In settings where there is no existence theory for a partial differential equation, one can still ask whether a solution of this equation, when it does exist, is unique. In this talk, we discuss this unique continuation problem for geometric wave equations, as well as the Carleman estimates which form the main analytical tool for solving these problems.
If time permits, we then discuss some recent results (joint with Spyros Alexakis and Volker Schlue) on unique continuation of waves from infinity for various asymptotically flat spacetimes. In particular, these results apply to a large family of spacetimes that are relevant in general relativity, including the Minkowski, Schwarzschild, and Kerr solutions, as well as many dynamical spacetimes.