Blow-up problems in geometric heat equations

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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.