Seminars and short courses

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 $p\gt 2$? 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 $A_2$ 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 seminal Edwards-Sokal coupling allows to express correlations in the Potts model via connection probabilities in the random-cluster model. In this talk we discuss how similar ideas can be applied in a number of other settings. Specifically: - the planar Potts model can be related to the Ashkin-Teller model, and this brings new results for both models, including convergence of interfaces to Brownian bridges and the wetting phenomenon (j.w. Moritz Dober and Sébastien Ott); - the loop O(n) model can be resampled via a divide-and-color procedure, and this can be used to establish a big part of its phase diagram. Another important ingredient is our proof of a conjecture of Benjamini and Schramm - in particular, we show that p_c is at least 1/2 on any unimodular invariantly amenable planar graph (j.w. Matan Harel and Nathan Zelesko); - we also show delocalisation of two-dimensional height functions related to these models (j.w. Piet Lammers).

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.