Free boundary regularity for the obstacle problem

The classical obstacle problem is to find the equilibrium position of an elastic membrane membrane whose boundary is fixed and forced to lie above a given obstacle.

According to Caffarelli's classical results, the free boundary is smooth outside a set of singular points. However, explicit examples show that the singular set can generally be as large as the regular set. This talk will introduce this beautiful problem and describe some classical and recent results on the regularity of the free boundary.