Geometric Aspects of Integrable Systems

三维空间中常负曲率曲面和等温曲面的方程都是孤立子方程。 孤子理论已成功地应用于研究这些几何方程。给定一个孤子方程，一个有趣的问题是是否存在一个几何对象，其控制方程是给定的孤子方程。 Terng建议研究Virasoro代数的几何方面的行动，子流形几何与孤子方程，哈密顿理论的空间-时间-时间-代数方程，并完成研究专著可积系统和微分几何和一本书的曲线和曲面。孤子方程自然出现在应用数学和微分几何。例如，非线性薛定谔方程（NLE）对光纤中的波的包络的运动进行建模。用于NLE的孤子波已经用于电信中。更好地理解孤子方程可以提供更多的应用和有趣的几何。Terng还为博士后学者，研究生和本科生的培训做出了贡献。自1994年以来，她还共同组织了IAS女性数学家的指导计划。 她计划继续开展这些活动。

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.