Some Geometric Inverse Problems

This thesis compiles my work on four projects.

The first project is the study of MP-systems and rigidity. In the second project, we study the linearized problem related to boundary action rigidity, that is, we study the X-ray transform associated to MP-systems. The third project studies scattering rigidity using timelike geodesics for standard stationary geometries. In the fourth project, we explore a Calderón's problem for harmonic maps.