Frequently, real economic agents do not follow purely rational strategies.  These individual non-rational behaviors (due to errors in judgment, incomplete information, emotional bias, etc.) can result in some fascinating organized large-scale structures, which depend on the degree of non-rational behavior.

We look at two such models for Potential Games [Shapley and Monderer]: a dynamical drift-diffusion model, and a static large deviation theory model based on Shannon information entropy and arbitrage.  The equilibrium measure in both cases is the Gibbs measure found in statistical mechanics.  We show that the variables that gauge non-rational behavior in both models are related to “temperature” by a fluctuation-dissipation relation.

A type of localized discrete Cournot oligopoly has a rich phase diagram with an "antiferromagnetic" checkerboard state, striped states and maze-like states with varying widths, and finally a "paramagnetic" unordered state. Such phases have economic implications as to how agents compete given various restrictions on how goods are distributed. 

The theory is also applied to a Speculative and Hedging Model in Oil and U.S. Dollar Markets [Carfi and Musolino] for a single multinational “airline” and many “bank” players.  Based on results for the Nash equilibrium (zero temperature) and preliminary results, there is a phase transition for which a single equilibrium exists at higher non-rational behavior (high temperature), and two equilibria at lower non-rational behavior (low temperature), when the “airline” makes no purchase of oil.  The low temperature phase is in the spirit of the Sonnenschein–Mantel–Debreu theorem, with the extra insight of symmetry-breaking to explain multiple equilibria.  Likewise, Huw Dixon’s result on the “inevitability of collusion” is shown to hold for a Cournot Oligopoly with a Veblen good.  Purely rational neoclassical theory (i.e., Nash equilibrium analysis) alone does not predict this, even though it is observed to occur in more general cases.

Given a locally compact second countable group G, the group von Neumann algebra L(G) is the algebra associated to the invariant subspace decomposition of the left regular representation. It is a natural, and quite difficult, question to address how much of the group structure is recoverable from L(G). That is if two groups have isomorphic group von Neumann algebras what algebraic structure do the groups have in common? In the case of infinite discrete groups, we will explain how if G is a direct product of "indecomposable" groups, such as nonabelian free groups or nonelementary hyperbolic groups, then the product structure can be fully recovered from L(G). This is joint work with Ionut Chifan and Rolando de Santiago.

Kloosterman sums play an important role in modern analytic number theory. I will give a brief survey of what is known about the classical Kloosterman sums and their connection to Maass cusp forms of weight 0 on the full modular group. I will then talk about recent progress toward bounding sums of Kloosterman sums of half-integral weight (joint with Scott Ahlgren) where the estimates are uniform in every parameter. Among other things, this requires us to develop a mean value estimate for coefficients of Maass cusp forms of half-integral weight. As an application, we obtain an improved estimate for the classical problem of bounding the size of the error term in Rademacher’s formula for the partition function.

In the first part of this talk I will do a brief introduction to the recent theory of Mean Field Games (MFG) initiated by J.-M. Lasry and P.-L. Lions. The main objective of the MFG theory is the study of the limit behavior of Nash equilibria for symmetric differential games with a very large number of “small” players. In its simplest form, as the number of players tends to infinity, limits of Nash equilibria can be characterized in terms of the solution of a coupled system of a Hamilton-Jacobi and Fokker-Planck (or continuity) equations. The first equation describes the evolution of the value function of a typical agent, while the second one characterizes the evolution of the agents’ density. In the second part, I will introduce a variational MFG model, where we impose a density constraint. From the modeling point of view, imposing this new constraint means that we are aiming to avoid congestion among the agents. We will see that a weak solution of the system contains an extra term, an additional price imposed on the saturated zones. I will show that this price corresponds to the pressure field from the models of incompressible Euler equations à la Brenier. If time permits, I will discuss the regularity properties of the pressure variable, which allows us to write optimality conditions at the level of single-agent trajectories and to define a weak notion of Nash equilibrium for our model. The talk is based on a joint work with P. Cardaliaguet (Paris Dauphine) and F. Santambrogio (Paris-Sud, Orsay).