Scattering theory is motivated physically by a desire to understand and control the propagation of waves such as sound or light waves. Moreover, measurements of waves can be used in the identification of objects or perturbations. The latter use inspires inverse scattering theory. The project will study several such inverse problems--one is the problem of determining a set with analytic boundary in Euclidean space from information about the scattering matrix in a very limited number of directions, and another is the problem of determining a radial potential from its resonances. Resonances are poles of certain meromorphic functions and are, in some sense, analogous to eigenvalues. Another problem involves further understanding the distribution of resonances in Euclidean scattering. Other projects involve manifolds with infinite cylindrical ends. In particular, a goal is to improve our understanding of how the geometric structure of such manifolds is reflected in analytic objects such as the scattering matrix or the set of resonances.
Radar, sonar, and medical imaging provide examples of inverse scattering theory in practical use. Resonances are of interest to physicists and chemists, for whom a resonance corresponds to a metastable state, containing information about both the frequency and the rate of decay. The class of manifolds with infinite cylindrical ends include waveguides, which are variously used in modeling semiconductors and the ocean. Thus advances in the mathematical understanding of these problems may lead to advances in other disciplines.
