Mathematical Sciences: Differential Equations and Differential Geometry

这项研究主要集中在微分方程组和微分几何。研究涉及无共轭点流形、不同维度流形的等周不等式、退化偏微分方程组(特别是指定曲率的方程)、等谱流形的几何、某些类极小子流形，以及几何学在生理学上的某些应用。总的来说，本研究涉及流形的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(切除胆囊术)。然而，近年来，碎石(碎石)和胆结石溶解作为一种替代方法得到了发展。该奖项的研究人员参与开发了一个复杂的数学模型，该模型预测了碎片在溶剂中随着时间的推移的行为。该模型预测了石头/溶剂界面的演化，是对微分几何中曲率流动的修正，许多研究人员对此进行了研究。

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.