Amick Lectures in Applied Mathematics

On Mean Field Games

Thursday, October 2, 4:30 PM, Eckhart 133, 1118 E. 58th St.

This talk will be a general presentation of Mean Field Games (MFG in short), a new class of mathematical models and problems introduced and studied in collaboration with Jean-Michel Lasry. Roughly speaking, MFG are mathematical models that aim to describe the behavior of a very large number of "agents" who optimize their decisions while taking into account and interacting with the other agents. The derivation of MFG, which can be justified rigorously from Nash equilibria for N player games, letting N go to infinity, leads to new nonlinear systems involving ordinary differential equations or partial differential equations. Many classical systems are particular cases of MFG like, for example, compressible Euler equations, Hartree equations, porous media equations, semilinear elliptic equations, Hamilton-Jacobi-Bellman equations, Vlasov-Boltzmann models... In this talk we shall explain in a very simple example how MFG models are derived and present some overview of the theory, its connections with many other fields and its applications.


Symmetric functions of a large number of variables

Friday, October 3, 4:00 PM, Eckhart 202, 1118 E. 58th St.

In this talk, we present the general mathematical tools needed to justify the derivation of Mean Field Games models. It turns out that these tools have many other applications : Large deviations for Stochastic Partial Differential Equations and applications to Physics, interacting systems of stochastic particles, Nonlinear Partial Differential Equations (NLPDE in short) in large dimensions, mass transportation theory... We shall present the natural setup for all these asymptotic problems is the space of probability measures (the so-called Wasserstein space), the differential calculus and NLPDE on this space deduced from such limits.


Mean Field Games : Analysis and Applications

Monday, October 6, 4:00 PM, Eckhart 202, 1118 E 57th St.

In this talk, we present some mathematical results on MFG systems both for Ordinary Differential Equations and Partial Differential Equations (existence, uniqueness, stability, approximation...). And we discuss briefly some applications to Economics and Finance.