Collaborative Research: Topological Dynamics of Hyperbolic and Fractal Lattices

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.