Mathematical Sciences: Singularities of Geometric Heat Flows

数学科学：几何热流的奇点

• 批准号: 8905937
• 负责人: Richard Hamilton
• 金额: $ 7.05万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1992-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8905937&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Singularities; Geometric; Heat

Essential computer studies will be used by the principal investigator to formulate theories concerning formation of singularities of maps between smooth Riemannian manifolds. These maps evolve both by harmonic map heat flow and mean curvature heat flow. Specifically, he will investigate time durations for singularity formation, the dependence of blow-up times on the singularity's codimension, and whether such heat flows may be extended in unique fashion past the singularity. The work extends known theories of nonsingular heat flows and the special case of non-positively curved target manifolds. Computers will be used extensively in this project. Equations modeling physical systems such as phase boundaries in chemical reactions and liquid crystals will be the subject of a systematic study. The principal investigator will use numerical studies to suggest quantities which in turn will be used as barrier functions for theoretical estimates.

主要研究者将使用基本的计算机研究来制定关于光滑黎曼流形之间映射奇点形成的理论。这些图是由调和图热流和平均曲率热流演变而来的。具体来说，他将研究奇点形成的时间持续时间，爆炸时间对奇点协维的依赖，以及这种热流是否可以以独特的方式延伸到奇点之后。推广了已知的非奇异热流理论和非正弯曲目标流形的特殊情况。计算机将在这个项目中广泛使用。模拟物理系统的方程，如化学反应和液晶中的相边界，将是系统研究的主题。首席研究员将使用数值研究来建议数量，这些数量反过来将用作理论估计的障碍函数。