Does knowledge of the volumes of embedded minimal hypersurfaces in a compact Riemannian manifold $(M,g)$ with boundary
uniquely determine the metric $g$? By measuring at the surface of the earth the response of an induced seismic wave, can one uniquely recover the speed of sound below ground? What bridges these otherwise unrelated inverse problems is that both naturally give rise to transforms that are Fourier integral operators (FIOs). Thereby motivated, upon presenting basic tools of analytic microlocal analysis relating support and analytic wavefront set, we develop a recipe that reduces uniqueness questions to the following analytic microlocal statement that we prove: one can recover the analytic wavefront set of a distribution from its image under a general class of elliptic analytic FIO. Exploiting this framework, we show that the conformal perturbation of an analytic metric can be recovered in a H\"older stable way from minimal hypersurface volume data, provided a sufficiently rich family of minimal hypersurfaces exists. For the seismic inversion problem we show that perturbations of an analytic sound speed are uniquely determined by boundary measurements under some geometric conditions. This talk is based partly on joint work with Tony Liimatainen, Mikko Salo and Leo Tzou.
