Evolution Equations in Geometry

摘要 提案：DMS-9803550主要研究者：Richard汉密尔顿 理查德汉密尔顿将致力于非线性抛物部分 微分方程和方程组在科学和 几何，包括黎曼度量的Ricci演化 曲率：曲面的平均曲率或高斯运动 曲率，多孔机构中的气体流动， 液晶等。 特别强调的是 奇点，使用精确的模型来显示其明确的 结构 这些方程模拟了科学中许多有趣的现象， 扩散发生。 例如：材料科学中的应用 固体和液体之间的边界运动，磨损 岩石在随机角度的冲击下，以及石油在 页岩;在生物学上， 独居动物的迁徙;在摄影中用于图像锐化;以及 宇宙的形状。

Abstract Proposal: DMS-9803550 Principal Investigator: Richard Hamilton Richard Hamilton will work on non-linear parabolic partial differential equations and systems of equations in science and geometry, including the evolution of a Riemannian metric by its Ricci curvature: the motion of a surface by its mean curvature or its Gauss curvature, the flow of gas in a porous mechanism, the motion of a liquid crystal, and others. Special emphasis will be on the formation of singularities, using precise models to show their explicit structure. These equations model many interesting phenomena in science where diffusion occurs. As examples: applications occur in material science to the motion of the boundary between a solid and a liquid, the wear of rocks under impact at a random angle, and the diffusion of oil in shale; in biology to the reproduction of sparse species and the migration of solitary animals; in photography to image sharpening; and in general relativity to the shape of the universe.