Mathematical problems from materials science

科恩1311833 该项目有三个主要的科学目标：（a）第一个也是最广泛的目标是在薄弹性片中形成弹性能量驱动的图案。 研究人员研究应力驱动的模式，包括皱纹，折叠，分层，和其他缺陷，特别强调的情况下，能量最小化的模式发展精细尺度的结构，因为板厚度趋于零。 他的方法是专注于最小能量如何与片材厚度和其他相关物理参数成比例。(b)第二个推力涉及表面能驱动的两相混合物的粗化。 研究者从物理学文献中研究了一族非定域演化，它推广了相对较好理解的“Cahn-Hilliard动力学”模型。“这里的重点是了解大时间粗化率。(c)第三个重点是关于专家建议的预测（机器学习文献中的一个主题）。 研究人员的目标是对一些基于后悔最小化的预测算法提供一个新的视角。 他认为遗憾最小化作为一个强大的控制问题，并认为一个合适的比例限制，其中相关的价值函数解决了微分方程。 研究人员研究三个跨学科的主题。 前两者位于数学与物理学和材料科学的接口，而第三个则位于与机器学习的接口。 在每个领域，来自应用的挑战推动了新数学方法的发展。 例如，对弹性薄板的研究有助于发展能量最小化模式的理论，就像上一代人对肥皂泡和肥皂膜的研究导致了最小表面理论一样。 薄的床单经常起皱或折叠，这当然是一个熟悉的事实：我们的皮肤起皱，我们的衣服起皱;树叶，花朵和悬挂的窗帘都有褶皱。 在受控环境中的物理实验可以量化这种现象，数值模拟可以在模型中演示模式如何发展。 但是，无论是实验还是模拟都不能告诉我们“为什么”一个系统会选择一个特定的模式。 该项目提供了一个有价值的补充，其他方法，通过显示，弹性能量最小化需要类型的模式。 该项目为研究生提供培训机会。

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.