Spin glasses are paradigmatic models of complex systems, originally invented by physicists to qualitatively describe properties of exotic magnetic alloys, and later found to be relevant to other complex phenomena in theoretical statistics and theoretical computer science. From a mathematical point of view their study is essentially the question of determining the properties of the extreme values of a highly correlated random field known as the Hamiltonian.

The study of spin glasses have spawned a multitude of methods in pure mathematics and theoretical physics. Some landmark results, like the Parisi formula, have been mathematically proven, but others remain out-of-reach for current mathematical methods.

A particularly attractive method was proposed by Thouless-Andersson-Palmer (TAP), but has never been fully developed as a stand-alone approach due to difficulties in its implementation.

This project aims to develop a comprehensive mathematically rigorous method to study spin glass models using a TAP approach.

Specifically during the grant period new results estimating the free energy in terms of the TAP energy will be proven, as well as the first mathematically rigorous estimates for the number of critical points of Hamiltonians on the sphere in settings where the expectation of the critical point count does not predict the true value (i.e. where "quenched" and "annealed" averages do not agree).

For this purpose the project will develop new geometric arguments and truncated moment methods using the machinery of probability theory, in the framework of pure mathematics. The project will lead to proofs of important unknown results and new proofs of known results, which will shed light on the mysterious hierarchical structure underlying the extreme values of spin glass Hamiltonians.