Mathematical Analysis of Multi-dimensional Topological Edge Modes

本提案的目标是开发新的数学技术来分析拓扑边缘模式，这将为超材料设计的快速发展提供信息。这一提议将产生新的分析方法，能够描述正在开发的丰富多样的多维超材料几何。这将导致在多维系统中拓扑边缘模式（局域化强度、鲁棒性、本征频率）与底层材料（拓扑指数、对称性）的性质相关的严格结果。这将揭示对基础物理学的基本新见解，可交付成果（定理，公式，代码）将用于快速准确地指导超材料设计。它还将克服对摄动方法的依赖。这将通过发展具有大于1的一般几何尺寸的几乎周期性（半无限，随机摄动，准周期性）结构的散射建模方法来实现。这将把我在边界积分方法方面的广泛专业知识与主持小组（由伦敦帝国理工学院的理查德·卡斯特领导）在Wiener-Hopf方法方面的专业知识结合起来。此外，来自主机组和帝国理工学院等离子体和超材料中心其他成员在拓扑波导设计方面独特的、世界领先的专业知识的投入，将有助于指导该项目实现高影响力、前沿的成果。这项奖学金将使我有机会开展一个独立的研究项目，并在帝国理工学院等离子体和超材料中心开展高调的合作，从而进一步推动我的职业生涯。我将培养应用数学研究所需的管理、沟通和技术技能，并实现我成为欧洲超材料数学分析领导者的目标。

The goal of this proposal is to develop novel mathematical techniques for analysing topological edge modes that will inform rapid advancements in metamaterial design. This proposal will yield new analytic methods capable of describing the rich variety of multi-dimensional metamaterial geometries being developed. This will lead to rigorous results relating the properties of topological edge modes (localization strength, robustness, eigenfrequency) with those of the underlying materials (topological indices, symmetry) in multi-dimensional systems. This will reveal fundamental new insight into the fundamental physics and the deliverables (theorems, formulas, codes) will be used to rapidly and accurately guide metamaterial design. It will also overcome the reliance on perturbative methods. This will be achieved by developing approaches for modelling scattering by almost periodic (semi-infinite, randomly perturbed, quasi-periodic) structures with general geometries in dimension greater than one. These will combine my extensive expertise in boundary integral methods with the expertise of the host group (led by Richard Craster, at Imperial College London) in the Wiener-Hopf method. Additionally, input from the unique, world-leading expertise in topological waveguide design possessed by the host group and other members of the Imperial College Centre for Plasmonics and Metamaterials will help guide the project towards achieving high-impact, cutting-edge results. This fellowship will further my career by giving me the opportunity to conduct an independent program of research and to develop high-profile collaborations within the Imperial College Centre for Plasmonics and Metamaterials. I will develop the management, communication and technical skills needed for a career of applied mathematical research and to achieve my goal of becoming a European leader on the mathematical analysis of metamaterials.