Scattering control and inverse problem for the wave equation with piecewise smooth wave speeds Maarten De Hoop Rice University We consider the wave equation with an unknown, piecewise smooth, wave speed on a bounded domain. The wave speed contains a discrete set of conormal singularities. We assume that we can probe the domain from outside with arbitrary Cauchy initial data and observe the wavefield outside the domain for sufficiently large times. We introduce the notion of an almost directly transmitted wave constituent generated by "localized" Cauchy initial values in an exact setting and a microlocal framework. We show that one can obtain exterior Cauchy initial data through scattering control that generate this constituent, at a time equal to some geodesic distance from the boundary of the domain erasing the multiple scattering that would probe the deeper part of the domain. This holds up to a harmonic extension of the first component of the (interior) Cauchy data at the above mentioned time. The scattering control can be implemented as an iteration of Neumann type, involving instantaneous time mirrors, which converges on a set that is dense in the space of all exterior Cauchy initial data. We then prove uniqueness, that is, the recovery of the locations of the discontinuities and the wave speeds in between them. Joint research with P. Caday, V. Katsnelson and G. Uhlmann.