Nonlinear heat equations applied to geometry and mechanics

PI打算继续研究微分几何和数学物理中非线性热方程的奇异性，例如里奇流、调和映射流、平均曲率流和多孔介质方程。可能的奇异性将通过匹配渐近展开进行分析。 除此之外，PI希望继续与医学成像领域的团体合作，至少作为这些团体的理论知识来源，并可能进行更深入的合作。非线性扩散方程在数学物理和工程以及纯数学（微分几何）中以多种形式出现。通过研究这些方程的奇点，可以了解这些方程的解。例如，反应器中模拟化学反应的方程的解的奇点可能对应于爆炸，而不是稳定反应。PI打算研究这种奇异性。 此外，PI打算继续与医学成像研究小组合作。如何自动处理三维fMRI图像的实际问题都属于微分几何的一个分支。微分几何的详细知识在过去的医学成像中被证明是非常有用的，并且有希望保持重要性。医学影像学问题可以作为新的数学问题和理论的集合。

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.