Institut de recherche mathématique avancée

Résumé : Abstract: About 20 years ago, Kontsevich & Suhov conjectured the existence and uniqueness of a family of measures on the set of Jordan curves, characterised by conformal invariance and another property called "conformal restriction". This conjecture was motivated by (seemingly unrelated) works of Schramm, Lawler & Werner on stochastic Loewner evolutions (SLE), and Malliavin, Airault & Thalmaier on "unitarising measures". The existence of this family was settled by works of Werner-Kemppainen and Zhan, using a loop version of SLE. The uniqueness was recently obtained in a joint work with Jego. I will start by reviewing the different notions involved before giving some ideas of our proof of uniqueness: in a nutshell, we construct a family of "orthogonal polynomials" which completely characterise the measure. In the remaining time, I will discuss the broader context in which our construction fits, namely the conformal field theory associated with SLE.

Résumé : There are many interesting interplays between representation theory of algebras and combinatorics. In this talk, I will give a tour of some of these interactions using friezes interpreted in different contexts. This presentation serves as an exposition of two collaborative works with K. Baur, L. Bittmann, E. Gunawan, and G. Todorov and E. Kantarcı Oğuz.

Résumé : The Boltzmann equation can be derived rigorously from a system of elastic hard spheres (Lanford’s theorem). In the case of large systems of particles that interact inelastically (sand, snow, interstellar dust), the derivation of the inelastic Boltzmann equation is still open. One major difficulty, already at the microscopic level, comes from the phenomenon of inelastic collapse, when infinitely many collisions take place in finite time.

Assuming that the restitution coefficient r is constant, we obtain general results of convergence and asymptotics concerning the variables of the dynamical system describing a collapsing system of particles. We prove a complete classification of the singularities when a collapse of three particles takes place, obtaining only two possible orders of collisions between the particles: either the particles arrange in a nearly-linear chain (studied in [3]), or they form a triangle, and we show that, after sufficiently many collisions, the particles collide according to a unique order of collisions, which is periodic. Finally, we construct explicit initial configurations leading to a nearly-linear collapse in a stable way, such that the angle between the particles at the time of collapse can be chosen a priori, with an arbitrary precision.

The results are taken from [1] and [2], obtained in collaboration with Juan J. L. Velázquez (Universität Bonn).

[1] Théophile Dolmaire, Juan J. L. Velázquez, “Collapse of inelastic hard spheres in dimension d ≥ 2”, to appear in Journal of Nonlinear Science, preprint arXiv:2402.13803v2 (02/2024).
[2] Théophile Dolmaire, Juan J. L. Velázquez, “Properties of some dynamical systems for three collapsing inelastic particles”, preprint arXiv:2403.16905 (03/2024).
[3] Tong Zhou, Leo P. Kadanoff, “Inelastic collapse of three particles”, Physical Review E, 54:1, 623–628 (07/1996).

Résumé : Le théorème de Maschke affirme que les représentations complexes d'un groupe fini sont complètement réductibles. Leur étude se ramène ainsi à celle des représentations irréductibles, et on dispose d'outils pour construire ces dernières. Les choses sont plus compliquées sur un corps de caractéristique positive. Ici, on doit classifier les représentations indécomposables. Ces dernières sont constituées de représentations irréductibles assemblées au moyen d'extensions. Dans les années 70, Gabriel a eu l'idée de représenter graphiquement ces extensions par des flèches, donnant naissance à la théorie des carquois. Mon ambition dans cet exposé est d'utiliser cette méthode pour décrire toutes les représentations indécomposables du groupe symétrique S_3 sur un corps de caractéristique 3.