A three dimensional acoustic medium (operator is $\partial_t^2 {-} \Delta_x + q(x)$) is probed by plane waves coming from infinity and the medium response is measured at infinity. One aims to recover the acoustic property (the function $q(x)$) from this type of measurement. Two such longstanding open problems are the “fixed angle” scattering problem and the “back-scattering” problem. Both these problems involve studying the injectivity and the inversion of some non-linear map from compactly supported smooth functions on $\mathbb{R}^3$ to distributions on $\mathbb{R}^3$. These maps are defined through non-explicit solutions of the perturbed wave equation - the existence and uniqueness of these solutions is well understood.

We will state these two problems, describe what was known and then state our (with Mikko Salo of University of Jyvaskyla, Finland) injectivity result for the “fixed angle scattering” problem. If time permits we will describe some of the ideas used to prove our result for the fixed angle scattering problem – Carleman estimates for the perturbed wave equation play a big role.