Mathematical problems from materials science

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Kohn1311833 This project has three main scientific thrusts: (a) The first and broadest thrust concerns elastic-energy-driven pattern formation in thin elastic sheets. The investigator studies stress-driven patterns involving wrinkles, folds, delamination, and other defects, with particular emphasis on situations where the energy-minimizing pattern develops fine-scale structure as the sheet thickness tends to zero. His approach is to focus on how the minimum energy scales with respect to the sheet thickness and other relevant physical parameters. (b) The second thrust concerns surface-energy-driven coarsening of two-phase mixtures. The investigator studies a family of nonlocal evolutions from the physics literature, which generalize the relatively well-understood model of "Cahn-Hilliard dynamics." The focus here is on understanding the large-time coarsening rate. (c) The third thrust concerns prediction with expert advice (a topic from the machine learning literature). The investigator's goal is a fresh perspective on some regret-minimization-based algorithms for prediction. He views regret minimization as a robust control problem and considers a suitable scaling limit in which the associated value function solves a differential equation. The investigator studies three interdisciplinary topics. The first two lie at the interface where mathematics meets physics and materials science, while the third lies at the interface with machine learning. In each area, challenges from applications drive the development of new mathematical methods. For example, the work on thin elastic sheets is helping develop a theory of energy-minimizing patterns, in much the same way that consideration of soap bubbles and soap films led to the theory of minimal surfaces a generation ago. 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. The project provides a valuable complement to other methods, by showing that elastic energy minimization requires types of patterns. The project provides training opportunities for graduate students.