Alessio Figalli is Director of the Forschungsinstitut für Mathematik (FIM), a research institute that was founded in 1964 by Beno Eckmann with the objective to promote and facilitate the exchange and cooperation between ETH Zürich and the international mathematical community. He is chaired Professor at ETH Zürich (Zurich, Switzerland). Figalli works in the broad areas of Calculus of Variations and Partial Differential Equations, with particular emphasis on optimal transportation, Monge-Ampère equations, functional and geometric inequalities, elliptic PDEs of local and non-local type, free boundary problems, Hamilton-Jacobi equations, transport equations with rough vector-fields, and random matrix theory. In 2018, Figalli was awarded the Fields Medal "for his contributions to the theory of optimal transport, and its application to partial differential equations, metric geometry, and probability".
ABSTRACT: The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is C^∞ outside a set of singular points. Explicit examples show that the singular set could be in general (n−1)-dimensional — that is, as large as the regular set. In a recent paper with Ros-Oton and Serra we show that, generically, the singular set has zero Hⁿ⁻⁴ measure (in particular, it has codimension 3 inside the free boundary), solving a conjecture of Schaeffer in dimension n ≤ 4. The aim of this talk is to give an overview of these results.