Nonlinear Partial Differential Equations

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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.