abstract: This may be subject to change. The localization problem concerns eigenfunctions for an elliptic operator (for instance, Laplacian on a complicated domain, or Schrodinger on a simple one). We (with M. Filoche, D. Jerison, S. Mayboroda) propose a free boundary problem to find subregions where the eigenfunctions are likely to be localized, and study regularity properties of the minimizers.