Blow-up problems in geometric heat equations

题目：几何热方程的爆破问题[j]: Sigurd Angnent， University of Wisconsin （Madison）摘要：该PI打算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给出精确的定量描述。这样的描述通常必须用所谓的“匹配渐近展开式”来给出。计算这种展开的方法在应用数学中是已知的，但是缺乏证明计算展开的有效性的技术（当它们实际上是有效的时候）。PI在过去对非线性热流（曲线缩短，平均曲率回流）的数量给出了严格的渐近描述。在目前的提案中，PI提到了一些他想要研究的新问题。这些新问题一方面与以前研究过的问题相似，但另一方面旧方法似乎不能直接适用。此外，以前的许多工作在某种意义上是“一维的”，因为对称假设是用来简化情况的。在接下来的几年里，PI想要解决不那么对称的问题，看看是否在真正的高维环境下也能得到匹配的渐近展开。除了对纯数学感兴趣外，PI还参与了与生物医学工程师的合作，其中涉及与微分几何和医学成像中的PDE相关的数学问题。更广泛的影响：由于他对医学成像的兴趣，PI为生物医学工程师（包括他在校外的合作者，以及威斯康星大学麦迪逊分校的医学工程学院和研究生）提供了一个资源。微分几何在医学成像和计算机视觉方面的新应用的知识增强了PI的本科课程（例如计划创建一门关于医学成像的数学方法的高级本科课程），并且从长远来看激发了纯数学的新问题。

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.