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.

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.

In the architecture of large-scale distributed systems, the challenge of distributing data across a dynamic set of nodes is fundamental. Traditional hashing methods often fail when nodes enter or leave the system, causing massive data reshuffling.
Consistent Hashing is a specialized hashing technique that provides hash table functionality in a way that the addition or removal of one slot (node) does not significantly alter the mapping of keys to slots. In few words, in a standard hashing scheme (using the modulo operation mod{n}), changing the number of slots n results in nearly every key being remapped. In contrast, consistent hashing ensures that only K/n keys need to be remapped on average, where K is the number of keys and n is the number of slots.Consistent Hashing emerged as a revolutionary solution to this problem that become a "building block" for many of the most successful distributed architectures in history. Among them, we mention the following applications.
- Content Delivery Networks (CDNs): As pioneered by Akamai, CDNs use it to map web requests to edge servers. It minimizes the need to re-cache content when servers are added to the network to handle peak loads.
- Distributed Databases and NoSQL: Modern distributed databases use Consistent Hashing to achieve horizontal scalability (sharding):
• Apache Cassandra & Amazon Dynamo: Use the hash ring to determine which node stores a particular row of data.
• Virtual Nodes (Vnodes): To solve the problem of uneven data distribution, many systems hash a single physical node to multiple points on the ring, ensuring a more uniform load.
- Load Balancing: Load balancers (like HAProxy or Nginx) use consistent hashing to ensure session persistence. By hashing the client’s IP address, the balancer ensures the user is always routed to the same backend server, even if the server pool changes slightly.
- Distributed Caching: Memcached, one of the most famous distributed memory caching systems, relies heavily on consistent hashing to manage cache hits and misses across a cluster of machines. Focusing on the mathematics and algorithms of consistent hashing, this course requires only an undergraduate STEM background in algebra, probability, and algorithm theory.

Main References
1. Karger, D., et al. (1997). Consistent Hashing and Random Trees: Distributed Caching Protocols for Relieving Hot Spots on the World Wide Web. Proceedings of the Twenty-ninth Annual ACM Symposium on Theory of Computing (STOC).
2. Lewin, D.M. (1998) Consistent hashing and random trees : algorithms for caching in distributed networks. Lewin, Daniel Mark. PhD Thesis, MIT.
3. Leighton, T., & Lewin, D. (1998). Algorithms for Content Delivery. Akamai Technologies Internal Documentation/Foundational Patents.

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].

Consider the connected components of the level sets of a smooth Gaussian field on the d-dimensional Euclidean space. For which levels does an unbounded connected component appear? When it exists, is it unique? What is the typical size of large bounded components and how does it depend on the level? What happens at the critical level? How do the answers to these questions depend on the dimension and on the correlation decay? These natural questions lie at the heart of level-set percolation of smooth gaussian fields, a topic that has seen significant progress over the past decade. In this talk, I will give an overview of several recent results in the field and highlight a number of open problems that remain to be addressed.

We study the plurality consensus problem in distributed systems where a population of extremely simple agents, each initially holding one of k opinions, aims to agree on the initially most frequent one. The h-Majority protocol is arguably the simplest and most studied approach: each agent samples the opinions of h random neighbors and updates its own to the most frequent value in the sample. We present the first upper bound on the convergence time to consensus of the h-Majority dynamics for non-constant values of both h and k, thereby sharpening the theoretical understanding of this classical process. Beyond this, we introduce DéjàVu, a novel and extremely simple mechanism in which an agent queries neighbors until it encounters an opinion for the second time, then adopts that opinion. This rule requires no counters, frequency estimation, or parameter choice (such as a sample size h); it relies solely on the ability to detect repetition. Our analysis shows that DéjàVu is competitive with h-Majority and, in some regimes, substantially more communication-efficient, providing a powerful and minimalist primitive for achieving plurality consensus.
These results are based on joint work with Francesco d'Amore (Gran Sasso Science Institute), George Giakkoupis (INRIA Rennes), Frédéric Giroire and Emanuele Natale.

TBA