Abstract

The classical inverse boundary value problem of Caldern consists in determining the conductivity inside a body from the Dirichlet-to-Neumann map. The problem is of interest in medical imaging and geophysics, where one seeks to image the conductivity of a body by making voltage and current measurements at its surface.

Bukhgeim-Uhlmann and Kenig-Sjstrand-Uhlmann have shown that (in dimensions three and higher) uniqueness in the above problem holds even if measurements are available on possibly very small subsets of the boundary. This course will explain in detail a constructive proof of these results, obtained in joint work with Brian Street.

The topics I hope to cover are: 1. Review of the reconstruction method for Caldern’s problem with full data. 2. New Green’s functions for the Laplacian. 3. Boundedness properties of the corresponding single layer operators. 4. New solutions of the Schrdinger equation. 5. Unique solvability of the main boundary integral equation involving only the partial Cauchy data.