Mathematical Sciences: Evolution Equations in Geometry

汉密尔顿9504528拟议的研究领域是在微分几何和物理科学中产生的非线性抛物型偏微分方程--例如，热方程。作者以前在抛物型微分方程解的定量和定性方面得到了一些结果，这些结果反过来又导致了在光滑流形理论中的应用。特别是，作者对具有正迷向曲率的四流形的微分同胚分类所做的工作被认为是该领域的一个重要结果。所提出的研究在物理科学中的应用包括材料固液界面的运动、材料在随机载荷作用下的磨损、润滑液的运动和液晶的演化。

Hamilton 9504528 The proposed research is in the area of nonlinear parabolic partial differential equations - for example, the heat equation - arising in differential geometry and the physical sciences. The proposer has previously obtained a number of results on the quantitative and qualitative aspects of the solutions to these parabolic differential equations; these results, in turn, led to applications in smooth manifold theory. In particular, the proposer's work on the diffeomorphism classification of four-manifolds with positive isotropic curvature is considered an important result in the field. Applications of the proposed research in the physical sciences include movement of solid-liquid interfaces of materials, wear of material bodies under random impart, motion of lubricating fluids, and evolution of liquid crystals.