Waves, Quantum Materials, Metamaterials, and Applications

The project is at the nexus of Applied Mathematics and the Applications of the Physics of Waves to Novel Material Media. The principal investigator’s (PI) goal is to develop mathematical methodologies – analytical to computational - for the prediction of phenomena in new generations of quantum materials (such as graphene) and quantum physics inspired metamaterial variants. These questions are central to understanding of the physical wave effects which play a role in emergent applications of wave phenomena in communication and computing technologies. The PI will study a range of problems in fundamental and applied mathematics. This project focuses on wave propagation in quantum materials, and their synthetic analogues (metamaterials). A part also explores the nonlinear interaction dynamics between fluids across a deforming interface. Some specific topics to be investigated are: (A) energy propagation as waves in bulk two-dimensional (2D) materials, and as ``edge states'' along line defects, such as domain walls within or sharp terminations of the bulk in 2D materials. This is studied in the context of topological materials, for which edge transport properties are completely determined by spectral properties of the bulk ``boundaryless'' structure. Specific topics include: (i) novel effects in quantum materials due to strong magnetic fields (ii) the pseudo-magnetic effect arising due to non-uniform deformations of non-magnetic classical or quantum wave media, and (iii) edge states in non-commensurate structures, e.g. non-periodic / non-translation invariant line defects. (B) nonlinear dispersive wave propagation in discrete and continuous media with novel underlying lattice structures, building on the PI's research on linear wave propagation in novel bulk media with special symmetries, and bulk media with defects. This work is informed by what the PI has learned about effective models in quantum materials governed by linear wave equations in periodic media with novel symmetry. (C) the partial differential equations (PDE)/free surface problem describing nonlinear deformations of a gas bubble immersed in an incompressible liquid, with a view toward extending our results on the radial symmetric case to general deformations. A further direction is one where the liquid is slightly compressible. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.