Forward and Inverse Problems for Topological Insulators and Kinetic Equations

拓扑绝缘体和动力学方程的正逆问题

• 批准号: 2306411
• 负责人: Guillaume Bal
• 金额: $ 37.5万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306411&HistoricalAwards=false
• 关键词: Forward; Inverse; Problems; Topological; Insulators

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.

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。主要原因是在隔开这些绝缘体的界面上观察到的传输特性：传输受到拓扑保护，因此不受缺陷和杂质的影响。这项建议的主要目的是加深对这类材料的定量理解，并根据散射测量确定其性质。作为第二个目标，首席研究人员将继续发展对几种传输模型的理论理解，这些模型直接应用于新的医学成像模式，如光声断层扫描或多能量计算机断层扫描。该项目还为研究生提供了研究培训机会。在数学上，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，特别是弗洛奎特拓扑绝缘体和由栅极扭曲的双层石墨烯片产生的绝缘体。这将涉及开发积分公式，以准确地模拟这些材料的传输特性。还将设计一种逆散射方法来表征来自远场散射测量的微分模型中的系数。该项目的另一个重要目的是提出一种半经典分析波包沿分隔这些绝缘子的导电界面传播的方法。在医学成像方面，这项建议的主要目标是分析逆传输模型，如福克-普朗克模型和出现在多能量计算机断层扫描中的积分几何设置。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。