Nonlinear Partial Differential Equations

龙卷风和飓风是自然界中奇点的两个例子。像所有的大气现象一样，它们必须遵守用偏微分方程式表示的数学形式的物理定律。从数学上讲，偏微分方程解的水平突变(“界面”)或龙卷风般的旋转(“涡旋”)被称为奇点，而理解奇点形成机制的挑战是当前的热门话题。需要丰富的创新方法。最近，我们开发并应用了“粘合方法”来产生引人注目的例子和回答长期悬而未决的问题。在接下来的几年里，我建议为这些方法建立一个统一的框架，并用它来研究来自几何、物理、生物、化学和社会互动的各种问题中的奇点形成。

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.