Geometric Aspects of Integrable Systems

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Equations for surfaces with constant negative curvature and isothermic surfaces in three space are soliton equations. Soliton theory have been applied successfully to study these geometric equations. Given a soliton equation, an interesting question is whether there is a geometric object whose controlling equation is the given soliton equation. Terng proposes to study the geometric aspects of the actions of Virasoro algebra, the submanifold geometry associated to soliton equations, the Hamiltonian theory of space-time monopole equations, and to finish a research monograph on integrable systems and differential geometry and a book on curves and surfaces.Soliton equations arise naturally in applied mathematics and differential geometry. For example, the non-linear Schrodinger equation (NLE) models the motion of the envelope of waves in optic fiber. Soliton waves for NLE have been used in telecommunications. Better understanding of soliton equations may provide more applications and interesting geometry. Terng has also contributed to the training of postdoctoral scholars, graduate students, and undergraduate students. She has also co-organized the Mentoring program for women mathematicians at IAS since 1994. She plans to continue these activities.