I will explain how the results of Bourgain, Burq and the speaker '13 can be used to obtain control and observability by rough functions and sets on 2-tori. We show that for the time dependent Schr\"odinger equation, any set of positive measure can be used for observability and controllability.

For non-empty open sets this follows from the results of Haraux '89 and Jaffard '90, while for sufficiently long times and rational tori this can be deduced from the results of Jakobson '97.

Other than tori (of any dimension; cf. Komornik '91, Anantharaman--Macia '14) the only compact manifolds for which observability holds for any non-empty open sets are hyperbolic surfaces. That follows from results of Bourgain--Dyatlov '16 and Dyatlov--Jin '17 and I will discuss the difficulty of passing to rougher rougher sets in that case. Joint work with N Burq.