At first sight the subject of elasticity is dull and prosaic. However, a closer look at one’s wastepaper basket, or a peek outside the window shows that there are actually many experimental phenomena challenging investigation – the structure of a crumpled sheet of paper, the rippling of a leaf, the coils of a pea tendril, the cracking of mud, and more –  all beckon the curious scientist willing to veer away from the beaten path, hoping for the occasional sweep of generalization arising from geometry and topology.

One early question we addressed was the approximate shape of a Mobius band, linking topology, geometry and elasticity.  Later studies involved elastic patterns and defects in thin filaments, plates and shells. We provided a  theoretical and experimental description of the fundamental singularity in thin elastic sheets, that is ubiquitous in crumpled paper, origami, drapery, etc. This has helped resolve how approximately isometric deformations of a two-dimensional surface help pack it in three-dimensions.  We also derived a general nonlinear theory of how fine scales such as wrinkling arise in slender structures combining geometry and physics, with predictions for the wavelength and amplitude that were verified by our experiments and data that range over 16 orders of magnitude in scale, from nanotube wrinkling to tectonic subduction.

An exploration of the implications of these instabilities in different physical situations followed, e.g. how nested self-similar wrinkles arise in polymer skins, a theory of self -organized origami to create foldable structures such as the Miura-ori and related fine-scale instabilities in supported and unsupported thin films. Our work on how singularities  and fine scales   form has led to substantial interest in these problems from the engineering community who now use the design principles we uncovered to build surfaces with unusual frictional, optical and other properties, and in the mathematical community interested in a rigorous analysis of the scaling laws.

Another theme is the study of soft material interfaces via the use of simple geometric and scaling arguments combined with approximate analysis and experiments, complementing the large-scale computational approaches that are commonly used. We uncovered a fundamentally new instability in the simplest of these systems, a half-space that is compressed uniformly (in-plane strain) can fold into a scale-free sulcus akin to the structures in the brain – unexplained for more than a century – and yet as ubiquitous and general as buckling and cracking. Our work used experiment, theory, and computations to show that it is akin to a first-order transition, but without a barrier, with a number of physical and mathematical implications that are only just beginning to be explored. In studying soft interfaces and thin solid films, we showed how a confined elastic meniscus can lose stability to the formation of fingers (digits)  in a manner that is similar to the classical Saffman-Taylor instability in hydrodynamics, but via a sub-critical instability that opens the way for patterning thin films digitally.