Mathematical seismology and inverse problems

The classical Newtonian view is that planets move on curved paths because of the force exerted by the Sun, but Einstein postulated that planets move straight in a curved geometry whose curvature is caused by the Sun. We propose a similar change of views to seismology: Sound waves from earthquakes are not curved due to variations in sound speed, but they move straight in a somewhat exotic geometry determined by the said variations. Such a geometrization of elasticity allows us to use the powerful tools of modern geometry to tackle problems like determining the interior structure of the Earth or Mars using measurements on the surface.