From Topological Insulators to Hybrid Inverse Problems

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One primary objective of this project is to study transport phenomena that afford a topological description, in the sense that some of their characteristic properties are immune to large classes of defects and inhomogeneities. The resulting asymmetric transport (with larger transport in one direction than the other one) has the potential to transform our communication capabilities in electronic and photonic structures at the nanometer scale as well as allowing us to understand large-scale eastward moving modes along the equator at the planetary scale. Similar mathematical models will be used in the second objective of this project, which aims to improve our understanding of novel high-contrast high-resolution medical imaging modalities such as photo-acoustic tomography or elastography. A major component of the project includes the training of graduate students and mentoring of postdoctoral researchers to work on the aforementioned tasks and the development of graduate courses in these areas of mathematics. More specifically, the topological protection will be modeled by partial differential operators and their mapping, using non-commutative geometry tools, to Fredholm operators with non-trivial index (topology). Large classes of random coefficients added to the differential model are then shown not to modify the topology, as a topological obstruction to the otherwise expected Anderson localization. Physical observables of the form of currents will then be assigned to the topological invariants to provide a concrete physical expression of the asymmetric transport. A major objective of the project is to consider the field of Floquet topological insulators, where the non-trivial topology is obtained by high frequency periodic time-dependent fluctuations of the material's properties. The above descriptions involve the analysis of a forward map: from heterogeneous coefficients in differential models to solutions of the models. The second objective of this project is to analyze the inverse problem, aiming to reconstruct such coefficients from information about the solutions. The principal investigator and his collaborators plan to do so for several hybrid inverse problems, and in particular those modeled by kinetic Fokker-Planck equations, which describe wave propagation in turbid media with highly peaked forward scattering and find applications in laser light through turbulent atmospheres as well as near-infra-red light through biological tissues.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.