Forward and Inverse Problems for Topological Insulators and Kinetic Equations

Topological insulators are novel materials with the potential to transform communications capabilities using electronic (spintronics) and photonic structures. The main reason is the transport properties observed at the interface separating such insulators: transmission is protected topologically and hence immune to defects and impurities. The main objective of this proposal is to further the quantitative understanding of such materials and to characterize their properties from scattering measurements. As a second objective, the principal investigator will continue to develop the theoretical understanding of several transport models that find direct applications in novel medical imaging modalities such as Photo-acoustic tomography or multi-energy computerized tomography. This project also provides research training opportunities for graduate students. Mathematically, new tools will be developed in the derivation and analysis of partial differential models to quantify such insulators, in particular Floquet topological insulators and insulators generated by gated twisted bilayer graphene sheets. This will involve the development of integral formulations to accurately simulate transport properties of these materials. An inverse scattering method will also be devised to characterize the coefficients in the differential models from far field scattering measurements. Another important thrust of the project is to propose a semiclassical analysis of wave-packet propagation along conducting interfaces separating such insulators. On the medical imaging side, the main objective of this proposal is to analyze inverse transport models such as the Fokker Planck and the integral geometric settings that appear in multi-energy computerized tomography. 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.