Week 08.12.2025 – 14.12.2025

In the first part of the seminar I will review the main features of the Hopfield model, providing as well an overview on its numerous and broad applications. Then, inspired by biological information-processing, I will enrich the framework and make the network able to successfully and cheaply handle structured datasets. In particular, I will recast reinforcement and remotion mechanisms occurring in mammal’s brain during sleep into suitable machine-learning hyperparameters. Finally, I will show that this framework is also effective to face a classical inference problem, that is, matrix factorisation. Results presented are both analytical and computational.

olymers with associative motifs—chains bearing specific “sticker” groups capable of forming reversible, finite-lifetime bonds—exhibit phase behaviour governed by a competition between connectivity, transient crosslinking, and polymer fluctuations. The classical reversible-gelation theory of Semenov and Rubinstein provides the fundamental thermodynamic description of these systems, showing how sticker–sticker associations generate a sol–gel transitions. Subsequent sequence-dependent sticker–spacer models, including those developed by Pappu and collaborators, demonstrated that the spatial arrangement and density of associative motifs modulate condensation behaviour in biomolecular polymers. At a more universal level, scale-free models of droplet formation by Maritan and coworkers revealed nucleation and coarsening dynamics that emerge independently of molecular details.
In this seminar, I will present a field-theoretic framework that aims to unify these descriptions within a single microscopic model. Starting from a polymer Hamiltonian with explicit sticker variables—treated consistently in both quenched and annealed ensembles—we derive a saddle-point theory that yields the bonding free energy directly, without relying on combinatorial assumptions. A preliminary analysis of the resulting effective action further shows how condensation behaviour naturally emerges from the interplay between reversible crosslinking and density fluctuations.