Differential Geometry and Analysis for Elastic Rigidity and Flexibility

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The principal investigator pursues several directions of research in mathematics that are of both analytical and geometrical nature and could have applications in nonlinear elasticity and materials science. The questions under study lie at the interface of differential geometry and analysis. This project aims to establish new connections between these disciplines that will be of interest to mathematicians as well as members of the solid mechanics and materials science community. The work aims to advance materials science, particularly regarding the behavior of pre-strained elastic bodies or growing tissues. It is anticipated that the results will have many potential applications, from constructing shape-forming gels to detecting cancerous tumors. This research project further explores the flexibility vs. rigidity dichotomy, particularly in the context of isometric immersions and solutions to Monge-Ampere equations of low regularity. The rigidity and flexibility of solutions to various classes of partial differential equations in geometry and continuum physics has been a longstanding mathematical problem. Recent developments in fluid dynamics have interested mathematicians and materials scientists in analytical and geometrical aspects of these problems. Meanwhile, a rigorous understanding of nonlinear elastic phenomena (including elastic rigidity and flexibility) requires a deep engagement with mathematical disciplines such as differential geometry and nonlinear analysis. The principal investigator plans to use techniques including convex integration, the theory of nonlinear partial differential equations, geometric function theory, and geometric measure theory to tackle these questions.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.