Collaborative Research: Topological Dynamics of Hyperbolic and Fractal Lattices

这笔赠款将用于研究，极大地扩大未来吸振材料和结构设计的设计空间，并应用于能量收集和声学面板技术，从而促进科学进步，促进国家繁荣。这种材料的波导性取决于单个振荡器元件的基本空间模式。虽然与使用规则多边形细分平面的简单图案相关联的行为是众所周知的，但我们对其他类型图案驾驭、引导和定位波浪的能力的了解存在差距。这个项目将通过发现与这种新的模式相关联的全新的波导物理来填补这一空白，包括在多个尺度上具有自相似特征的分形图。这项工作的实验部分将揭示制造具有所需图案的声晶体以及表征给定晶体图案的问题的解决方案，从而在材料科学、声学和力学领域开辟新的研究方向。该项目的合作研究生态系统，其中纯数学满足计算建模和物理验证，将为本科生和研究生以及博士后研究人员提供独特的培训机会。推广项目将通过专门的实践活动让初中生和教师接触到几何学、拓扑学和动力学方面的高级主题。本研究旨在为波导型超材料的数学理论以及为实验验证或设计而物理实现此类结构的能力做出基础性贡献。它将通过为拓扑动力学的分类和体积边界原理在双曲和分形格中的可能表现形式制定一个理论框架来实现这一结果。这项研究将进一步展示如何控制这类晶格的固有自由度，以实现新形式的波控、相位控制、边缘和体模局域化以及拓扑泵浦。这项实验工作将展示生物灵感包装和设计解决方案，用于大规模制造非周期晶格。该项目将扩展我们关于晶格集体动力学的知识，并将提供分析工具、数学模型和实验平台，帮助绘制新型晶格几何图形的复杂景观，以及它们可能在未来材料和结构设计中的应用。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。

This grant will fund research that dramatically enlarges the design space for future vibration absorbing materials and structural designs, with applications to energy harvesting and acoustic panel technologies, thereby promoting the progress of science and advancing the national prosperity. The wave guiding properties of such materials depend on an underlying spatial pattern of individual oscillator elements. While the behavior associated with simple patterns that tessellate the plane using regular polygons is well understood, there is a gap in our knowledge of the ability of other classes of patterns to steer, guide, and localize waves. This project will fill this gap by discovering radically new wave-guiding physics associated with such new classes of patterns, including fractals with self-similar features at multiple scales. The experimental part of this work will uncover solutions to the problems of fabricating acoustic crystals with a desired pattern, as well as characterizing the pattern of a given crystal, opening up new research directions in materials science, acoustics, and mechanics. The project’s collaborative research ecosystem, where pure mathematics meets computational modeling and physical validation, will provide unique training opportunities for both undergraduate and graduate students, as well as for postdoctoral researchers. Outreach programs will expose middle- and high-school students and teachers to advanced topics in geometry, topology, and dynamics through dedicated and hands-on activities.This research aims to make fundamental contributions to the mathematical theory of wave-guiding metamaterials that can be characterized as hyperbolic or fractal lattices, as well as to the ability to physically realize such structures for experimental validation or design. It will achieve this outcome by formulating a theoretical framework for the classification of topological dynamics and of the possible manifestations of the bulk-boundary principle in hyperbolic and fractal lattices. The research will further demonstrate how intrinsic degrees of freedom of such lattices may be controlled to achieve new forms of wave steering, phase control, edge and bulk mode localization, and topological pumping. The experimental effort will demonstrate bioinspired packing and design solutions for large-scale fabrication of aperiodic lattices. The project will expand our knowledge about the collective dynamics of lattices, and will deliver analysis tools, mathematical models, and experimental platforms that will help chart the complex landscape of novel lattice geometries and their possible application for future material and structural designs.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.