A fundamental question raised by A. Calderon in 1980 asks whether or not one can determine the conductivity of a body from knowledge of steady state direct current measurements at the boundary. This problem has attracted much attention in the past few years and forms the basis for work to be done on this project. The mathematical model of conductivity is a nonlinear elliptic differential operator, defined in a domain in a space of any dimension, with prescribed boundary values. The boundary values are viewed as the induced voltage potential; the solution as the resulting potential inside the domain. This is a standard partial differential equation. The real problem occurs when one does not have the boundary potential but instead is given the current flux density across the boundary. This is a quantity which can actually be measured in the physical world. Significant progress has been made in this area in the case of domains of dimension three or greater, especially when there is considerable smoothness to the boundary. In dimension two, the inverse problem is more difficult. Good results are only available for conductivity which is close to constant. The most recent work in two dimensions will be continued in this project. Mathematically, the problem consists of showing that the flux density and conductivity are in a one-one relationship. What is now known is that a local univalency implies global. Actually, this only holds in a dense set of problems. The work to be done is to establish the local-global result in general and to determine methods for testing the local condition.
