Nonlinear heat equations applied to geometry and mechanics

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

The PI intends to continue the study of singularities of nonlinear heat equations in differential geometry and mathematical physics, such as The Ricci flow, Harmonic Map flow, Mean curvature flow, and the Porous medium equation. Possible singularities will be analyzed by means of matched asymptotic expansions. In addition to this the PI hopes to continue his collaboration with groups in medical imaging, serving at least as a source of theoretic knowledge for such groups and perhaps collaborating more intensely. Nonlinear diffusion equations occur in many forms in mathematical physics and engineering, as well as in pure mathematics (differential geometry). Understanding of the solutions to such equations can be gained by studying their singularities. E.g. singularities of a solution to an equation modeling a chemical reaction in a reactor may correspond to explosion as opposed to steady reaction. The PI intends to study such singularities. In addition, the PI intends to continue his collaboration with a research group in Medical Imaging. The practical problems of how to automatically process 3D fMRI images all belong to a branch of mathematics called differential geometry. A detailed knowledge of differential geometry has proved to be very useful in medical imaging in the past, and promisses to stay important. Conversely, medical imaging problems may serve as a source of new mathematical problems and theories.