Nonlinear Heat Equations in Geometry, Mechanics and Imaging

几何、力学和成像中的非线性热方程

• 批准号: 0705431
• 负责人: Sigurd Angenent
• 金额: $ 16.91万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2011-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0705431&HistoricalAwards=false
• 关键词: Nonlinear; Heat; Equations; Geometry; Mechanics

The PI intends to study the asymptotics of singular solutions to heat equations which occur in Riemannian geometry, such as Ricci flow and mean curvature flow, and also free boundary problems involving fourth order equations. Further problems of interest to the PI are related to the formulation of the Monge-Kantorovich mass-transfer problem, and in particular variations thereof which may appear in medical imaging problems.Differential Geometry and partial differential equations (PDE) play a central role in modern science and engineering. The PI is interested both in studying these subjects from the "pure-mathematics" point of view, and in collaborating with engineers and scientists to find new applications of geometry and PDE to practical problems. Particular applied areas of interest to the PI involve the automated processing of images (brain scans, virtual colonoscopy, etc) generated in the medical sciences.

PI打算研究黎曼几何中热方程奇异解的渐近性，例如里奇流和平均曲率流，以及涉及四阶方程的自由边界问题。 PI 感兴趣的其他问题与 Monge-Kantorovich 质量传递问题的表述有关，特别是医学成像问题中可能出现的其变体。微分几何和偏微分方程 (PDE) 在现代科学和工程中发挥着核心作用。 PI 既有兴趣从“纯数学”的角度研究这些学科，也有兴趣与工程师和科学家合作，寻找几何和偏微分方程在实际问题中的新应用。 PI 感兴趣的特定应用领域涉及医学科学中生成的图像（脑部扫描、虚拟结肠镜检查等）的自动处理。