"Fast and accurate structure-preserving algorithms in computational scattering, and interdisciplinary modeling and simulation"

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Mathematics provides an elegant framework to understand physical and biological phenomena, and the tools with which to probe the detailed behaviour of these systems. In this research program, I will be investigating some specific physical and biological systems from a mathematical viewpoint. A key goal is to develop computational techniques to help further our understanding. We have all experienced wave scattering in our lives. Wireless technology, light refracting in fibre-optic cable, oceanic currents or radar waves bouncing off airplanes - scattering is ubiquitous. Scattering theory describes the interaction of waves- acoustic, electromagnetic, elastic, fluid- with obstacles. The proposed research develops numerical algorithms to simulate these interactions in novel situations, study the resultant physics, and investigate novel applications without expensive experimentation. The development of these computational tools presents many interesting mathematical and computational challenges, which we address in the program. In the other part of this research program, we bring mathematical tools to bear on problems arising in biology. We continue our efforts to understand the growth and behaviour of bone cells in the body, including the profound role of these cells in bone-metastasizing cancers. We seek to understand at a fundamental level why muscles bulge. Mathematical and computational tools will be developed to ask: what is the effect of changed muscle architecture and fiber properties on muscle force output? In both instances, the biological and physical situation is quite complex, and experiments to understand these are expensive, time-consuming and frequently impractical. Mathematics can be used to shed light on the behaviour of these systems. We can decide which parameters are truly important, and then develop efficient computer algorithms. Interesting mathematical issues will arise, which will warrant a more rigorous theoretical investigation.