The PI intends to study the singularities of solutions to nonlinear heat equations which occur in differential geometry. The PI is particularly interested in giving precise quantitative descriptions of these singularities. Such descriptions must generally be given in terms of so-called "matched asymptotic expansions." Methods for computing such expansions are known in applied mathematics, but there is a shortage of techniques for proving the validity of the computed expansions (when they actuallly are valid).
The PI has in the past given rigorous asymptotic descriptions for a number of nonlinear heat flows (curve shortening, mean curvature flow). In the current proposal the PI mentions a number of new problems which he would like to study. These new problems are on one hand similar to the previously studied problems, but on the other hand the old methods don't seem directly applicable. In addition, much of the previous work was in a sense "one-dimensional" in that symmetry assumptions were used to simplify the situation. In the next few years the PI would like to address less symmetric problems, to see if the matched asymptotic expansions can also be obtained in truly higher dimensional settings.
Apart from his interest in pure mathematics, the PI is also involved in a collaboration with biomedical engineers, in which mathematical problems related to differential geometry and PDE in Medical Imaging are addressed.
Broader impact: Due to his interest in medical imaging, the PI serves as a resource for biomedical engineers (both his collaborators who are off campus, and the medical engineering faculty & grad students on the UW Madison campus). Knowledge of newer applications of differential geometry in medical imaging and computer vision enhances the PI's undergraduate classes (e.g. the planned creation of an advanced undergraduate course on mathematical methods of medical imaging) and in the long run inspires new problems in pure mathematics.
