The theory of mean field games (MFG for short) aims to study limiting behavior of Nash equilibria of (stochastic) differential games when the number of agents tends to infinity. While in general existence of MFG Nash equilibria can be established under fairly general assumptions, uniqueness is the exception rather than the rule.

How does an elastic membrane lie on a given obstacle? This naive looking question hides a beautiful mathematical theory integrating powerful tools with far-reaching applications. In this mini-course we will discuss the obstacle problem as a free boundary model. All necessary tools will be carefully constructed from scratch. We will mainly focus on: optimal regularity of solutions, non-degeneracy estimates, weak geometric-measure properties of the free boundary, classification of global profiles, and differentiability of the free boundary.

The theory of mean field games (MFG for short) aims to study limiting behavior of Nash equilibria of (stochastic) differential games when the number of agents tends to infinity. While in general existence of MFG Nash equilibria can be established under fairly general assumptions, uniqueness is the exception rather than the rule.

How does an elastic membrane lie on a given obstacle? This naive looking question hides a beautiful mathematical theory integrating powerful tools with far-reaching applications. In this mini-course we will discuss the obstacle problem as a free boundary model. All necessary tools will be carefully constructed from scratch. We will mainly focus on: optimal regularity of solutions, non-degeneracy estimates, weak geometric-measure properties of the free boundary, classification of global profiles, and differentiability of the free boundary.

The theory of mean field games (MFG for short) aims to study limiting behavior of Nash equilibria of (stochastic) differential games when the number of agents tends to infinity. While in general existence of MFG Nash equilibria can be established under fairly general assumptions, uniqueness is the exception rather than the rule. For finite horizon mean field games uniqueness typically holds if the time horizon is small (or the data satisfies other smallness conditions).

How does an elastic membrane lie on a given obstacle? This naive looking question hides a beautiful mathematical theory integrating powerful tools with far-reaching applications. In this mini-course we will discuss the obstacle problem as a free boundary model. All necessary tools will be carefully constructed from scratch. We will mainly focus on: optimal regularity of solutions, non-degeneracy estimates, weak geometric-measure properties of the free boundary, classification of global profiles, and differentiability of the free boundary.