Mathematical Analysis of Multi-dimensional Topological Edge Modes

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.