I work in nonlinear partial differential equations and the calculus of variations. Much of my work concerns problems from physics and materials science. One current theme is the study of elastic-energy-driven pattern formation in thin elastic sheets; this work brings a variational perspective to phenomena such as wrinkling, folding, and delamination. It is of course a familiar fact that thin sheets often wrinkle or fold: our skin wrinkles and our clothes wrinkle; leaves, flowers, and hanging drapes have folds. Physical experiments in controlled settings can quantify such phenomena, and numerical simulations can demonstrate within a model how the patterns develop. But neither experiment nor simulation can tell us "why" a system chooses a particular pattern. My variational perspective (applied in recent work with Jacob Bedrossian, Peter Bella, Jeremy Brandman, and Hoai-Minh Nguyen) provides a valuable complement to other methods, by showing that elastic energy minimization requires certain types of patterns.
A rather different theme involves prediction with expert advice -- a widely used paradigm for machine learning. Here nonlinear partial differential equations arise by considering continuum limits and taking advantage of analogies with optimal control. A current project with Kangping Zhu applies this viewpoint to the "stock prediction problem" -- a model problem involving prediction of a binary sequence -- using ideas from my 2006 work with Sylvia Serfaty on motion by curvature.