Abstract: Semiclassical resolvent norms relate dynamics of a particle scattering problem to regularity and decay of waves in a corresponding wave scattering problem. In my talk I will discuss the effect that geometric trapping of particles has on resolvent norms. I will focus in particular on the phenomena of propagation of singularities and quantum tunneling, in the setting of scattering by a compactly supported smooth function in Euclidean space. This talk is based in part on joint works with Long Jin and Andras Vasy.