Dear all, Next week RoMaDS will host two seminars and a mini-course at the Department of Mathematics of the University of Rome, Tor Vergata (this is the link <https://www.mat.uniroma2.it/~rds/events.php> to RoMaDS webpage with all future events. We have some technical problems at the moment, but you can safely access the website ignoring the warning!). All abstracts can be found at the end of this email. Best, Michele

Monday, 11th May, Seminar

Johannes Alt (University of Bonn) "Spectra of critical Erdős-Rényi graphs" 14h00-15h00, Department of Mathematics, Aula Dal Passo. Teams link <https://teams.microsoft.com/meet/35777624543024?p=boIC6CbJ8QfDutJzUV>.

Thursday, 14th May, Seminar

Isabella Ziccardi (IRIF, Université Paris Cité) "Electing a Leader with Limited Resources" 16h00-17h00, Department of Mathematics, Aula D’Antoni. Teams link <https://teams.microsoft.com/meet/389989257555244?p=IMwo46pwx3w3fSzhOm>.

12-13-14th May, Mini-course

Cyril Letrouit (CNRS, Laboratoire de Mathématiques d'Orsay) "The Mathematics of Transformers: From Particle Systems to Control Theory" 12, 13, 14 May, 11h00-12h30, Department of Mathematics, Aula Dal Passo For all further information you can contact Prof. Caponigro at caponigro@mat.uniroma2.it <mailto:caponigro@mat.uniroma2.it> ABSTRACTS Abstract Alt “Spectra of critical Erdős-Rényi graphs" We consider the Erdős-Rényi graph G in its critical regime when its expected degree d scales like the logarithm of its number of vertices. On this critical scale, G undergoes a connectivity transition through the formation of isolated vertices. Moreover, localized eigenvectors emerge. The time evolution of a free quantum particle on G is governed by the adjacency matrix A of G through the Schrödinger equation. We determine the solution to this Schrödinger equation by comparison to an infinite tree. As A possesses localized and delocalized eigenvectors, the solution is in general a mixture of localized and scattering waves Abstract Ziccardi “Electing a Leader with Limited Resources” A distributed system consists of n independent entities—such as computers, servers, or even biological organisms—connected by a communication graph G. In such a system, nodes operate autonomously using their own local memory. A fundamental challenge in this setting is the Leader Election problem, where nodes must coordinate to elect a single leader. Electing a leader is fundamental because it serves as a central control, simplifying many other coordination tasks. Recently, much attention has been focused on leader election within weak communication models, like the beeping or stone age models. While many algorithms have been proposed, they typically require nodes to have large memory capacities, require knowledge of global quantities (such the total number of nodes n) or only work on specific families of graphs. In this talk, I introduce a very simple randomized algorithm that almost surely elects a leader on any network G. Our approach requires only constant memory and zero prior knowledge about G’s size or topology; nodes interact by sending only a 1-bit signal. We show that this protocol is efficient, reaching a stable leader configuration in O(D^2 log n) rounds with high probability, where D is the diameter of G. Finally, I will address the issue of resilience. Because our algorithm requires all nodes to start from the same, fixed initial state, it cannot guarantee leader election if the system starts in an arbitrary configuration reached due to local faults. I will discuss how to extend this algorithm to be self-stabilizing—meaning it can recover and converge from any initial state. Our self-stabilizing solution uses only O(log log n) bits of memory, representing an exponential improvement over previous literature that required O(log n) bits. This presentation is based on joint work with Robin Vacus [PODC 2025] and Lélia Blin and Sylvain Gay [PODC 2026] Abstract Letroiut “The Mathematics of Transformers: From Particle Systems to Control Theory” Since their introduction in 2017, Transformers have profoundly transformed large language models and, more generally, deep learning. This success largely relies on the mechanism known as "self-attention". In this course, I will introduce a mathematical framework that allows self-attention to be viewed as a system of interacting particles. I will explain certain remarkable properties of the associated dynamics in the space of probability measures, with particular emphasis on cluster formation, the preservation of Gaussian distributions, the subtleties of the associated mean-field limit, and the great "expressivity" of these neural networks, proved thanks to control theory.