**Title:** Inside Out: Inverse Boundary Problems

**Speaker:** Professor Gunther Uhlmann

**Speaker Info:** University of Washington

**Brief Description:**

**Special Note**:

**Abstract:**

Inverse boundary problems are a class of problems in which one seeks to determine the internal properties of a medium by performing measurements along the boundary of the medium. These inverse problems arise in many important physical situations, ranging from geophysics to medical imaging to the non-destructive evaluation of materials.The appropriate mathematical model of the physical situation is usually given by a partial differential equation (or a system of such equations) inside the medium. The boundary measurements are then encoded in a certain boundary map. The inverse boundary problem is to determine the coefficients of the partial differential equation inside the medium from knowledge of the boundary map.

The prototypical example of an inverse boundary problem is the inverse conductivity problem, also called electrical impedance tomography, first proposed by A. P. Calder\'on. In this case the boundary map is the voltage to current map; that is, the map assigns to a voltage potential on the boundary of a medium the corresponding induced current flux at the boundary of the medium. The inverse problem is to recover the electrical conductivity of the medium from the boundary map.

This problem can be recast in geometric terms as determining a Riemannian manifold from the DN map associated to the Laplace-Beltrami operator. We will discuss recent results concerning the determination of the manifold itself from this information.

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