In seismology the travel times of elastic waves can give information about the different layers of the earth. In medical imaging a two-dimensional image of the electrical conductivity can help localizing tumors or strokes. What these problems have in common is that they are so called inverse problems, where one wants to recover material parameters, layers of the earth and conductivity, from boundary measurements.

In this talk I will touch upon inverse problems related to the elastic wave equation and the conductivity equation. The stiffness tensor is a measure of the elasticity of the material. It is modelled as a fourth rank tensor and I will address how this tensor behaves under coordinate transformations (that maintain the symmetries of the tensor) and what information of this tensor can be recovered at the boundary. Additionally, I consider an inverse problem for the conductivity equation, where we want to recover the conductivity at the boundary of an object from interior data. The interior data corresponds to two boundary functions imposed as a boundary condition to the PDE. I address under which conditions on these boundary functions a reconstruction is possible. This is also addressed in a limited view setting, where only a limited part of the boundary of the object is accessible.