Electrical impedance tomography and Calderón problem

We survey mathematical developments in the inverse method of Electrical Impedance Tomography which consists in determining the electrical properties of a medium by making voltage and current measurements at the boundary of the medium. In the mathematical literature this is also known as Calderón’s problem from Calderón’s pioneer contribution [23]. We concentrate this article around the topic of complex geometrical optics solutions that have led to many advances in the field. In the last section we review some counterexamples to Calderón’s problems that have attracted a lot of interest because of connections with cloaking and invisibility