Mathematical Sciences: Microlocal Analysis and Nonlinear Waves

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This project involves the use and development of microlocal techniques to study three categories of nonlinear wave phenomena: diffraction by smooth obstacles, diffraction by nonsmooth obstacles, and transmission from one medium to another. In all three cases we are concerned with the behavior of singularity- bearing rays as well as with progressing waves. The aim is to produce propagation of regularity theorems along with examples demonstrating the optimality of those theorems. In the latter two categories, the research addresses some basic questions concerning existence which remain open even in the linear case.