Silver Professor of Mathematics

kohn@cims.nyu.edu

212-998-3217

Warren Weaver Hall, Office 502

Ph.D., Mathematics, Princeton University, USA, 1979.

M.Sc., Mathematics, University of Warwick, UK, 1975.

A.B., Mathematics, Harvard University, USA, 1974.

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.

P. Bella and R.V. Kohn,
"Wrinkles as the result of compressive stresses in an annular thin film",
Communications on Pure and Applied Mathematics
67, 693-747 (2014)

R.V. Kohn and H.-M. Nguyen,
"Analysis of a compressed thin film bonded to a compliant substrate: the energy scaling law",
Journal of Nonlinear Science
23, 343-362 (2013)

R.V. Kohn and S. Serfaty,
"A deterministic-control-based approach to motion by curvature",
Communications on Pure and Applied Mathematics
59, 344-407 (2006)