Given an L2-normalized eigenfunction with eigenvalue λ2 on a Riemannian manifold (M,g) and a nonempty open set Ω⊂M, what lower bound can we prove on the L2-mass of the eigenfunction on Ω? The unique continuation principle gives a bound for any Ω which is exponentially small as λ→∞. On the other hand, microlocal analysis gives a λ-independent lower bound if Ω is large enough, i.e. it satisfies the geometric control condition. This talk presents a λ-independent lower bound for any set Ω in the case when M is a hyperbolic surface. The proof uses microlocal analysis, the chaotic behavior of the geodesic flow, and a new ingredient from harmonic analysis called the Fractal Uncertainty Principle. Applications include control for Schrödinger equation and exponential decay of damped waves. Joint work with Jean Bourgain, Long Jin, and Joshua Zahl.