Abstract:
-- Please note the unusual time for this seminar --

A basic fact of geometry is that there are no length-preserving maps from a sphere to the plane. But what happens if you confine a thin elastic shell, which prefers to be a curved surface but can deform in an approximately isometric way, to reside nearby a plane? It wrinkles, and forms a remarkable pattern of peaks and troughs, the arrangement of which is sometimes random, sometimes not depending on the shell. After a brief introduction to the mathematics of thin elastic shells, this talk will focus on a new set of simple, geometric rules we have discovered for predicting wrinkle patterns driven by confinement. These rules are the latest output from an ongoing study of highly wrinkled shells using the tools of Gamma-convergence and convex analysis. The asymptotic expansions they encode reveal a beautiful and unexpected connection between opposite curvatures --- apparently, surfaces with positive or negative intrinsic Gaussian curvatures can be paired according to the way that they wrinkle when confined. Our predictions match the results of numerous experiments and simulations done with Eleni Katifori (U. Penn) and Joey Paulsen (Syracuse). Underlying their analysis is a certain class of interpolation inequalities of Gagliardo-Nirenberg type, whose best prefactors encode the optimal patterns.