Geometry of Integrable Systems and Submanifold Geometry

邓楚莲的提案包含三个项目。第一个项目是与德克萨斯大学奥斯汀分校的KarenUhlenbeck联合开展的。Terng和Uhlenbeck在过去的几年里一直在合作，给出可积系统中解析和代数结构的几何解释和应用。他们建议继续研究可积系统的几何方面、Virasoro作用量和拓扑共形场理论。第二个项目是关于对称空间中Gauss-Codazzi方程是可积系统的子流形。Terng在这个领域做出了一些贡献，当子流形位于空间形式时，她计划继续研究对称空间的情况。第三个项目是与Gudlagur Thorbergsson(科隆大学)联合开展的。Terng和Thorbergsson对空间形式的等参子流形和对称空间的等焦子流形理论作出了贡献。在这个项目中，他们建议研究复数和四元数n-空间中的子流形的几何，这些空间的U(N)和Sp(N)不变量在各种意义上都是常量。这个项目的成功应该会更好地理解厄米特和四元数Kahler对称空间中的子流形几何。可积系统理论与力学、动力学、应用数学、代数、理论物理、偏微分方程组、代数几何和微分几何有着深刻的联系。它还被用于其他科学。例如，(1)Sine-Gordon方程，即三维常负高斯曲率表面的Gauss-Codazzi方程，也出现在等离子体物理中；(2)非线性薛定谔方程模拟光纤中波包的特性。拟议项目的成功将使人们更好地了解不可拆分系统的结构及其几何化。

The proposal of Chuu-Lian Terng contains threeprojects. The first project is joint with KarenUhlenbeck (University of Texas at Austin). Terng andUhlenbeck have collaborated for the past several yearsto give geometric interpretations and applications ofanalytic and algebraic constructions in integrablesystems. They propose to continue their investigationof geometric aspects of integrable systems, Virasoroactions and topological conformal field theory. Thesecond project is on submanifolds in symmetric spaceswhose Gauss-Codazzi equations are integrable systems.Terng has made several contributions to this area whensubmanifolds lie in a space form, and she plans tocontinue the research for the symmetric space case. The third project is joint with Gudlaugur Thorbergsson(University of Koln). Terng and Thorbergsson havemade contributions to the theory of isoparametric submanifolds in space forms and equifocal submanifoldsin symmetric spaces. In this project, they proposeto study the geometry of submanifolds in complex andquaternionic n-space whose U(n) and Sp(n)-invariantsare constant in various senses. The success of thisproject should give better understanding ofsubmanifold geometry in Hermitian and quaternionicKahler symmetric spaces. The theory of integrable systems has deep relationswith mechanics and dynamics, applied mathematics,algebra, theoretical physics, partial differentialequations, algebraic geometry, and differentialgeometry. It has also been used in other sciences. For example, (1) the sine-Gordon equation, which isthe Gauss-Codazzi equation for constant negativeGaussian curvature surfaces in 3-space, also arises inplasma physics; (2) the non-linear Schrodingerequation models the propergation of wave envelope inoptic fiber. Success of the proposed projects willgive better understanding of the structure ofintegrable systems and their geometrizations.