Idisplays

The Boltzmann equation can be derived rigorously from a system of elastic hard spheres (Lanford?s theorem, [11], [8]). Kinetic theory may also be fruitfully used to model large systems of particles that interact inelastically (sand, snow, interstellar dust, see [3], [10], [4]). Such materials are known as granular media. The theory enables for instance to explain the onset of inhomogeneities, as well as to quantify the decay of the temperature (Haff?s law, [9], [3]). In this case, the derivation of the inelastic Boltzmann equation is still open, mainly due to the complicated dynamics of the particles. In particular, it is still unknown if the dynamics of such particle systems is well-posed. One major difficulty comes from the phenomenon of inelastic collapse. A system of particles is said to experience an inelastic collapse when infinitely many collisions take place in finite time. It is known that inelastic collapse may take place for systems of only three particles [12]. We studied systems of three particles, in dimension d ? 2. Assuming that the restitution coefficient r is constant, we obtained 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. In the first case we recover that the particles arrange in a nearly-linear chain, already studied by Zhou and Kadanoff [13], and in the second case we obtain that the particles arrange in 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 an initial configuration leading to a nearly-linear collapse, stable under perturbations, and such that the angle between the particles at the time of collapse can be chosen a priori, with an arbitrary precision. Another important question is the following: since inelastic collapse can take place, is it possible to continue the dynamics of the particles anyway? We report also partial results in this direction. Considering on the other hand another law of collision, prescribing that a fixed quantity of kinetic energy is lost during each collision, we obtained results on systems of an arbitrary number of particles interacting according to this law, that look a priori contradictory. Namely, we proved that the flow of such a system of particles conserves the measure in the phase space, whereas the kinetic energy is not conserved. From these results, we deduce an Alexander?s theorem [1] for such systems of particles: for almost every initial datum, the dynamics of such systems is globally well-posed. To the best of our knowledge, this is the first result of global well-posedness concerning the dynamics of systems of inelastic particles. The results are taken from [5], [7], [6], obtained in collaboration with Juan J. L. Velázquez (Universität Bonn).

The study of wave propagation on black hole spacetimes has been an intense field of research in the past decades. This interest has been driven by the stability problem for black holes and by questions related to scattering theory. On Kerr black holes, the analysis of Maxwell's equations and the equations of linearized gravity, can be simplified by introducing the Teukolsky equation, which offers the advantage of being scalar in nature. After explaining this reduction, I will present a result providing the large time leading-order term for initially localized and regular solutions of the Teukolsky equation, valid for the full subextremal range of black hole parameters and for all spins. I will explain how such a development follows naturally from the precise analysis of the resolvent operator on the real axis. Recent advances in microlocal analysis are used to establish the existence and mapping properties of the resolvent.

This talk is about a joint work with Aris Papadopoulos and Blaise Boissonneau [BPT]. Starting from any first order structure S, Mekler constructs in [M] a 2-nilpotent group of prime exponent M=(G, ·) which interprets, in the pure language of groups, the structure S. This 2-nilpotent group shares numerous model theoretical properties with the structure S, notably in terms of dividing lines: M is Stable (resp. Simple, NIP_n for every n, Strong, NTP2...) if and only if S satisfies this property. See [CH]. I will motivate these results and show how one can generalise some of them, by considering a uniform hierarchy of dividing lines, introduced in [GHS]: the NC_K-hierarchy, which rises from coding (or not coding) Ramsey classes of structures K . I will also state a transfer principle for stably embedded pairs of Mekler groups (all these notions will be defined). Our method, that I will briefly sketch, was to establish new relative quantifier elimination results, and was inspired by a step-by-step approach for proving transfer principles in valued fields. [BPT] Boissonneau, Papadopoulos and T., Mekler's Construction and Murphy's Law for 2-Nilpotent Groups, arXiv:2403.20270. [GHS] Guingona, Hill and Scow, Characterizing model-theoretic dividing lines via collapse of generalized indiscernibles. [M] Mekler, Stability of nilpotent groups of class 2 and prime exponent. [CH] Chernikov and Hempel, Mekler's construction and generalized stability.

Already Euler computed the values $\zeta(2), \zeta(4), \zeta(6),\ldots$ of the Riemann zeta function $\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$ to be \begin{equation*} \zeta(2k)=-\frac{(2\pi i)^{2k}}{2(2k!)}B_{2k} \end{equation*} where $B_{2k}\in \mathbb{Q}$ are the Bernoulli numbers. This formula can be seen as the easiest case of a vast conjecture by Deligne from 1977, which relates special values of $L$-functions of arithmetic varieties and their periods. In this talk we want to give a non-technical introduction to the Deligne conjecture, aimed at general mathematical audience. In the end we discuss very recent developments, which lead to a complete proof in the case of Hecke $L$-functions.

Given a finitely generated, residually finite group $G$, we ask which properties can be detected from the set of its finite quotients, encoded in its profinite completion. After investigating recent advances in the context of profinite rigidity, we will focus on the interplay between profinite completions and the notion of amenability. We will see, following a construction by S. Kionke and E. Schesler, that amenability is not a profinite invariant by using tools related to automorphisms of rooted trees