Geodesic X-ray tomography arises in geomathematics as the linearized travel time problem of planets. Planets have non-smooth geometry as the sound speed is generally non-smooth, can have jump discontinuities and other extreme behavior. In this talk we consider the question: How non-smooth can Riemannian geometry be for the X-ray transform of scalar functions (and tensor fields) to remain injective? We prove that the X-ray transform is (solenoidally) injective on Lipschitz functions (tensor fields) when the Riemannian geometry is simple $C^{1,1}$. The class $C^{1,1}$ is the natural lower bound on regularity to have a well-defined X-ray transform. Our proofs are based on energy estimates derived from a Pestov identity, which lives on the non-smooth unit sphere bundle of the manifold. The talk is based on joint work with Joonas Ilmavirta.

In this talk we address the Euclidean elastic wave equation under change of variables and extend this to Riemannian geometry. This is inspired by previous research that has concerned the principal behavior of the Euclidean elastic wave equation under coordinate transformations. Further research has concerned how the density normalized stiffness tensor gives rise to a Finsler metric. With this in mind we will touch upon what one can say about the density and stiffness tensor fields that give rise to the same Finsler metric. In this context we will talk about how this will affect the full elastic wave equation and not only the principal behavior.

Seismic waves can be modeled by the elastic wave equation, which has two material parameters: the stiffness tensor and the density. The inverse problem is to reconstruct these two fields from boundary data, and the stiffness tensor can be anisotropic. I will discuss how this problem can be tackled by geometric methods and how that leads to geometric inverse problems in Finsler geometry. This talk is related several other talks in the same minisymposium.

Stiffness tensors serve as a fingerprint of a material. We describe how to harness anisotropy, using standard tools from algebraic geometry (e.g., generic geometric integrality, upper-semicontinuity of some standard functions, and Gröbner bases) to uniquely reconstruct the stiffness tensor of a general anisotropic material from an analytically small neighborhood of its corresponding slowness surface.