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

数学为理解物理和生物现象提供了一个优雅的框架，也为探索这些系统的详细行为提供了工具。在这个研究项目中，我将从数学的角度研究一些具体的物理和生物系统。一个关键的目标是发展计算技术来帮助我们进一步理解。** **我们在生活中都经历过波散射。无线技术、光纤电缆中的光折射、海流或飞机反射的雷达波——散射无处不在。散射理论描述了声波、电磁波、弹性波、流体波与障碍物的相互作用。提出的研究开发了数值算法来模拟这些相互作用在新的情况下，研究由此产生的物理，并研究新的应用，而不需要昂贵的实验。这些计算工具的发展提出了许多有趣的数学和计算挑战，我们在程序中解决。**在这个研究项目的另一部分，我们使用数学工具来解决生物学中出现的问题。我们继续努力了解骨细胞在体内的生长和行为，包括这些细胞在骨转移性癌症中的深刻作用。我们试图从根本上理解肌肉膨胀的原因。将开发数学和计算工具来询问：改变肌肉结构和纤维特性对肌肉力量输出的影响是什么？在这两种情况下，生物和物理情况都相当复杂，为了了解这些情况而进行的实验既昂贵又耗时，而且往往不切实际。数学可以用来阐明这些系统的行为。我们可以确定哪些参数是真正重要的，然后开发有效的计算机算法。有趣的数学问题将出现，这将保证更严格的理论研究。**************

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.**************