Consider two systems, wave propagation in space and waves interacting with an object or medium. In scattering theory one shows that under appropriate circumstances the interacting system behaves like the free one as time goes to infinity and minus infinity. In inverse scattering the goal is to determine the object or medium that produced the scattering from measurements of the scattered waves.

Inverse scattering, in fact, is how we obtain a large part of our information about the world. An example is human vision: from the measurements of scattered light that reaches our retinas, our brains construct a detailed three-dimensional map of the world around us. We know about the interior structure of the Earth by solving inverse acoustic problems, the structure of DNA from solving inverse X-ray diffraction problems, and the structure of the atom and its constituents from studying the scattering when materials are bombarded with particles. The medical imaging technique of ultrasound also uses inverse acoustic scattering.

In the first quarter we will develop in detail the mathematical theory of scattering for the acoustic wave equation and the wave equation plus potential. In the second quarter we will consider the inverse problem for these two cases.

Prerequisites: The Linear Analysis Course or the Real Analysis course. We will assume knowledge of the theory of distributions and the Fourier transform.