Nonlinear Partial Differential Equations

龙卷风和飓风是自然界奇异现象的两个例子。像所有的大气现象一样，它们必须遵守物理定律，以偏微分方程的数学形式表示。在数学上，PDE的解决方案，显示突然变化的水平（“界面”），或龙卷风般的旋转（“漩涡”）被称为奇点，和理解奇点形成的机制的挑战是当前的热门话题。需要大量的创新方法。最近，我们开发并应用了“胶合方法”，以产生引人注目的例子并回答长期悬而未决的问题。在未来的几年里，我建议为这些方法建立一个统一的框架，并使用它来研究奇点形成的各种问题所产生的几何，物理，生物学，化学和社会相互作用。

Tornadoes and hurricanes are two examples of singularities in nature. Like all atmospheric phenomena, they must comply with the laws of Physics, expressed in mathematical form by Partial Differential Equations. Mathematically, solutions to PDE's that display abrupt changes in level (``interfaces''), or tornado-like rotation (``vortices'') are called singularities, and the challenge of understanding the mechanisms of singularity formation is a hot current topic. A rich collection of innovative methods is required. Recently we have developed and applied ``gluing methods''to produce striking examples and answer long-open questions. In the years ahead, I propose to build a unified framework for these methods and use it to study singularity formation in a diverse range of problems arising from geometry, physics, biology, chemistry, and social interactions.