Geometry of Integrable Systems and Submanifold Geometry

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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.