The lecture, a continuation of last year's, continues the presentation of a new theory called " mean-field games theory ", developed in collaboration with Mr. Jean-Michel Lasry. The aim of this theory is to introduce, justify, analyze and apply in different contexts a new class of mathematical models for studying the collective behavior of a very large number of interacting agents (or players in the sense of game theory), all of whom wish to " optimize " their decisions.
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This year's lecture focused on the theoretical mathematical framework needed to justify (rigorously) the limit when the number of players N tends towards infinity. This mathematical framework enables us to understand the asymptotic behavior of symmetric functions of a large number of variables and their differential calculus, as well as symmetric solutions of very high-dimensional PDEs. A. Grunbaum's earlier work [1] contains a sketch of the abstract general framework we are introducing, and A. Matytsin's heuristic considerations [2]. Finally, let us point out that the results we present apply to many subjects other than mean-field game theory : PDEs with N variables when N tends to infinity, large deviations for stochastic PDEs, transport theory and interacting (possibly stochastic) particle systems.