MARK KAC SEMINAR

Introduced as a model of associative memory, the Hopfield model has become a paradigmatic framework at the interface of statistical physics, theoretical neuroscience, and machine learning. Its simple definition already captures key features of high-dimensional systems with quenched disorder, frustration, and emergent collective behavior, making it a natural benchmark for both physical and computational questions. From the neuroscience perspective, biologically motivated mechanisms of learning and memory have long inspired algorithmic developments, and the Hopfield model provides a natural setting in which such ideas can be analyzed in a mathematically controlled way. At the same time, its formal equivalence with Boltzmann machines connects it to several themes in machine learning, including representation learning, regularization strategies, and energy-based optimization. From the viewpoint of statistical mechanics, the Hopfield model remains a fundamental laboratory for rigorous and heuristic methods developed for spin glasses and random systems. These methods can also be extended beyond the classical setting to structured or correlated data, where new phenomena arise.

Jamming transitions comprise a broad class of nonequilibrium mechanical phenomena in which material systems abruptly transition from floppy or fluid phases to rigid or solid ones. In this lecture, I will review some recent theoretical and computational advances toward understanding jamming transitions in soft-matter systems. Starting from the simplest system of disordered spring networks, I will then describe the mechanics of soft-sphere packings, the rheology of dense non-Brownian suspensions, and finally the stiffening of biopolymer networks subjected to large deformations. I will show how the scaling behavior near the respective jamming transitions in these systems can be obtained by perturbing their corresponding critical states.