Forward and Inverse Problems for Topological Insulators and Kinetic Equations

拓扑绝缘体是一种新型材料，具有利用电子（自旋电子学）和光子结构改变通信能力的潜力。主要原因是在分离这种绝缘体的界面上观察到的传输特性：传输在拓扑上受到保护，因此不受缺陷和杂质的影响。本提案的主要目的是进一步定量了解这些材料，并通过散射测量来表征它们的特性。作为第二个目标，首席研究员将继续发展对几种传输模型的理论理解，这些模型可以直接应用于新型医学成像模式，如光声断层扫描或多能计算机断层扫描。本项目也为研究生提供了研究训练的机会。数学上，将开发新的工具来推导和分析偏微分模型，以量化这种绝缘体，特别是Floquet拓扑绝缘体和门控扭曲双层石墨烯片产生的绝缘体。这将涉及到积分公式的发展，以准确地模拟这些材料的输运性质。还将设计一种逆散射方法来表征远场散射测量的微分模型中的系数。该项目的另一个重要推力是提出波包沿分离这种绝缘体的导电界面传播的半经典分析。在医学成像方面，本提案的主要目标是分析反输运模型，如福克普朗克模型和多能计算机断层扫描中出现的积分几何设置。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。

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.