Seminários e cursos curtos

By Alexander's theorem, every link in the 3-sphere can be represented as the closure of a braid. Lorenz links and twisted torus links are two families that have been extensively studied and are well-described in terms of braids. In this talk, we will present a natural generalization of Lorenz links and twisted torus links that produces all links in the 3-sphere. This provides a simpler braid description for all links in the 3-sphere.

How do competing pathogen strains evolve within and across a population of hosts? We propose a simple stochastic model in which the type composition within each host evolves according to a family of Markov kernels. When hosts evolve independently, the model reveals a moment duality with genealogies related to the Ancestral Selection Graph and, under suitable scaling, converges to a Wright–Fisher diffusion with drift. When hosts interact through the population distribution, the system becomes weakly interacting. We prove propagation of chaos and show that the dynamics of a typical host converge to a McKean–Vlasov diffusion. As an illustration, we consider mutation rates depending on the current population state and study ergodicity of the resulting mean-field dynamics. This talk is based on join work with Leonardo Videla (Universidad de Santiago) and Héctor Olivero (Universidad de Valparaiso).

Given a weight w on the unit circle, consider the orthogonal polynomials on the unit circle generated by w. Steklov famously conjectured that if w is bounded below, then the polynomials all ought to be uniformly bounded above. While false, this conjecture begs the follow-up question: under what regularity conditions on w are the polynomials uniformly bounded in L^p(w) for some p>2? Building upon a preliminary answer given by Nazarov for when w is bounded above and below, we provide a positive answer when w is an A2 weight. This is joint work with Alexander Aptekarev and Sergey Denisov.

In this talk, we address the localization of general nonlocal functionals of double-integral type with fractional dependence on the state variable, inspired by peridynamics. Localization is carried out as the interaction horizon among particles tends to zero. As a main result, we obtain an explicit formulation of the local $Γ$-limit, also covering the vectorial case. Applications of this result to nonlinear elasticity and the p-Laplacian eigenvalue problem will be discussed.

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.