Mathematical Theory of Spin Glasses

Mean-field spin glasses are paradigmatic models of complex systems. Originally introduced to explain the puzzling magnetic properties of certain exotic materials, closely related models have since appeared in fields such as theoretical computer science and theoretical statistics, suggesting that the complex phenomena they exhibit are fundamental.

This project aims to develop a mathematically rigorous, stand-alone theory of mean-field spin glasses based on the Thouless-Anderson-Palmer (TAP, 1977) free energy. The TAP functional captures the trade-off between energy and entropy, fitting elegantly into the basic framework of statistical mechanics. However, conceptual difficulties specific to spin glasses have so far confined TAP analysis to a complementary role.

By addressing these difficulties directly, the project seeks to build a stand-alone TAP theory of mean-field spin glasses, providing new proofs and insights for known models, and solving models beyond the reach of current mathematical methods.