Recent progress in the Calderon problem

The inverse conductivity problem, posed by A.P. Calderon in 1980, consists in determining the coefficient $A$ in the elliptic PDE $div(A\mathrm{\nabla }u)=0$ from the Cauchy data of its solutions. This problem is the mathematical model for Electrical Impedance Tomography. Various harmonic analysis, PDE and geometric techniques come into play in its study, and the Calderon problem remains a central question in the theory of inverse problems. We will survey known results and open questions, focusing on issues with low regularity, partial data and matrix coefficients.