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.
Alessio Figalli is professor of mathematics at ETH Zurich and Director of the Forschungsinstitut für Mathematik (FIM). He obtained a doctorate in 2007 and, in the same year, was nominated as a researcher in the National Centre for French Scientific Research (CNRS); the following year he taught at the École Polytechnique in Paris and, in 2009, in the United States at the University of Texas in Austin. In 2013 he became Robert Lee Moore chaired full professor there. Since 2016, he has been a professor in Switzerland where he obtained a chair at the Federal Institute of Technology in Zurich. In 2018 he won the Fields Medal, the most prestigious award in matemathics, for his work on optimal transport theory.
The talk will take place Thursday July 2nd at 4PM.
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Photo © École polytechnique - J.Barande