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).

The classical Sylvester-Gallai theorem states the following: Given a finite set of points in the 2-dimensional Euclidean plane, not all collinear, there must exist a line containing exactly 2 points (referred to as an ordinary line). In a recent result, Green and Tao were able to give optimal lower bounds on the number of ordinary lines for large finite point sets.

In this talk we will consider the situation over the complex numbers. While the Sylvester-Gallai theorem as stated is false in the complex plane, Kelly's theorem states that if a finite point set in 3-dimensional complex space is not contained in a plane, then there must exist an ordinary line. Using techniques developed for bounding ranks of design matrices, we will show that either such a point set must determine at least 3n/2 ordinary lines or at least n-1 of the points are contained in a plane. (Joint work with Z. Dvir, S. Saraf and C.Wolf.)

Geometric analysis in differential geometry is a powerful tool in Riemannian geometry. It has been used to solve many problems in Riemannian geometry. In the field of several complex variables, it was not the most popular weapon to attack questions. One of the reasons is that many problems in the several complex variables relates to some types of differential equations of complex-valued functions which is currently not well understood. In this talk, we consider problems in the Diederich-Forn\ae ss index with a viewpoint of geometric analysis and see what we obtain. This is a joint work with Krantz and Peloso.

Some prominent conjectures in the theory of C*-algebras ask whether or not every C*-algebra of a particular form embeds into an ultrapower of a particular C*-algebra.  For example, the Kirchberg Embedding Problem asks whether every C*-algebra embeds into an ultrapower of the Cuntz algebra O_2.  In this series of lectures, we show how techniques from model theory, most notably model-theoretic forcing, can be used to give nontrivial reformulations of these conjectures.  We will start from scratch, assuming no knowledge of C*-algebras nor model theory.