The motivations for this works come from understanding how solutions to some PDE’s or eigenfunctions of Laplace operators can concentrate on small sets. I will describe two kinds of results. First on rational tori when we observe the $L^2$ norms on space time measurable sets and second in more general geometries, under geometric control conditions or on negatively curved compact surfaces when we observe solutions to Schödinger equations on time measurable sets. All the results presented rely on obtaining precise high frequency description/estimates. This is based on joint works with Hui Zhu (NYU Abu Dhabi).
