Seminars, for informal dissemination of research results, exploratory work by research teams, outreach activities, etc., constitute the simplest form of meetings at a Mathematics research centre.
CAMGSD has recorded the calendar of its seminars for a long time, this page serving both as a means of public announcement of forthcoming activities but also as a historic record.
The Hopfield Neural Network has played, ever since its introduction in 1982 by John Hopfield, a fundamental role in the inter-disciplinary study of storage and retrieval capabilities of neural networks, further highlighted by the recent 2024 Physics Nobel Prize.
From its strong link with biological pattern retrieval mechanisms to its high-capacity Dense Associative Memory variants and connections to generative models, the Hopfield Neural Network has found relevance both in Neuroscience, as well as the most modern of AI systems.
Much of our theoretical knowledge of these systems however, comes from a surprising and powerful link with Statistical Mechanics, first established and explored in seminal works of Amit, Gutfreund and Sompolinsky in the second half of the 1980s: the interpretation of associative memories as spin-glass systems.
In this talk, we will present this duality, as well as the mathematical techniques from spin-glass systems that allow us to accurately and rigorously predict the behavior of different types of associative memories, capable of undertaking various different tasks.
In this talk, we will present some recent results concerning the vector-valued mean-field spherical model in a random external field, which is a mean-field generalization of the Berlin-Kac model subject to an additional external random field term in the Hamiltonian. Through the random field term, the Gibbs measures become random Gibbs measures, and their convergence in the infinite volume limit can be studied in different probabilistic modes. For this particular model, we are able to characterize exactly the limit points, convergence in distribution, and convergence in empirical distributions. The properties of the model are closely connected to corresponding limit theorems for random walks, and they exhibit similar dimensional dependence for phenomena like transience and recurrence. Time permitting, we will also discuss the spin glass features of the model. This talk is based on the joint work arXiv:2505.16843 with Professor Christof Külske.
