The proposal of Chuu-Lian Terng contains three projects. The first project is joint with Karen Uhlenbeck (University of Texas at Austin). Terng and Uhlenbeck have collaborated for the past several years to give geometric interpretations and applications of analytic and algebraic constructions in integrable systems. They propose to continue their investigation of geometric aspects of integrable systems, Virasoro actions and topological conformal field theory. The second project is on submanifolds in symmetric spaces whose Gauss-Codazzi equations are integrable systems. Terng has made several contributions to this area when submanifolds lie in a space form, and she plans to continue the research for the symmetric space case. The third project is joint with Gudlaugur Thorbergsson (University of Koln). Terng and Thorbergsson have made contributions to the theory of isoparametric submanifolds in space forms and equifocal submanifolds in symmetric spaces. In this project, they propose to study the geometry of submanifolds in complex and quaternionic n-space whose U(n) and Sp(n)-invariants are constant in various senses. The success of this project should give better understanding of submanifold geometry in Hermitian and quaternionic Kahler symmetric spaces.
The theory of integrable systems has deep relations with mechanics and dynamics, applied mathematics, algebra, theoretical physics, partial differential equations, algebraic geometry, and differential geometry. It has also been used in other sciences. For example, (1) the sine-Gordon equation, which is the Gauss-Codazzi equation for constant negative Gaussian curvature surfaces in 3-space, also arises in plasma physics; (2) the non-linear Schrodinger equation models the propergation of wave envelope in optic fiber. Success of the proposed projects will give better understanding of the structure of integrable systems and their geometrizations.
