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.

The Becker-Döring equation is among the wide class of coagulation-fragmentation equations and is one of the earliest descriptions of particle growth in the theory of nucleation from supersaturated vapour. This model describes the growth and decay of clusters, consisting of identical monomers, only by the addition and removal of monomers.

In this talk, motivated by enzymatic reactions in biology, we will introduce and analyse a variant of the Becker-Döring equations; this model incorporates irreversible fragmentation and monomer injection. After some recollection of coagulation-fragmentation equations, we will first present theoretical results on our model; we will establish global existence and uniqueness under some suitable conditions, and then we will focus on the long-time behaviour of our solution. Finally, we will present an efficient scheme that preserves the asymptotic and allows fast computation by sub-sampling the clusters.

We study a special case of the invasion fitness matrix in a replicator equation: the invader-driven case. In this replicator, each species is defined by its unique active invasiveness potential (initial growth rate when rare), upon invading any other species, independently of the partner. We derive explicit expressions and theorems to fully characterize the steady-states of this system, including its unique interior coexistence regime, reached for positive species traits, or alternative boundary exclusion states, reached for negative species traits. We study the internal stability of coexistence steady-states, and the system’s stability to outsider invasion, relevant for system assembly. We provide detailed analytical results for critical diversity thresholds, and for the special case of random uniform species traits, we analytically compute the probability of stable $k-$ species coexistence in a random pool of size $N$, and show that the mean number of co-existing species can be approximated as $\mathbb{E}[n]\sim \sqrt{2N}$. We also derive explicit mathematical conditions for invader traits and invasion outcomes (augmentation, rejection, and replacement), dependent on the history of system assembly.

In this talk, I will talk about the tagged particle in the symmetric simple exclusion process (SSEP) in an extreme density regime, where the initial distribution is the Bernoulli product measure with a density either vanishing or converging to one. I will explain the limits of the tagged particle under a proper space-time scaling in different scenarios, which are either Fractional Brownian motion, Brownian motion or Brownian motion with Brownian barriers. Based on an on-going project with Tertuliano Franco and Patrícia Gonçalves.

The motion of a charged particle in an electromagnetic field is governed by the Lorentz-force equation (LFE), a classical model independently introduced by Poincaré and Planck in the early twentieth century. Despite being an ordinary differential equation, the LFE presents important analytical challenges that have delayed a fully general mathematical treatment for over a century. First, the equation is vector-valued, rather than scalar. Moreover, the relativistic acceleration term becomes singular as the particle's velocity approaches the speed of light, leading to a non-smooth action functional. In addition, the action depends on the particle's velocity through a dot product, resulting in a sign-indefinite term which complicates the variational treatment of the equation. In this talk, I will survey recent variational methods developed to overcome these difficulties and to establish the existence of periodic solutions to the LFE. Some of these results are part of a joint work with Manuel Garzón (ICMAT, Madrid). I will conclude with a discussion of several open problems.