Rough Solutions of Hyperbolic Monge-Ampere Equations With Applications to Non-Euclidean Elasticity.

Soft materials—such as leaves, flowers, sea slugs, and corals—bend, twist, and ripple into complex shapes that are both beautiful and functional. These systems belong to a broader class of materials known as soft matter, which are characterized by their ability to deform easily and organize themselves into larger structures with collective, emergent behavior. This project investigates a particularly intriguing subset of soft materials called hyperbolic non-Euclidean plates—thin sheets with built-in curvature that causes them to spontaneously buckle into wavy or ruffled shapes. These forms are not only common in nature, but also offer new possibilities for the design of smart materials and soft robots. By uncovering the rules that govern the shapes and behaviors of these systems, the project advances our understanding of geometry in natural design and supports the development of next-generation materials inspired by biology. This research contributes to the national interest by promoting the progress of science through the development of new mathematical and computational tools for studying nonlinear systems and emergent behavior. It strengthens the connection between mathematics, physics, and engineering, while also offering applications in biology and materials science. The project supports education and workforce development by providing research training for undergraduates, graduate students, and postdoctoral scholars. Through its interdisciplinary scope and training opportunities, the project fosters innovation and builds capacity for a competitive STEM workforce for the nation. The investigator studies the mechanics of non-Euclidean elastic thin sheets, with a focus on hyperbolic geometries that lead to spontaneously formed complex shapes. Recent work by the investigator has identified universal behaviors in these systems that are mediated by geometric defects—line-like and point-like singularities that emerge as non-smooth solutions to hyperbolic Monge-Ampère-type equations. These singular structures provide a framework for understanding the shape selection and evolution of thin hyperbolic sheets. The project develops a suite of analytical and numerical tools grounded in differential geometry, topology, and functional analysis to describe and predict these singular structures. These methods are applied to problems in biomechanics—including plant morphogenesis and marine invertebrate forms—as well as the design of soft robotic actuators and reconfigurable materials. The research advances the theory of singular solutions to nonlinear PDEs and provides a unified framework to understand geometric frustration in thin structures across biology and engineering. 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.