Particle channeling occurs when an energetic charged particle beam is aligned with a major crystal axis or plane. Channeling has contributed in important ways to our understanding of particle motion in solids at the fundamental level and just as significantly has led to numerous applications in physics and technology. Furthermore, channeling is a new source of interesting and challenging mathematical problems in dynamical systems and stochastic processes. A complete understanding of channeling requires detailed knowledge of the channeling particle motion and the spatial and momentum densities of the particles as they penetrate the crystal. Ellison's mathematical approach to the study of channeling will continue including: channeling effects in strain-layer superlattices and other distorted crystals, channeling radiation and related quantum electrodynamical processes; multiple scattering effects viewed as particle motion in a random media; and channeling in perfect crystals viewed within the context of dynamical systems. This research will deepen our knowledge of channeling, extend its usefulness and lead to new applications. Furthermore, the dynamical systems and probabilistic methods developed to solve these problems will be of general applicability and will have importance in other areas of mathematical science.
