Mathematical Analysis of Multi-dimensional Topological Edge Modes

该提案的目标是开发新的数学技术，用于分析拓扑边缘模式，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多维超材料几何形状的丰富多样性。这将导致严格的结果拓扑边缘模式（本地化强度，鲁棒性，本征频率）与那些底层材料（拓扑指数，对称性）在多维系统中的属性。这将揭示对基础物理学的基本新见解，并且可交付成果（定理，公式，代码）将用于快速准确地指导超材料设计。它还将克服对微扰方法的依赖。这将通过开发方法来模拟散射的几乎周期性（半无限，随机扰动，准周期性）结构的一般几何尺寸大于一个。这些将结合联合收割机我在边界积分方法的广泛专业知识和主办小组（由理查德·克拉斯特领导，在帝国理工学院伦敦）在维纳-霍普夫方法的专业知识。此外，主办方和帝国理工学院等离子体和超材料中心的其他成员在拓扑波导设计方面拥有的独特的世界领先的专业知识将有助于指导该项目实现高影响力的尖端成果。这个奖学金将进一步我的职业生涯，让我有机会进行一个独立的研究计划，并在帝国理工学院等离子体和超材料中心内发展高调的合作。我将开发应用数学研究的职业生涯所需的管理，沟通和技术技能，并实现我的目标，成为超材料数学分析的欧洲领导者。

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.