Mathematical Sciences: Singularities of Geometric Heat Flows

基本的计算机研究将由校长使用 研究人员制定理论的形成 光滑黎曼流形之间映射的奇异性 这些 调和映射热流和平均曲率共同作用下的映射演化 热流具体来说，他将研究持续时间， 奇点的形成，爆破时间对 奇点的余维，以及这种热流是否可能是 以独特的方式延伸到奇点之外 工作 推广了已知的非奇异热流理论， 非正弯曲目标流形的情况。 在这个项目中将广泛使用计算机。 方程模拟物理系统，例如 化学反应和液晶将是一个主题， 系统研究 主要研究者将使用数值 研究建议的数量，这反过来将被用作 理论估计的障碍函数。

Essential computer studies will be used by the principal investigator to formulate theories concerning formation of singularities of maps between smooth Riemannian manifolds. These maps evolve both by harmonic map heat flow and mean curvature heat flow. Specifically, he will investigate time durations for singularity formation, the dependence of blow-up times on the singularity's codimension, and whether such heat flows may be extended in unique fashion past the singularity. The work extends known theories of nonsingular heat flows and the special case of non-positively curved target manifolds. Computers will be used extensively in this project. Equations modeling physical systems such as phase boundaries in chemical reactions and liquid crystals will be the subject of a systematic study. The principal investigator will use numerical studies to suggest quantities which in turn will be used as barrier functions for theoretical estimates.