We propose a mean field kinetic model for systems of rational agents interacting in a game theoretical framework. This model is inspired from non-cooperative anonymous games with a continuum of players and Mean-Field Games. The large time behavior of the system is given by a macroscopic closure with a Nash equilibrium serving as the local thermodynamic equilibrium. Applications of the presented theory to social and economical models will be given.[-]

In this talk we first quickly present a classical and simple model used to describe flow in porous media (based on Darcy's Law). The high heterogeneity of the media and the lack of data are taken into account by the use of random permability fields. We then present some mathematical particularities of the random fields frequently used for such applications and the corresponding theoretical and numerical issues.
After giving a short overview of various applications of this basic model, we study in more detail the problem of the contamination of an aquifer by migration of pollutants. We present a numerical method to compute the mean spreading of a diffusive set of particles representing a tracer plume in an advecting flow field. We deal with the uncertainty thanks to a Monte Carlo method and use a stochastic particle method to approximate the solution of the transport-diffusion equation. Error estimates will be established and numerical results (obtained by A.Beaudoin et al. using PARADIS Software) will be presented. In particular the influence of the molecular diffusion and the heterogeneity on the asymptotic longitudinal macrodispersion will be investigated thanks to numerical experiments. Studying qualitatively and quantitatively the influence of molecular diffusion, correlation length and standard deviation is an important question in hydrogeolgy.[-]

Consider a problem of Markovian trajectories of particles for which you are trying to estimate the probability of a event.
Under the assumption that you can represent this event as the last event of a nested sequence of events, it is possible to design a splitting algorithm to estimate the probability of the last event in an efficient way. Moreover you can obtain a sequence of trajectories which realize this particular event, giving access to statistical representation of quantities conditionally to realize the event.
In this talk I will present the "Adaptive Multilevel Splitting" algorithm and its application to various toy models. I will explain why it creates an unbiased estimator of a probability, and I will give results obtained from numerical simulations.[-]

The Ceresa class is the image under a cycle class map of a canonical algebraic cycle associated to a curve in its Jacobian. This class vanishes for all hyperelliptic curves, and is known to be non-vanishing for the generic curve of genus at least 3. It is necessary for the Ceresa class to have infinite order for the Galois action on the fundamental group of a curve to have big image. We will present an algorithm for certifying that a curve over a number field has infinite order Ceresa class.

N.B. This is preliminary joint work with Jordan Ellenberg, Adam Logan and Akshay Venkatesh.[-]

It is of soaring demand to develop statistical analysis tools that are robust against contamination as well as preserving individual data owners' privacy. In spite of the fact that both topics host a rich body of literature, to the best of our knowledge, we are the first to systematically study the connections between the optimality under Huber's contamination model and the local differential privacy (LDP) constraints. We start with a general minimax lower bound result, which disentangles the costs of being robust against Huber's contamination and preserving LDP. We further study four concrete examples: a two-point testing problem, a potentially-diverging mean estimation problem, a nonparametric density estimation problem and a univariate median estimation problem. For each problem, we demonstrate procedures that are optimal in the presence of both contamination and LDP constraints, comment on the connections with the state-of-the-art methods that are only studied under either contamination or privacy constraints, and unveil the connections between robustness and LDP via partially answering whether LDP procedures are robust and whether robust procedures can be efficiently privatised. Overall, our work showcases a promising prospect of joint study for robustness and local differential privacy.
This is joint work with Mengchu Li and Yi Yu.[-]

Dans cet exposé, nous introduirons certaines chaînes de Markov simples, dites “montantes-descendantes”, sur les permutations et les graphes. Une étape de la chaîne consiste à dupliquer un élément aléatoire de la permutation ou un sommet aléatoire du graphe (pas montant), puis à supprimer un autre élément/sommet aléatoire (pas descendant). Nous prouvons que ces chaînes convergent dans la limite des grandes tailles et après renormalisation du temps vers une diffusion de Feller sur l'espace des permutons et des graphons, respectivement. Nous obtenons également une formule explicite pour la distance de séparation entre la distribution des chaînes après n pas, excluant l'apparition d'un phénomène de “cut-off”. Notre approche fonctionne dans un cadre plus général : il est basé sur des relations de commutation entre les opérateurs des pas montants et descendants, et s'inspire des travaux de Fulman, Olshanski et Borodin–Olshanski sur l'espace des partitions et le simplex de Thoma. Je ne supposerai aucune connaissance préalable des permutons, graphons, diffusions de Feller, distances de séparation, seuils, ... Travail joint (et encore en cours) avec Kelvin Rivera-Lopez, Gonzaga University.[-]