Calderon's inverse conductivity problem in the plane

We show that the Dirichlet to Neumann map for the equation del center dot sigma del u = 0 in a two-dimensional domain uniquely determines the bounded measurable conductivity a. This gives a positive answer to a question of A. P. Calderon from 1980. Earlier the result has been shown only for conductivities that are sufficiently smooth. In higher dimensions the problem remains open.