Nonlinear heat equations applied to geometry and mechanics

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The PI intends to continue the study of singularities of nonlinear heatequations in differential geometry and mathematical physics, such as TheRicci flow, Harmonic Map flow, Mean curvature flow, and the Porous mediumequation. Possible singularities will be analyzed by means of matchedasymptotic expansions. In addition to this the PI hopes to continue his collaboration withgroups in medical imaging, serving at least as a source of theoreticknowledge for such groups and perhaps collaborating more intensely.Nonlinear diffusion equations occur in many forms in mathematical physicsand engineering, as well as in pure mathematics (differential geometry).Understanding of the solutions to such equations can be gained by studyingtheir singularities. E.g. singularities of a solution to an equationmodeling a chemical reaction in a reactor may correspond to explosion asopposed to steady reaction. The PI intends to study such singularities. In addition, the PI intends to continue his collaboration with aresearch group in Medical Imaging. The practical problems of how toautomatically process 3D fMRI images all belong to a branch of mathematicscalled differential geometry. A detailed knowledge of differential geometryhas proved to be very useful in medical imaging in the past, and promissesto stay important. Conversely, medical imaging problems may serve as asource of new mathematical problems and theories.