Understanding deformations of macroscopic thin plates and shells has a long and rich history, culminating with the Foeppl-von Karman equations in 1904, characterized by a dimensionless coupling constant (the "Foeppl-von Karman number") that can easily reach vK = 10^7 in an ordinary sheet of writing paper. However, thermal fluctuations in thin elastic membranes fundamentally alter the long wavelength physics. We discuss the remarkable properties of free-standing graphene sheets (with vK = 10^13!) at room temperature, where enhancements of the bending rigidity by factors of ~4000 compared to T = 0 values have now been observed. Thermalized elastic membranes can undergo a crumpling transition when the microscopic bending stiffness is comparable to kT. We argue that the crumpling temperature can be dramatically reduced by inserting a regular lattice of laser-cut perforations. These expectations are confirmed by extensive molecular dynamics simulations, which also reveal a remarkable “frame crumpling transition” triggered by a simple large hole inserted into a graphene sheet. We show finally that thin amorphous spherical shells with a background Gaussian curvature are inevitably (in the absence of a stabilizing pressure difference) crushed by thermal fluctuations beyond a critical size, of order 160 nm for graphene at room temperature.

Professor Nelson obtained PhD degree in Physics from Cornell University under the supervision of Prof. Michael E. Fisher, at age 24. Among numerous honors and awards, he was the recipient of the Bardeen Prize in 2003 and the APS Oliver E. Buckley Prize in 2004. He was elected a member of the National Academy of Sciences (USA) in 1994. His 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. 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. His 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. He has also studied the flux line entanglement in the high temperature superconductors, in particular the melted flux liquid whose physics is important for many of the proposed applications. His current interests include vortex physics, the statistical mechanics of polymers, topological defects on frozen topographies and biophysics.