Waves, Novel Two-Dimensional Materials, and Applications

从通信网络到量子计算，许多已经实现和设想的科学技术进步的核心是在非常小的空间和时间尺度上强有力地传输、引导和操纵各种形式的能量，例如电子和光学能量的能力。这是通过制造新的介质来实现的，在这种介质中，以波的形式传播的能量与工程微特征相互作用。一种新颖的、自然发生的、引起人们强烈兴趣的介质的当前例子是石墨烯，这是一种单原子厚度的单层碳原子，以蜂窝结构在平面内排列，延伸到宏观尺度。石墨烯是常温下已知的最具导电性的物质(电子和热学)。这种材料一直处于凝聚态物理和材料科学革命的中心，也就是众所周知的二维材料。在石墨烯中观察到的许多新现象与波在具有新对称性的介质中的一般数学性质有关，因此物理学家和工程师们设计和研究了具有类似一般性质的材料，称为“人造石墨烯”，着眼于在光学、通信、信息存储和计算方面的应用。皮温斯坦将研究基础数学和应用数学中的一系列问题，这些问题与波在这种新介质中的存在、稳定性和传输特性有关，蜂窝结构是2D材料的范例，以及与非线性系统中波的动力学有关的问题。这项工作与NSF的十大想法之一Quantum Leap紧密结合在一起。该奖项将支持一名研究生接受使用建模、分析和计算的多方面方法进行应用数学研究的培训，重点是复杂介质中的波现象对重要科学问题的应用。Pi Weinstein将研究(A)在由连续介质PDE模型、拓扑保护态和周期介质的整体性质与边缘效应之间的关系支配的新型2D周期介质中沿线缺陷的能量传播，以及(B)离散和连续非线性波动系统中相干结构的动力学，特别是(I)在非线性格子中的传输和(Ii)可压缩流体中的自由边界问题的动力学。(A)2D材料的许多显著性质(例如，非凡的导电性)与其能带结构的微妙性质(Floquite-Bloch理论的色散面和本征模的集合)有关，这决定了能量输运。石墨烯色散表面的锥形奇点(Dirac点)是其显著电子迁移率的核心，其能带结构的拓扑性质提供了有关石墨烯与自由空间或其他材料界面时可能发生的能量传输类型的信息。因此，正是通过分析、偏微分方程和拓扑学的融合概念，人们才能使过程理解这些新材料及其人工类似物。(B)这项研究是PI在非线性波方面的长期研究计划的一部分，该计划现在强调i)相干结构在离散非线性色散系统中的传播，以及ii)具有无限多个内部自由度的相干结构的非线性PDE动力学，例如流体中气泡的非线性振荡。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。

At the heart of many realized and envisioned advances in science and technology --from communication networks to quantum computation -- is the ability to robustly transport, channel, and manipulate forms of energy, e.g. electronic and optical, on very small spatial and temporal scales. This is achieved by fabricating novel media in which energy, which propagates as waves, interacts with engineered microfeatures. A current example of a novel, naturally occurring, medium of intense interest is graphene a one-atom thick mono-layer of carbon atoms, arranged in-plane in a honeycomb structure, extending to the macro-scale. Graphene is the most conductive substance known at room temperature (electronically and thermally). This material has been at a center of a revolution in Condensed Matter Physics and Materials Science, known as the Two-Dimensional Material. Many of the novel phenomena observed in graphene relate to general mathematical properties of waves in media with novel symmetry, and hence physicists and engineers have engineered and investigated materials with similar general properties, dubbed "artificial graphene," with a view on applications to optics, communications, information storage and computing. PI Weinstein will study a range of problems in fundamental and applied mathematics related to the existence, stability and transport properties of waves in such novel media, with honeycomb structures as a paradigm for 2D materials, as well as questions concerning the dynamics of waves in nonlinear systems. This work is firmly aligned with Quantum Leap, one of the NSF's 10 Big Ideas. This award will support one graduate student to be trained in a multifaceted approach to applied mathematical research using modeling, analysis and computation, with emphasis on applications of wave phenomena in complex media to the scientifically important questions. PI Weinstein will investigate (A) Energy propagation along line-defects in novel 2D periodic media governed by continuum PDE models, topologically protected states and the relation between bulk properties of periodic media and edge effects, and (B) Dynamics of coherent structures in discrete and continuous nonlinear wave systems, in particular (i) transport in nonlinear lattices and (ii) dynamics of a free boundary problem in compressible fluids. (A) Many remarkable properties of 2D materials (e.g. extraordinary conductivity) are related to the subtle properties of its band structure (the collection of dispersion surfaces and eigenmodes of Floquet-Bloch theory) which dictate energy transport. The conical singularities (Dirac points) of the dispersion surfaces of graphene are at the core of its remarkable electron mobility, and the topological properties of its band structure properties give information about the types of energy-transport can occur when graphene is interfaced with free-space or another material. Thus, it is through a convergence notion from Analysis, PDE and Topology that one can make process understanding these novel materials and their artificial analogues. (B) This research is part of the PI's long-standing research program in nonlinear waves, which now emphasizes i) coherent structure propagation in discrete nonlinear dispersive systems and ii) the nonlinear PDE dynamics of coherent structures with infinitely many internal degrees of freedom, as exemplified by the nonlinear oscillations of a bubble in the fluid.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.