Differential Geometry and Integrable Systems

摘要奖：DMS-9972172主要研究员：Chuu-Lian Terng主要研究人员与Karen Uhlenbeck共同提出了下列与几何可积系统有关的项目：(1)了解孤子方程的对称性和几何方面，(2)研究Lax对、双哈密顿结构、无限多交换哈密顿量、Backlund变换和Darboux变换以及逆散射变换之间的关系，(3)将孤子理论的技术应用于几何问题，特别是对称空间中的子流形、调和映射和自对偶杨-米尔场。最著名的三个孤子方程是KDV方程、SGE方程和NLS方程。KdV方程描述浅水管道中的波动；SGE方程源于等离子体物理；NLS方程描述光在光缆中的传播，目前用于电信。这些方程也出现在19世纪的微分几何中。几何联系为构造更多的孤子方程和解提供了有趣的方法。数学物理和几何中许多有趣的问题现在都与孤立子方程有关。这项建议的主要目的是用几何方法研究孤子方程，并将孤子方程理论应用于微分几何问题。

AbstractAward: DMS-9972172Principal Investigator: Chuu-Lian TerngThe principal investigator proposes jointly with Karen Uhlenbeckto work on the following projects concerning geometric integrablesystems: (1) to understand the symmetries and geometric aspectsof soliton equations, (2) to study the relations among Lax pairs,bi-Hamiltonian structures, infinitely many commutingHamiltonians, Backlund and Darboux transformations, and theinverse scattering transform, (3) to apply the techniques ofsoliton theory to geometric problems, in particular, submanifoldsin symmetric spaces, harmonic maps and self-dual Yang Millsfields.The three most well-known soliton equations are KdV, SGE and NLSequations. The KdV equation describes wave motion in a shallowcanal; the SGE equation arose from plasma physics; and the NLSequation describes the propogation of light in a fiber opticcable and is currently used in telecommunication. Theseequations also arose in the 19 century differential geometry.The geometric connection gives interesting methods to constructmore soliton equations and solutions. Many interesting problemsin mathematical physics and geometry are now known to be relatedto soliton equations. The main goal of this proposal is to usegeometric method to study soliton equations and apply the theoryof soliton equations to problems in differential geometry.