Blow-up problems in geometric heat equations

几何热方程中的爆炸问题

• 批准号: 0405084
• 负责人: Sigurd Angenent
• 金额: $ 12.6万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405084&HistoricalAwards=false
• 关键词: Blow; up; problems; geometric; heat

DMS-0405084Title: Blow-up problems in geometric heat equationsPI: Sigurd Angnent, University of Wisconsin (Madison)ABSTRACTThe PI intends to study the singularities of solutions to nonlinearheat equations which occur in differential geometry. The PI isparticularly interested in giving precise quantitative descriptions ofthese singularities. Such descriptions must generally be given interms of so-called "matched asymptotic expansions." Methods forcomputing such expansions are known in applied mathematics, but thereis a shortage of techniques for proving the validity of the computedexpansions (when they actuallly are valid).The PI has in the past given rigorous asymptotic descriptions for anumber of nonlinear heat flows (curve shortening, mean curvatureflow). In the current proposal the PI mentions a number of newproblems which he would like to study. These new problems are on onehand similar to the previously studied problems, but on the other handthe old methods don't seem directly applicable. In addition, much ofthe previous work was in a sense "one-dimensional" in that symmetryassumptions were used to simplify the situation. In the next few yearsthe PI would like to address less symmetric problems, to see if thematched asymptotic expansions can also be obtained in truly higherdimensional settings.Apart from his interest in pure mathematics, the PI is also involvedin a collaboration with biomedical engineers, in which mathematicalproblems related to differential geometry and PDE in Medical Imagingare addressed.Broader impact: Due to his interest in medical imaging, the PI servesas a resource for biomedical engineers (both his collaborators who areoff campus, and the medical engineering faculty & grad students on theUW Madison campus). Knowledge of newer applications of differentialgeometry in medical imaging and computer vision enhances the PI'sundergraduate classes (e.g. the planned creation of an advancedundergraduate course on mathematical methods of medical imaging) andin the long run inspires new problems in pure mathematics.

DMS-0405084标题：几何热方程中的爆破问题PI：西古尔德Angnent，威斯康星州大学（麦迪逊）摘要PI旨在研究微分几何中出现的非线性热方程解的奇异性。 PI特别感兴趣的是对这些奇点进行精确的定量描述。这种描述一般必须用所谓的“匹配渐近展开式”给出。计算这种展开式的方法在应用数学中是已知的，但是缺乏证明计算出的展开式的有效性的技术（当它们实际上是有效的时）。PI在过去对许多非线性热流（曲线缩短，平均曲率流）给出了严格的渐近描述。在目前的提案中，PI提到了一些他想研究的新问题。这些新问题一方面与以前研究的问题相似，另一方面旧的方法似乎不直接适用。此外，许多以前的工作在某种意义上是“一维”的，因为大量的假设被用来简化情况。在接下来的几年里，PI希望解决不太对称的问题，看看匹配的渐近展开是否也可以在真正的高维环境中获得。除了对纯数学感兴趣之外，PI还参与了与生物医学工程师的合作，其中涉及微分几何和PDE在医学成像中的相关问题。更广泛的影响：由于他对医学成像的兴趣，PI成为生物医学工程师的资源（包括他在校外的合作者和华盛顿大学麦迪逊校区的医学工程系&格拉德生）。微分几何在医学成像和计算机视觉中的新应用的知识增强了PI的本科生课程（例如，计划创建一个关于医学成像数学方法的高级本科课程），从长远来看，激发了纯数学的新问题。