Mathematical Sciences: Differential Equations and Differential Geometry

数学科学：微分方程和微分几何

• 批准号: 9203362
• 负责人: Christopher Croke
• 金额: --
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1996-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9203362&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Differential; Equations; Geometry

This research focuses on differential equations and differential geometry. The research involves the study of manifolds without conjugate points, sharp isoperimetric inequalities for manifolds of various dimensions, degenerate systems of partial differential equations (especially equations of prescribed curvature, the geometry of isospectral manifolds, certain classes of minimal submanifolds, and certain applications of geometry to physiology. In general, this research concerns mathematical models of the Riemannian geometry of manifolds. Some of this research has application to physiology; for example, the traditional treatment of gallstones is cholecystectomy (removal of the gallbladder). However, lithotripsy (fragmentation) and dissolution of gallstones has been advanced as an alternative in recent years. The investigators on this award are involved in developing a sophisticated mathematical model which predicts the behavior of fragments in solvent over time. The model predicts the evolution of the stone/solvent interface and is a modification of the curvature flow in differential geometry studied by many researchers.

本研究的重点是微分方程和微分几何。本研究涉及无共轭点流形的研究，不同维流形的尖锐等周不等式，偏微分方程的退化系统（特别是规定曲率方程），等谱流形的几何，某些类型的极小子流形，以及几何学在生理学上的某些应用。总的来说，本研究涉及流形黎曼几何的数学模型。其中一些研究应用于生理学；例如，胆结石的传统治疗方法是胆囊切除术（切除胆囊）。然而，近年来，碎石（碎裂）和胆结石溶解已成为一种替代方法。该奖项的研究人员致力于开发一种复杂的数学模型，该模型可以预测溶剂中碎片随时间的行为。该模型预测了石/溶剂界面的演化，是对微分几何中曲率流动的修正。