Constructing Hyperbolic Leaves and Flowers

Series: Program in Applied Mathematics Colloquium
Location: Math 501
Presenter: Toby Shearman, Program in Applied Mathematics, University of Arizona

Leaves, much like the edges of torn plastic sheets, often display intricate buckling patterns. This fine-scale wrinkling found in thin objects from Nature is postulated to be the result of minimizing an elastic energy, an energy with a range of relevant length scales. In this way, elasticity represents a prototypical variational problem with inherent multiple-scales phenomena where oscillations are a natural feature.

As we explore such elastic surfaces, I will present a novel mechanism through which these complex configurations can arise in a purely geometric setting, i.e. in the absence of stretching. And in true Al Scott fashion, we tackle the fully nonlinear geometry.

Using a discrete geometric construction, I will motivate the role smoothness plays in the extrinsic geometry as well as the energetic consequences of introducing "monkey saddle" weak-singularities or "branch points." Moreover, we will explore a nontrivial topological degree associated with such points and discuss its computational importance. Finally, I will mention some work towards understanding growth of hyperbolic surfaces.

Al Scott Prize & Lecture