From Topological Insulators to Hybrid Inverse Problems

这个项目的一个主要目标是研究运输现象，提供一个拓扑描述，在这个意义上说，他们的一些特性是免疫大类的缺陷和不均匀性。由此产生的不对称传输（在一个方向上的传输比另一个方向更大）有可能改变我们在纳米尺度上的电子和光子结构中的通信能力，并使我们能够理解行星尺度上沿沿着向东移动的大规模模式。类似的数学模型将用于本项目的第二个目标，其目的是提高我们对新型高对比度高分辨率医学成像方式的理解，如光声断层扫描或弹性成像。该项目的一个主要组成部分包括培训研究生和指导博士后研究人员从事上述任务，并编制这些数学领域的研究生课程。更具体地说，拓扑保护将通过偏微分算子及其映射来建模，使用非交换几何工具，到具有非平凡指数（拓扑）的Fredholm算子。大类的随机系数添加到微分模型，然后示出不修改拓扑结构，作为一个拓扑障碍，否则预期的安德森本地化。然后将电流形式的物理可观测量分配给拓扑不变量，以提供非对称输运的具体物理表达。该项目的一个主要目标是考虑Floquet拓扑绝缘体领域，其中非平凡的拓扑结构是通过材料特性的高频周期性随时间变化的波动获得的。上述描述涉及到一个前向映射的分析：从微分模型中的异质系数到模型的解。本项目的第二个目标是分析反问题，旨在从解的信息中重建这些系数。首席研究员和他的合作者计划这样做的几个混合逆问题，特别是那些由动力学福克-普朗克方程，其描述了具有高峰值前向散射的混浊介质中的波传播，并在激光通过湍流大气以及近红外中找到了应用，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.