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Modèles d'EDP en Sciences Sociales

Les sciences sociales posent des défis importants à la modélisation mathématique, en particulier pour concevoir des méthodes prédictives. Les Equations aux Dérivées Partielles sont aujourd'hui un outil majeur dans ce domaine.

Ce workshop a présenté plusieurs cas d'étude de modèles d'EDP en sciences sociales : dynamiques d'opinion et réalisation de consensus, mouvements de foule, économie et distribution de la richesse.

Cet événement a été organisé par la Fondation Sciences Mathématiques de Paris et a constitué la clôture de la Chaire d'excellence de Peter Markowich (lauréat en 2011) au LJLL et au CEREMADE. Cette journée a eu lieu le Vendredi 24 Janvier 2014 au Laboratoire Jacques‑Louis Lions, UPMC.

Programme scientifique

    Laurent Boudin (LJLL, UPMC) : Mathematical Models for Opinion Dynamics Marie‑Therese Wolfram (KAUST, KSA) : On the Mathematical Modeling and Simulation of Crowd Motion Guillaume Carlier (CEREMADE Dauphine) : Equilibria in Games with a Continuum of Agents and Transport Giuseppe Toscani (Universita’ di Pavia, Italy) : Knowledge and Ingenuity: A Kinetic Approach Pierre‑Emmanuel Jabin (University of Maryland, USA) : Convergence to Consensus in Models with a Finite Range of Interactions Peter Markowich (KAUST, KSA) : Allocution de clôture


Jean Dolbeault (CEREMADE), Peter Markowich (KAUST) et Benoît Perthame (LJLL)

Les exposés

Mathematical Models for Opinion Dynamics par Laurent Boudin
About a decade ago mathematicians started to study opinion dynamics models. Various viewpoints have been proposed. In this talk, we shall mostly focus on kinetic models for opinion formation and discuss various phenomena which may affect the evolution of opinions inside a closed community.
Retrouvez ici les diapositives de cet exposé.

On the Mathematical Modeling and Simulation of Crowd Motion par Marie‑Therese Wolfram
Pedestrian crowds exhibit complex and coordinated behaviors, which result from the social interactions among individuals. The understanding of these microscopic interactions as well as the complex behavior on the macroscopic level pose challenging problems for the modeling, analysis and numerical simulations.
In this talk we present mathematical modeling approaches for pedestrian motion on the microscopic level and discuss their mean field limit as the number of individuals tends to infinity. The resulting macroscopic models are in general highly nonlinear partial differential equations or systems thereof.
Hence we focus on particular analytic aspects and numerical simulations of the limiting equations to understand and capture the complex behavior of pedestrian crowds.
Retrouvez ici les diapositives de cet exposé.

Equilibria in Games with a Continuum of Agents and Transport par Guillaume Carlier
In this talk, I will describe several models with a continuum of agents where one can obtain equilibria by minimization arguments and in particular tools from optimal transport. I will address matching problems for teams (joint with Ivar Ekeland) and Cournot‑Nash equilibria (joint with Adrien Blanchet). Existence, uniqueness and characterization of equilibria will be discussed and some numerical simulations will also be presented.
Retrouvez ici les diapositives de cet exposé.

Knowledge and Ingenuity: A Kinetic Approach par Giuseppe Toscani
In recent years the distribution of wealth in multi‑agent societies has been investigated by resorting to classical methods of kinetic theory of rariefed gases. In analogy with the Boltzmann equation, the change of wealth in these models is due to microscopic binary tradings among agents. Surprisingly, other important aspects linked to dfferent types of human wealth, like knowledge (information) and ingenuity, have not been taken into consideration. In this lecture, we aim to present an effective model to study both the distribution of knowledge and ingenuity in a society of agents, based on microscopic rules of change, which include both the increase of information and the necessity to discard the unnecessary part of it. The so‑obtained kinetic model is then studied in detail.
Retrouvez ici les diapositives de cet exposé.

Convergence to Consensus in Models with a Finite Range of Interactions par Pierre‑Emmanuel Jabin
We study the long time behavior of some opinion dynamics models. The evolution of the opinion of one individual depends on the opinions of other individual through a given influence function. The models under consideration here are a generalization of the so‑called Krause model which makes the interaction between individuals non symmetric. Because of this loss of symmetry, the long time behavior was essentially unknown in the realistic case of interactions with a finite range. We are able to show the convergence to an equilibrium consisting of several local consensus which do not interact anymore. This is a joint work with S. Motsch.

Allocution de clôture de Peter Markowich

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