Identifying Blaschke-Santal´o diagrams is an important topic that
essentially consists in determining the image Y = F (X) of a map F : X →
Rd, where the dimension of the source space X is much larger than the
one of the target space. In some cases, that occur for instance in shape
optimization problems, X can even be a subset of an infinite-dimensional
space. The usual Monte Carlo method,...
The goal of this work is to use a phase field method to approximate the notorious Plateau problem, which consists of finding a surface of minimum area that spans a given curve. To this aim, we want to generalise to Plateau’s problem, using a Reifenberg formulation, the functional introduced by M. Bonnivard, A. Lemenant, and F. Santambrogio for Steiner’s problem (searching for the shortest path...
In this talk I will introduce a branched transport problem with weakly imposed boundary conditions. This problem was first derived as a reduced model for pattern formation in type-I superconductors in [1]. For minima of the reduced model with weak boundary conditions, it is conjectured in [2] that the dimension of the boundary measure is non-integer. The conjecture was linked to local scaling...
The Mean Field Game we consider is motivated by the modelization of the housing dynamics where each inhabitant can move from one place to another. In particular, the trajectories of the agents are piecewise constant and they minimize a cost consisting in the number of jumps (or relocations) and two terms depending on the density: the first one is variational and the other one is...
In this talk, we start from a neural-net based approach developed by
Bretin et al (Bretin, Denis, Masnou, Terii, 2022), which extends the
classical phase-field method for the mean curvature flow of boundaries
to the case of non-oriented interfaces. We introduce an analytical,
energy-based approach which yields an evolution PDE reproducing the
numerical experiments obtained with the neural...
We consider zero-sum dynamic games of information revelation. Each player controls a martingale of beliefs by choosing in each period a particular splitting (balayée) of the current belief, and the payoffs are determined by the limit beliefs. We introduce constraints on the set of available splittings, and characterize the value of the game as the unique solution of an extended...
Decision makers act based on the data they observe. However, the nature of the true data-generating process is often only partially known: we model such partial knowledge as a set of moment conditions. Given the partial information available, we consider an heuristic model of belief formation derived from the maximization of the Shannon entropy. This paper characterizes the outcomes that can...
In this talk, I will introduce the problem of learning in zero-sum game, and especially for the problem of "last-iterate" convergence, unlike the traditional literature that looks at the average convergence (we argue it makes more sense). The interesting property is that the optimal rate is T^{-1/4} which is quite unusual (and unexpected) in this literature.
I will motivate the study of optimal control problems of systems described by positive measures, namely for optimization and large deviations of mean field systems. I will explain why the associated Hamilton-Jacobi equation plays a crucial role in these problems, as well as the main mathematical challenges it raises. I will focus in particular on the difficulties arising when trying to prove a...
Une conséquence bien connue de l’inégalité de Prékopa–Leindler est la préservation de la log-concavité par le semi-groupe de la chaleur. Cette propriété ne s’étend cependant pas aux semi-groupes plus généraux. Nous étudions donc une notion de log-concavité plus faible qui peut être propagée le long de semi-groupes de chaleur généralisés.
Nous en déduisons des plusieurs propriétés. Ici, je...
We introduce a new class of perfect information repeated zero-sum games in which the payoff of one player is the escape rate of a switched dynamical system which evolves according to a nonexpansive nonlinear operator depending on the actions of both players. This is motivated by applications to population dynamics (growth maximization and minimization).
Considering order preserving finite...
Nous étudions des problèmes variationnels dans des espaces de Banach faisant intervenir des énergies sous-modulaires. Nous étendons la notion de substituabilité à ce cadre de dimension infinie et montrons qu’elle est en dualité avec la sous-modularité. Ces deux notions nous permettent de dériver des principes de comparaison de manière abstraite. Nous appliquons ensuite nos résultats aux...
Ce séminaire portera sur des avancées récentes sur une inégalité isopérimétrique quantitative dans le plan, avec une contrainte géométrique.
Plus précisément si $\Omega$ est un ouvert borné, nous étudierons l'inégalité de type
$$
\delta(\Omega)\geq C \lambda_0^2(\Omega,B)\,,
$$
où $\delta(\Omega)$ est le déficit isopérimétrique (différence entre le périmètre de $\Omega$ et le...
