for anti-alignment. These models, often called "spin glasses", are already surprisingly difficult to

analyze when all spins directly interact with each other. I will describe a fundamental result of the theory called the Parisi formula. I

will then explain how this result can be recast using suitable Hamilton-Jacobi equations, and what benefits this new point of view may

bring to the topic.