From Topological Insulators to Hybrid Inverse Problems

该项目的一个主要目标是研究提供拓扑描述的输运现象，从某种意义上说，它们的一些特征性质不受大类缺陷和非均质性的影响。由此产生的不对称传输（一个方向上的传输比另一个方向上的传输更大）有可能改变我们在纳米尺度上的电子和光子结构的通信能力，并使我们能够理解沿赤道在行星尺度上的大规模向东移动模式。类似的数学模型将用于该项目的第二个目标，其目的是提高我们对新型高对比度高分辨率医学成像模式的理解，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，并在这些数学领域开发研究生课程。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，到具有非平凡索引（拓扑）的Fredholm算子。添加到微分模型中的大量随机系数不会改变拓扑结构，这是对预期的安德森定位的拓扑障碍。然后将电流形式的物理观测值分配给拓扑不变量，以提供不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，其中非平凡拓扑是通过材料性质的高频周期性时间相关波动获得的。上述描述涉及到从微分模型的非均质系数到模型解的正演图的分析。该项目的第二个目标是分析逆问题，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对几个混合反问题进行这样的研究，特别是那些由动力学福克-普朗克方程建模的问题，该方程描述了波在具有高峰值前向散射的浑浊介质中的传播，并在通过湍流大气的激光以及通过生物组织的近红外光中得到应用。该奖项反映了美国国家科学基金会的法定使命，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.