Appears in collection : Not Only Scalar Curvature Seminar

The classical obstacle problem involves determining the equilibrium state of an elastic membrane that is restricted to remain above a predetermined obstacle. By classical results of Caffarelli, the 'free boundary' — the demarcation line where the obstacle and membrane meet — is smooth outside a set of singular points. Also, explicit examples show that the singular set could be, in general, as large as the regular set. This talk aims to introduce this beautiful problem and describe some classical and recent results on the regularity of the free boundary.