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

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · University of Arizona (AZ) · $240,000

## Abstract

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 oppor

## Key facts

- **NSF award ID:** 2511503
- **Awardee organization:** University of Arizona (AZ)
- **SAM.gov UEI:** ED44Y3W6P7B9
- **PI:** Shankar C Venkataramani
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $240,000
- **Funds obligated:** $240,000
- **Transaction type:** Standard Grant
- **Period:** 07/15/2025 → 06/30/2028

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2511503

## Citation

> US National Science Foundation, Award 2511503, Rough Solutions of Hyperbolic Monge-Ampere Equations With Applications to Non-Euclidean Elasticity.. Retrieved via AI Analytics 2026-06-08 from https://api.ai-analytics.org/grant/nsf/2511503. Licensed CC0.

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