Title: Spherical spin glass models

Abstract: In the 70s, physicists proposed several models fordisordered magnetic alloys called spin glass models. Mathematically, thespherical models are random functions on the sphere in high-dimensions, andmany of the questions physicists are interested in can be phrased aspurely mathematical questions about geometric properties, extreme values,critical points, and Gibbs measures of random functions and the interplaybetween them.

After overviewing the model and its basic properties, I willdescribe a certain geometric description for the Gibbs measure achieved byanalysis of the critical points and the local behavior around them, for the"pure" spherical models. For general (non pure) models, the analysisof critical points fails, but a similar description of the Gibbs measure can beachieved using another (softer) geometric approach. I will explain thisapproach, and describe how it makes rigorous and generalizes the famousThouless-Anderson-Palmer approach from physics.