Understanding deformations of macroscopic thin plates and shells has a long and rich history, culminating with the nonlinear Foeppl-von Karman equations in 1904. However, thermal fluctuations in thin elastic membranes fundamentally alter the long wavelength physics, leading to strongly scale-dependent elastic constants, consistent with experiments that twist and bend atomically-thin free-standing graphene sheets. With thermally excited graphene sheets, one can study as well the quantum mechanics of two dimensional Dirac massless fermions in a fluctuating curved space whose dynamics resembles a simplified form of general relativity. We also describe recent measurements of a scale-dependent bending rigidity for rippled nanometer-thick cantilevers of Al_2O_3. We then move on to analyze the physics of sheets mutilated with puckers and stitches. Puckers and stitches lead to Ising-like phase transitions that strongly affect the physics of the fluctuating sheet. Thermal fluctuations also cause thin spherical shells beyond a certain critical radius to spontaneously collapse.

David Nelson's research focuses on collective effects in the physics and chemistry of condensed matter. He has been interested, in particular, in the interplay between fluctuations, geometry and statistical mechanics. In collaboration with his Harvard colleague, Bertrand I. Halperin, he is responsible for a theory of dislocation-mediated melting in two dimensions. The prediction of Halperin and Nelson of a fourth "hexatic" phase of matter, interposed between the usual solid and liquid phases, has now been confirmed in experiments on thin films and bulk liquid crystals. Nelson's research includes a theory of the structure and statistical mechanics of metallic glasses and investigations of "tethered surfaces", which are two-dimensional generalizations of linear polymer chains.