Removing a properly embedded ray from a (noncompact) manifold does not affect the topology nor the diffeotype. What about the symplectic analogue of this fact? And can we go beyond rays? I will show how to use incomplete Hamiltonian flows to excise interesting subsets: the product of a ray with a Cantor set, a "box with a tail", and - more generally - epigraphs of lower semicontinuous functions. This is based on joint work with Xiudi Tang, in which we answer a question of Alan Weinstein.